Horofunction Compactifications and Duality

Abstract

We study the global topology and geometry of the horofunction compactification of classes of symmetric spaces under Finsler distances in three settings: bounded symmetric domains of the form $B=B_1\times \cdots \times B_r$, where $B_i$ is an open Euclidean ball in ${\mathbb {C}}^{n_i}$, with the Kobayashi distance, symmetric cones with the Hilbert distance, and Euclidean Jordan algebras with the spectral norm. For these spaces we show, that the horofunction compactification is naturally homeomorphic to the closed unit ball of the dual norm of the Finsler metric in the tangent space at the basepoint. In each case we give an explicit homeomorphism. For finite dimensional normed spaces the link between the geometry of the horofunction compactification and the dual unit ball was suggested by Kapovich and Leeb, which we confirm for Euclidean Jordan algebras with the spectral norm. Our results also show that this duality phenomenon not only occurs in normed spaces, but also in a variety of noncompact type symmetric spaces with invariant Finsler metrics.

1 Introduction

Compactifications of symmetric spaces is a rich subject which has been studied extensively [9, 27]. Recently it was shown that various compactifications of noncompact type symmetric spaces $X=G/K$ can be realised as horofunction compactifications with respect to G-invariant Finsler metrics. For the generalised Satake compactifications this was shown by Haettel et al. [28], and for the Martin compactification this was established by Schilling [53]. The realisation of the maximal Satake compactification as a horofunction compactification was given by Kapovich and Leeb [34].

For symmetric spaces with nonpositive sectional curvature it is well-known that the horofunction compactification with respect to the Riemannian distance is homeomorphic to a Euclidean ball, see [12, 16, 17]. For various finite dimensional normed spaces it was observed that the horofunction compactification is naturally related to the closed dual unit ball. As a matter of fact, Kapovich and Leeb [34, Question 6.18] asked if for finite dimensional normed spaces the horofunction compactification (with its natural stratification) is homeomorphic to the closed unit ball of the dual normed space. This was confirmed by Ji and Schilling [32, 33] for normed spaces with a polyhedral unit ball.

In an analogous manner one can ask for noncompact type symmetric spaces if the horofunction compactification with respect to an invariant Finsler metric is naturally homeomorphic to the closed dual unit ball of the Finsler metric in the tangent space at the basepoint.

The main goal of this paper is to confirm this duality phenomenon for two classes of noncompact type symmetric spaces and a class of normed spaces. More specifically, we will consider bounded symmetric domains of the form $B_1\times \cdots \times B_r$, where $B_i$ is an open Euclidean ball in ${\mathbb {C}}^{n_i}$, with the Kobayashi distance, symmetric cones with the Hilbert distance, and Euclidean Jordan algebras with the spectral norm, i.e., finite dimensional JB-algebras.

The bounded symmetric domains $B=B_1\times \cdots \times B_r$ with the Kobayashi distance are prime examples of noncompact type symmetric spaces with invariant Finsler metric. In this case the open unit ball of the Finsler metric in the tangent space at the origin coincides with B. We will show that the horofunction compactification of B is homeomorphic to the dual unit ball, i.e., the polar of B. In fact, we shall work in a slightly more general domains $D = D_1\times \cdots \times D_r$, where $D_{i}$ is the open unit ball of a norm with a strongly convex $C^3$-boundary, even though these domains no longer correspond to symmetric spaces. The horofunction compactification of these spaces was studied in [40]. It should be noted that for general bounded convex domains $D\subset {\mathbb {C}}^n$, with the Kobayashi distance, various smoothness conditions on D are known that imply that the identity map extends as a homeomorphism from the horofunction compactification of D onto the norm closure of D, see [5, Theorem 1.2] and [7, 10, 11, 59]. In our setting, however, the domains are not smooth, and the identity does not extend as a homeomorphism.

Symmetric cones with the Hilbert distance are another interesting class of symmetric spaces with invariant Finsler metric. A prime example is the symmetric space ${\textrm{SL}}(n,{\mathbb {C}})/{\textrm{SU}}(n)$, which corresponds to the projective symmetric cone consisting of positive definite $n\times n$ Hermitian matrices. More precisely, one can realise this space as $\{A\in {\textrm{Herm}}(n,{\mathbb {C}}):\,\, A\,\,\text{ is } \text{ positive } \text{ definite } \text{ with } \text{ trace }\,\,n\}$. In our analysis we use the cone version of the Hilbert distance, see [42], which provides a convenient way to analyse its Finsler structure [48] and the dual unit ball. The horofunction compactification of symmetric cones with the Hilbert distance was determined in [44, Theorem 5.6] and is naturally described in terms of the Euclidean Jordan algebra associated to the symmetric cone, which will be exploited.

Euclidean Jordan algebras V with the spectral norm, i.e., finite dimensional JB-algebras [4], are an important class of real normed vector spaces. A prime example is the real vector space ${\textrm{Herm}}(n,{\mathbb {C}})$ consisting of $n\times n$ Hermitian matrices with the spectral norm, $\Vert A\Vert =\max \{ |\lambda |:\lambda \text{ eigenvalue } \text{ of } A\}$. We use the Jordan algebra structure to give a complete description of the horofunctions for these spaces and provide an explicit homeomorphism between the horofunction compactification and the closed dual unit ball. These normed spaces are related to an invariant metric on symmetric cones coming from the Thompson distance. More precisely, for a symmetric cone C in a Euclidean Jordan algebra V, the Finsler metric of the Thompson distance in the tangent space V at the unit is the spectral norm, see [48].

In a sequel [41] to this paper the first author has shown for the Hilbert distance and Thompson distance on symmetric cones that the exponential map at the unit extends as a homeomorphism between the horofunction compactification of the normed space at the unit with the Finsler metric, and the horofunction compactification of the symmetric cone with the Finsler distance. It would be interesting to know if this holds more generally for noncompact type symmetric spaces with invariant Finsler distances.

The origins of the horofunction compactification go back to Gromov [6, 21] who associated a boundary at infinity to any locally compact geodesic metric space. It has found numerous applications in diverse areas of mathematics including, geometric group theory [12], noncommutative geometry [51], complex analysis [1, 5, 7, 10, 11, 59], Teichmüller theory [15, 20, 36, 46, 56], dynamical systems and ergodic theory [8, 19, 35, 44] and in the study of compactifications of noncompact type symmetric spaces [28, 34, 53]. A general set up for metric spaces was discussed by Rieffel [51]. It should, however, be noted that if the metric space is not proper, then the embedding into its horofunction compactification need not be a homeomorphism. So, in that case, the horofunction compactification would not be a compactfication in the usual topological sense.

The horofunction compactification is a particularly powerful tool to study isometry groups of metric spaces and isometric embeddings between metric spaces, see [40, 43, 57, 58]. Especially useful in this context are the so-called Busemann points in the horofunction compactification, which are limits of almost geodesics. They were introduced by Rieffel [51], who asked whether every horofunction is a Busemann point in a finite dimensional normed space. Walsh [54] showed that in general this is not the case and found necessary and sufficient conditions for a finite dimensional normed space to have the property that all its horofunctions are Busemann points.

For the metric spaces considered in this paper, we show that all horofunctions are Busemann points. As a consequence we get that the horofunction boundary has a partition coming from the detour distance. Indeed, on the set of Busemann points the detour distance [2, 43] is a metric, where two Busemann points can lie at infinite distance from each other. This yields a partition of the set of Busemann points into so-called parts where two Busemann points lie in the same part if the detour distance between them is finite. As all horofunctions are Busemann points for our spaces, it follows from [57, Proposition 4.5] that this partition coincides with the partition of the horofunction boundary into subsets, where two horofunctions h and g lie in the same subset if $\sup _x |h(x) -g(x)|$ is bounded.

On the other hand, the closed dual unit ball $B_1^*$ is the disjoint union of the relative interiors of its nonempty faces, see [52, Theorem 18.2]. In each of our settings we will give an explicit homeomorphism that maps the metric space onto the interior of $B_1^*$, and each part in the horofunction boundary onto the relative interior of a boundary face of $B_1^*$. It is this property of the homeomorphism that ’naturally’ connects the geometry of the horofunction compactification to the closed dual unit ball for our spaces. The homeomorphisms we use are modifications of the maps used by Ji and Schilling [32] in the setting of polyhedral normed spaces. As pointed out there, the homeomorphisms resemble moment maps from symplectic geometry and Lie group actions, but the exact connection is not well understood at present. Similar maps were also used in the study of Satake compactifications of symmetric spaces in [31, 39].

The results are consistent with what is known for the horofunction compactification of the symmetric spaces with nonpositive sectional curvature under the Riemannian distance. In that case all horofunctions are Busemann points and each part is a singleton, which reflects the fact that each point in the boundary of the Euclidean ball is a relatively open face, as it is an extreme point.

In general it is difficult to determine the horofunction compactification of a metric space explicitly and only in relatively few spaces has this been done. For ${\textrm{CAT}}(0)$ spaces the horofunction compactification is well understood, see [12, Chap. II.8] and coincides with the visual boundary. Gutièrrez [22,23,24] computed the horofunction compactification of several classes of $L_p$-spaces. It has also been identified for finite dimensional polyhedral normed spaces, see [13, 28, 33, 37]. For arbitrary (possibly infinite dimensional) normed spaces the Busemann points in the horofunction boundary have been characterised by Walsh [58]. For Hilbert metric spaces there exists a characterisation of the Busemann points [55]. For the Hilbert distance on a symmetric cone in a Euclidean Jordan algebra, the horofunction compactification was obtained in [44], the cone in a (possibly infinite dimensional) spin factor was discussed in [14], and results for the p-metrics, with $1\le p<\infty$, on the symmetric cone in ${\textrm{Herm}}_n({\mathbb {C}})$ were obtained in [26].

2 Metric Geometry Preliminaries

We start by recalling the construction of the horofunction compactification and the detour distance.

Let (M, d) be a metric space and let ${\mathbb {R}}^M$ be the space of all real functions on M equipped with the topology of pointwise convergence. Fix $b\in M$, which is called the basepoint, and let ${\textrm{Lip}}^c_b(M)$ denote the set of all functions $h\in {\mathbb {R}}^M$ such that $h(b)=0$ and h is c-Lipschitz, i.e., $|h(x)-h(y)|\le cd(x,y)$ for all $x,y\in M$.

Then ${\textrm{Lip}}^c_b(M)$ is a compact subset of ${\mathbb {R}}^M$. Indeed, the complement of ${\textrm{Lip}}^c_b(M)$ is open, so ${\textrm{Lip}}^c_b(M)$ is closed subset of ${\mathbb {R}}^M$. Moreover, as $|h(x)|= |h(x)-h(b)|\le cd(x,b)$ for all $h\in {\textrm{Lip}}^c_b(M)$ and $x\in M$, we get that ${\textrm{Lip}}_b^c(M)\subseteq [-cd(x,b),cd(x,b)]^M$, which is compact by Tychonoff’s theorem.

For $y\in M$ define the real valued function,

$$\begin{aligned} h_{y}(z) = d(z,y)-d(b,y)\quad \text{ with } z\in M. \end{aligned}$$

(2.1)

Then $h_y(b)=0$ and $|h_y(z)-h_y(w)| = |d(z,y)-d(w,y)|\le d(z,w)$. Thus, $h_y\in {\textrm{Lip}}_b^1(M)$ for all $y\in M$. Using the previous observation one now defines the horofunction compactification of (M, d) to be the closure of $\{h_y:y\in M\}$ in ${\mathbb {R}}^M$, which is a compact subset of ${\textrm{Lip}}_b^1(M)$ and is denoted by ${\overline{M}}^h$. Its elements are called metric functionals, and the boundary $\partial {\overline{M}}^h= {\overline{M}}^h{\setminus } \{h_y:y\in M\}$ is called the horofunction boundary. The metric functionals in $\partial {\overline{M}}^h$ are called horofunctions, and all other metric functionals are said to be internal points.

The topology of pointwise convergence on ${\textrm{Lip}}_b^1(M)$ coincides with the topology of uniform convergence on compact sets, see [47, Sect. 46]. In general the topology of pointwise convergence on ${\textrm{Lip}}_b^1(M)$ is not metrisable, and hence horofunctions are limits of nets rather than sequences. If, however, the metric space is separable, then the pointwise convergence topology on ${\textrm{Lip}}_b^1(M)$ is metrizable and each horofunction is the limit of a sequence. It should be noted that the embedding $\iota :M \rightarrow {\textrm{Lip}}_b^1(M)$, where $\iota (y) = h_y$, may not have a continuous inverse on $\iota (M)$, and hence the metric compactification is not always a compactification in the strict topological sense. If, however, (M, d) is proper (i.e. closed balls are compact) and geodesic, then $\iota$ is a homeomorphism from M onto $\iota (M)$. Recall that a map $\gamma$ from a (possibly unbounded) interval $I\subseteq {\mathbb {R}}$ into a metric space (M, d) is called a geodesic path if

$$\begin{aligned} d(\gamma (s),\gamma (t)) = |s-t|\quad \text{ for } \text{ all } s,t\in I. \end{aligned}$$

The image, $\gamma (I)$, is called a geodesic, and a metric space (M, d) is said to be geodesic if for each $x,y\in M$ there exists a geodesic path $\gamma :[a,b]\rightarrow M$ connecting x and y, i.e, $\gamma (a)=x$ and $\gamma (b) = y$. We call a geodesic $\gamma ([0,\infty ))$ a geodesic ray.

The following fact, which is slightly weaker than [51, Theorem 4.7], will be useful in the sequel.

Lemma 2.1

If (M, d) is a proper geodesic metric space, then $h\in \partial {\overline{M}}^h$ if and only if there exists a sequence $(x^n)$ in M with $d(b,x^n)\rightarrow \infty$ such that $(h_{x^n})$ converges to $h\in {\overline{M}}^h$ as $n\rightarrow \infty$.

A sequence $(x^n)$ in (M, d) is called an almost geodesic sequence if for all $\varepsilon >0$ there exists a $N\ge 0$ such that

$$\begin{aligned} d(x^n,x^m) +d(x^m,x^0) - d(x^n,x^0) <\varepsilon \quad \text{ for } \text{ all } n\ge m\ge N. \end{aligned}$$

The notion of an almost geodesic sequence goes back to Rieffel [51] and was further developed by Walsh and co-workers in [2, 40, 43, 58]. In particular, every unbounded almost geodesic sequence yields a horofunction in a proper geodesic metric space [58].

Lemma 2.2

Let (M, d) be a proper geodesic metric space. If $(x^n)$ is an unbounded almost geodesic sequence in M, then

$$\begin{aligned} h(z) = \lim _n d(z,x^n)-d(b,x^n) \end{aligned}$$

exists for all $z\in M$ and $h\in \partial {\overline{M}}^h$.

Given a proper geodesic metric space (M, d), a horofunction $h\in {\overline{M}}^h$ is called a Busemann point if there exists an almost geodesic sequence $(x^n)$ in M such that $h(z) = \lim _{n} d(z,x^n)- d(b,x^n)$ for all $z\in M$. We denote the collection of all Busemann points by ${\mathcal {B}}_M$.

Suppose that $h,h'\in \partial {\overline{M}}^h$ be horofunctions and (M, d) is a proper geodesic metric space. Let $W_h$ be the collection of neighbourhoods of h in ${\overline{M}}^h$. The detour cost is given by

$$\begin{aligned} H(h,h') = \sup _{W\in W_h}\left( \inf _{x:\iota (x)\in W} d(b,x) +h'(x)\right) . \end{aligned}$$

The detour distance is given by

$$\begin{aligned} \delta (h,h') = H(h,h')+H(h',h). \end{aligned}$$

(2.2)

It is known [58] that if $(x^n)$ is an almost geodesic sequence converging to a horofunction h, then

$$\begin{aligned} H(h,h') = \lim _n d(b,x^n) +h'(x^n) \end{aligned}$$

(2.3)

for all horofunctions $h'$. Moreover, on the set of Busemann points ${\mathcal {B}}_M$ the detour distance is a metric where points can be at infinite distance from each other, see [58]. The detour distance yields a partition of ${\mathcal {B}}_M$ into equivalence classes, called parts, where h and $h'$ are equivalent if $\delta (h,h') <\infty$. The equivalence class of h is denoted by ${\mathcal {P}}_h$. So $({\mathcal {P}}_h,\delta )$ is a metric space, and ${\mathcal {B}}_M$ is the disjoint union of metric spaces under the detour distance.

It is known [57, Proposition 4.5] that two Busemann points h and g in the horofunction boundary are in the same part if and only if $\sup _{x\in M} |h(x)-g(x)|<\infty$. Furthermore, any isometry on M extends as an isometry to the set of Busemann points under the detour distance, see [43].

For symmetric spaces with nonpositive sectional curvature, all horofunctions with respect to the Riemannian metric are Busemann points and each part is a singleton. For the spaces under consideration in this paper we show that all horofunctions are Busemann points, but the parts can be nontrivial.

3 Bounded Symmetric Domains

In this section we analyse the geometry and topology of the horofunction compactification of bounded symmetric domains of the form $B=B_1\times \cdots \times B_r$, where $B_i=\{z\in {\mathbb {C}}^{n_i}:|z_1|^2+\cdots +|z_{n_i}|^2<1\}$, under the Kobayashi distance. In fact, we shall consider slightly more general product domains where each $B_i$ is the open unit ball of a norm on ${\mathbb {C}}^{n_i}$ with a strongly convex $C^3$-boundary. Even though these domains no longer correspond to noncompact type symmetric spaces we shall see that there still exists a homeomorphism between the horofunction compactification and the closed dual unit ball of the Finsler metric at the origin. We will start by recalling some basic concepts.

3.1 Product Domains and Kobayashi Distance

On a convex domain $D\subseteq {\mathbb {C}}^n$ the Kobayashi distance is given by

$$\begin{aligned} k_D(z,w) =\inf \{ \rho (\zeta ,\eta ):\exists f:\Delta \rightarrow D \text{ holomorphic } \text{ with } f(\zeta )=z \text{ and } f(\eta )=w\} \end{aligned}$$

for all $z,w\in D$, where

$$\begin{aligned} \rho (z,w) = \log \frac{ 1+\left| \frac{w-z}{1-{\bar{z}}w}\right| }{1-\left| \frac{w-z}{1-{\bar{z}}w}\right| }=2\tanh ^{-1}\left( 1 -\frac{(1-|w|^2)(1-|z|^2)}{|1-w{\bar{z}}|^2}\right) ^{1/2} \end{aligned}$$

is the hyperbolic distance on the open disc $\Delta =\{z\in {\mathbb {C}}:|z|<1\}$.

It is known, see [1, Proposition 2.3.10], that if $D\subset {\mathbb {C}}^n$ is bounded convex domain, then $(D,k_D)$ is a proper metric space, whose topology coincides with the usual topology on ${\mathbb {C}}^n$. Moreover, $(D,k_D)$ is a geodesic metric space containing geodesic rays, see [1, Theorem 2.6.19] or [38, Theorem 4.8.6].

For the Euclidean ball $B^n =\{(z_1,\ldots ,z_n)\in {\mathbb {C}}^n:\Vert z\Vert ^2<1\}$, where $\Vert z\Vert ^2 = \sum _i |z_i|^2$, the Kobayashi distance satisfies

$$\begin{aligned} k_{B^n}(z,w) = 2\tanh ^{-1}\left( 1 -\frac{(1-\Vert w\Vert ^2)(1-\Vert z\Vert ^2)}{|1-\langle z,w\rangle |^2}\right) ^{1/2} \end{aligned}$$

for all $z,w\in B^n$, see [1, Chaps. 2.2 and 2.3].

In our setting we will consider product domains $B=\prod _{i-1}^r B_i$, where each $B_i$ is an open unit ball of a norm in ${\mathbb {C}}^{n_i}$, and we will use the product property of $k_B$, which says that

$$\begin{aligned} k_B(z,w) = \max _{i=1,\ldots ,r} k_i(z_i,w_i), \end{aligned}$$

where $k_i$ is the Kobayashi distance on $B_i$, see [38, Theorem 3.1.9]. So for the polydisc $\Delta ^r=\{(z_1,\ldots ,z_r)\in {\mathbb {C}}^r:\max _i |z_i|<1\}$, the Kobayashi distance satisfies

$$\begin{aligned} k_{\Delta ^r}(z,w) =\max _i \rho (z_i,w_i)\quad \text{ for } \text{ all }\, w=(w_1,\ldots ,w_r), z=(z_1,\ldots ,z_r)\in \Delta ^r. \end{aligned}$$

For the Euclidean ball, $B^n$, it is well known that the horofunctions of $(B^n,k_{B^n})$, with basepoint $b=0$, are given by

$$\begin{aligned} h_\xi (z) = \log \frac{ |1 -\langle z,\xi \rangle |^2}{1-\Vert z\Vert ^2}\quad \text{ for } \text{ all } z\in B^n, \end{aligned}$$

(3.1)

where $\xi \in \partial B^n$. Moreover, each horofunction $h_\xi$ is a Busemann point, as it is the limit induced by the geodesic ray $t\mapsto \frac{e^t-1}{e^t+1}\xi$, for $0\le t<\infty$.

Moreover, if B is a product of Euclidean balls, then the horofunctions are known, see [1, Proposition 2.4.12] and [40, Corollary 3.2]. Indeed, for a product of Euclidean balls $B^{n_1}\times \cdots \times B^{n_r}$ the Kobayashi distance horofunctions with basepoint $b=0$ are precisely the functions of the form

$$\begin{aligned} h(z) = \max _{j\in J} \left( h_{\xi _j}(z_j)-\alpha _j\right) , \end{aligned}$$

where $J\subseteq \{1,\ldots ,r\}$ nonempty, $\xi _j\in \partial B^{n_j}$ for $j\in J$, and $\min _{j\in J} \alpha _{j}=0$. Moreover, each horofunction is a Busemann point.

The form of the horofunctions of the product of Euclidean balls is essentially due to the product property of the Kobayashi distance and the smoothness and convexity properties of the balls. Indeed, more generally, the following result holds, see [40, Sect. 2 and Lemma 3.3].

Theorem 3.1

If $D_i\subset {\mathbb {C}}^{n_i}$ is a bounded strongly convex domain with $C^3$-boundary, then for each $\xi _i\in \partial D_i$ there exists a unique horofunction $h_{\xi _i}$ which is the limit of a geodesic $\gamma$ from the basepoint $b_i\in D_i$ to $\xi _i$. Moreover, these are all the horofunctions. If $D = \prod _{i=1}^r D_i$, where each $D_i$ is a bounded strongly convex domain with $C^3$-boundary, then each horofunction h of $(D,k_D)$ with respect to the basepoint $b=(b_1,\ldots ,b_r)$ is of the form

$$\begin{aligned} h(z) = \max _{j\in J} \left( h_{\xi _j}(z_j)-\alpha _j\right) , \end{aligned}$$

(3.2)

where $J\subseteq \{1,\ldots ,r\}$ nonempty, $\xi _j\in \partial D_j$ for $j\in J$, and $\min _{j\in J} \alpha _{j}=0$. Furthermore, each horofunction is a Busemann point, and the part of h, where h is given by (3.2), consists of those horofunctions $h'$ of the form,

$$\begin{aligned} h'(z) = \max _{j\in J} \left( h_{\xi _j}(z_j)-\beta _j\right) , \end{aligned}$$

with $\min _{j\in J}\beta _j =0$.

Now let $D = \prod _{i=1}^r D_i$, where each $D_i$ is a bounded strongly convex domain with $C^3$-boundary. Given $J\subseteq \{1,\ldots ,r\}$ nonempty, $\xi _j\in \partial D_j$ for $j\in J$, and $\alpha _j\ge 0$ for $j\in J$ with $\min _{j\in J} \alpha _{j}=0$, we can find geodesic paths $\gamma _j:[0,\infty )\rightarrow D_j$ from $b_j$ to $\xi _j$, and form the path $\gamma :[0,\infty )\rightarrow D$, where

$$\begin{aligned} \gamma (t)_j = \left[ \begin{array}{ll} \gamma _j(t-\alpha _j) &{} \text{ for } \text{ all }\, j\in J \text{ and } t\ge \alpha _j \\ b_j &{} \text{ otherwise. } \end{array}\right. \end{aligned}$$

(3.3)

Lemma 3.2

The path $\gamma :[0,\infty )\rightarrow D$ in (3.3) is a geodesic path, and $h_{\gamma (t)}\rightarrow h$ where h is given by (3.2).

Proof

Let $k_i$ denote the Kobayashi distance on $D_i$. By the product property we have that

$$\begin{aligned} k_D(\gamma (s),\gamma (t)) =\max _i k_i(\gamma (s)_i,\gamma (t)_i) \end{aligned}$$

for all $s\ge t\ge 0$. By construction $k_i(\gamma (s)_i,\gamma (t)_i)\le k_i(\gamma _i(s),\gamma _i(t))=s-t$ for all i and $s\ge t\ge 0$. For $j\in J$ with $\alpha _j=0$ we have that $k_j(\gamma (s)_j,\gamma (t)_j) = k_j(\gamma _j(s),\gamma _j(t)) =s-t$ for all $s\ge t\ge 0$, and hence

$$\begin{aligned} k_D(\gamma (s),\gamma (t)) =\max _i k_i(\gamma (s)_i,\gamma (t)_i) = s-t \end{aligned}$$

for all $s\ge t\ge 0$.

Note that for $z\in D$ we have

$$\begin{aligned} \lim _{t\rightarrow \infty }h_{\gamma (t)}(z)= & {} \lim _{t\rightarrow \infty } k_D(z,\gamma (t)) - k_D(\gamma (t),b) \\= & {} \lim _{t\rightarrow \infty } \max _i (k_i(z_i,\gamma (t)_i) - t)\\= & {} \lim _{t\rightarrow \infty } \max _{j\in J} (k_j(z_j,\gamma (t)_j) - t)\\= & {} \lim _{t\rightarrow \infty } \max _{j\in J} (k_j(z_j,\gamma _j(t-\alpha _j)) - k_j(\gamma _j(t-\alpha _j),b_j)-\alpha _j)\\= & {} \max _{j\in J} \left( h_{\xi _j}(z_j)-\alpha _j\right) , \end{aligned}$$

which shows that $h_{\gamma (t)}\rightarrow h$. $\square$

Consider $B=\prod _{i=1}^r B_i\subseteq {\mathbb {C}}^n$, where each $B_i$ is an open unit ball of a norm in ${\mathbb {C}}^{n_i}$. Then B is the open unit ball of the norm $\Vert \cdot \Vert _B$ on ${\mathbb {C}}^n$. In fact,

$$\begin{aligned} \Vert w\Vert _B =\max _{i=1,\ldots ,r} \Vert w_i\Vert _{B_i}, \end{aligned}$$

where $\Vert \cdot \Vert _{B_i}$ is the norm on ${\mathbb {C}}^{n_i}$ with open unit ball $B_i$.

To analyse the dual norm of $\Vert \cdot \Vert _B$ we identify the dual space of ${\mathbb {C}}^{n_1}\times \cdots \times {\mathbb {C}}^{n_r}$ with itself using the standard inner-product

$$\begin{aligned} \langle x,y\rangle = \sum _{i=1}^r \langle x_i,y_i\rangle \quad \text{ for } x =(x_1,\ldots ,x_r),y=(y_1,\ldots ,y_r)\in {\mathbb {C}}^{n_1}\times \cdots \times {\mathbb {C}}^{n_r}. \end{aligned}$$

So, $y\in {\mathbb {C}}^{n_1}\times \cdots \times {\mathbb {C}}^{n_r} \mapsto \langle \cdot ,y\rangle \in ({\mathbb {C}}^{n_1}\times \cdots \times {\mathbb {C}}^{n_r})^*$. Note that the dual norm $\Vert \cdot \Vert _B^*$ satisfies

$$\begin{aligned} \Vert y\Vert _B^*= & {} \sup _{\Vert x\Vert _B=1} {\textrm{Re}} \langle x,y\rangle = \sup _{\Vert x\Vert _B=1} \sum _{i=1}^r {\textrm{Re}} \langle x_i,y_i\rangle \\= & {} \sum _{i=1}^r \Vert y_i\Vert _{B_i}^*\qquad \text{ for } y=(y_1,\ldots ,y_r) \in {\mathbb {C}}^{n_1}\times \cdots \times {\mathbb {C}}^{n_r}, \end{aligned}$$

as $\Vert x\Vert _B= \max _i \Vert x_i\Vert _{B_i}$. So we see that the closed dual unit ball is given by

$$\begin{aligned} B_1^*= & {} \{ y\in {\mathbb {C}}^{n_1}\times \cdots \times {\mathbb {C}}^{n_r}:{\textrm{Re}} \langle x,y\rangle \le 1 \text{ for } \text{ all } x\in \overline{B}\}\\= & {} \{y\in {\mathbb {C}}^{n_1}\times \cdots \times {\mathbb {C}}^{n_r}:\sum _{i=1}^r \Vert y_i\Vert _{B_i}^*\le 1\}. \end{aligned}$$

Now suppose that each $B_i$ is strictly convex and smooth. The closed ball $B_1^*$ has extreme points $p(\xi ^*_i)= (0,\ldots ,0,\xi _i^*,0,\ldots ,0)$, where $\xi _i^*\in {\mathbb {C}}^{n_i}$ is the unique supporting functional at $\xi _i\in \partial B_i$, i.e., ${\textrm{Re}} \langle \xi _i,\xi _i^*\rangle = 1$ and ${\textrm{Re}} \langle w_i,\xi _i^*\rangle < 1$ for $w_i\in \overline{B_i}$ with $w_i\ne \xi _i$.

The relatively open faces of $B_1^*$ are the sets of the form

$$\begin{aligned} F(\{\xi _j\in \partial B_j:j\in J\}) = \left\{ \sum _{j\in J}\lambda _jp(\xi _j^*):\sum _{j\in J}\lambda _j =1 \text{ and } \lambda _j>0 \text{ for } \text{ all } j\in J\right\} , \end{aligned}$$

where $J\subseteq \{1,\ldots ,r\}$ is nonempty and $\xi _j\in \partial B_j$ for $j\in J$ are fixed. Here the relative topology is taken with respect to the affine span of $\{p(\xi _j^*):j\in J\}$.

On B the Kobayashi distance has a Finsler structure in terms of the infinitesimal Kobayashi metric, see e.g., [1, Chap. 2.3]. Indeed, we have that

$$\begin{aligned} k_B(z,w) =\inf _\gamma L(\gamma ), \end{aligned}$$

where the infimum is taken over all piecewise $C^1$-smooth paths $\gamma :[0,1]\rightarrow B$ with $\gamma (0)=z$ and $\gamma (1) =w$, and

$$\begin{aligned} L(\gamma ) = \int _0^1 \kappa _B(\gamma (t),\gamma '(t)){\textrm{d}}t, \end{aligned}$$

with

$$\begin{aligned} \kappa _B(u,v) = \inf \{|\xi |:\exists \varphi \in {\textrm{Hol}}(\Delta ,B) \text{ such } \text{ that } \varphi (0) = u \text{ and } ({\textrm{D}}\varphi )_0(\xi ) = v\}. \end{aligned}$$

Proposition 3.3

[1, Proposition 2.3.24] If B is the open unit ball of a norm on ${\mathbb {C}}^n$, then

$$\begin{aligned} \kappa _B(0,v) = \Vert v\Vert _B\quad \text{ for } \text{ all } v\in {\mathbb {C}}^n. \end{aligned}$$

For $z\in B$ and $i=1,\ldots ,r$, if $z_i\ne 0$, then we let $z_i' = \Vert z_i\Vert _{B_i}^{-1}z_i\in \partial B_i$ and we write $p(z_i^*) = (0,\ldots ,0,z_i^*,0,\ldots ,0)$, where $z_i^*$ is the unique supporting functional at $z_i'\in \partial B_i$. If $z_i=0$, we set $p(z_i^*)=0$.

We now define a map $\varphi _B:{\overline{B}}^h\rightarrow B_1^*$ and show in the remainder of this section that it is a homeomorphism. For $z\in B =B_1\times \cdots \times B_r$ let

$$\begin{aligned} \varphi _B(z) = \frac{1}{\sum _{i=1}^r e^{k_i(z_i,0)} + e^{-k_i(z_i,0)}} \left( \sum _{i=1}^r (e^{k_i(z_i,0)} - e^{-k_i(z_i,0)})p(z^*_i)\right) , \end{aligned}$$

where $k_i$ is the Kobayashi distance on $B_i$. For a horofunction h given by (3.2) we define

$$\begin{aligned} \varphi _B(h) = \frac{1}{\sum _{j\in J} e^{-\alpha _j}}\left( \sum _{j\in J} e^{-\alpha _j}p(\xi _j^*)\right) . \end{aligned}$$

More precisely we prove the following theorem.

Theorem 3.4

If $B = \prod _{i=1}^r B_i$, where each $B_i$ is the open unit ball of a norm on ${\mathbb {C}}^{n_i}$ which is strongly convex and has a $C^3$-boundary, then $\varphi _B:{\overline{B}}^h\rightarrow B_1^*$ is a homeomorphism, which maps each part of $\partial {\overline{B}}^h$ onto the relative interior of a boundary face of $B_1^*$.

3.2 The Map $\varphi _B$: Injectivity and Surjectivity

Throughout the remainder of this section we assume that $B = \prod _{i=1}^r B_i$ and each $B_i$ is the open unit ball of a norm on ${\mathbb {C}}^{n_i}$, which is strongly convex and has a $C^3$-boundary. So for each $\xi _i\in \partial B_i$ there exists a unique $\xi _i^*\in {\mathbb {C}}^{n_i}$ such that

$$\begin{aligned} {\textrm{Re}}\langle \xi _i,\xi _i^*\rangle = 1 \text{ and } {\textrm{Re}}\langle w_i, \xi _i^*\rangle < 1 \text{ for } \text{ all } w_i\in \overline{B_i} \text{ with } w_i\ne \xi _i, \end{aligned}$$

as $\overline{B_i}$ is strictly convex and smooth.

We start with the following basic observation.

Lemma 3.5

For each $z\in B$ we have that $\varphi _B(z) \in {\textrm{int}}\, B_1^*$, and $\varphi _B(h) \in \partial B_1^*$ for all $h\in \partial {\overline{B}}^h$.

Proof

Note that for $z\in B$ and $w\in \overline{B}$ we have that

$$\begin{aligned} {\textrm{Re}} \langle w,\varphi _B(z)\rangle= & {} \frac{1}{\sum _{i=1}^r e^{k_{i}(z_i,0)} + e^{-k_i(z_i,0)}} \left( \sum _{i=1}^r (e^{k_i(z_i,0)} - e^{-k_i(z_i,0)}){\textrm{Re}}\langle w_i,z^*_i\rangle \right) \\\le & {} \frac{1}{\sum _{i=1}^r e^{k_i(z_i,0)} + e^{-k_i(z_i,0)}} \left( \sum _{i=1}^r e^{k_i(z_i,0)} - e^{-k_i(z_i,0)}\right) \\< & {} 1-\delta \end{aligned}$$

for some $0<\delta <1$, which is independent of w. Thus, $\sup _{w\in \overline{B}} {\textrm{Re}} \langle w,\varphi _B(z)\rangle<1-\delta <1$, hence $\varphi _B(z) \in {\textrm{int}}\, B_1^*$.

To see that $\varphi _B(h)\in \partial B_1^*$, note that for $w =\sum _{j\in J}p(\xi _j)\in \overline{B}$, where $p(\xi _j) = (0,\ldots ,0,\xi _j,0,\ldots ,0)$, we have that ${\textrm{Re}}\langle w,\varphi _B(h)\rangle = 1$. $\square$

To show that $\varphi _B$ is injective on B, we need the following basic calculus fact, which can be found in [32, Sect. 4].

Lemma 3.6

If $\mu :{\mathbb {R}}^r\rightarrow {\mathbb {R}}$ is given by $\mu (x_1,\ldots ,x_r) = \sum _{i=1}^r e^{x_i} +e^{-x_i}$, then $x\mapsto \nabla \log \mu (x)$ is injective on ${\mathbb {R}}^r$.

Note that

$$\begin{aligned} (\nabla \log \mu (x))_j = \frac{e^{x_j}-e^{-x_j}}{\sum _{i=1}^r e^{x_i}+e^{-x_i}}\quad \text{ for } \text{ all } j. \end{aligned}$$

Lemma 3.7

The map $\varphi _B$ is a continuous bijection from B onto ${\textrm{int}}\, B_1^*$.

Proof

Cleary $\varphi _B$ is continuous on B and $\varphi _B(z) =0$ if and only if $z=0$. Suppose that $z,w\in B\setminus \{0\}$ are such that $\varphi _B(z) = \varphi _B(w)$. For simplicity write

$$\begin{aligned} \alpha _j = \frac{e^{k_j(z_j,0)} - e^{-k_j(z_j,0)}}{\sum _{i=1}^r e^{k_i(z_i,0)} + e^{-k_i(z_i,0)}}\ge 0 \quad \text{ and } \quad \beta _j = \frac{e^{k_j(w_j,0)} - e^{-k_j(w_j,0)}}{\sum _{i=1}^r e^{k_i(w_i,0)} + e^{-k_i(w_i,0)}}\ge 0. \end{aligned}$$

Note that $\alpha _jp(z_j^*)=0$ if and only if $z_j=0$, and $\beta _jp(w_j^*)=0$ if and only if $w_j=0$. Thus, $z_j=0$ if and only if $w_j=0$. Now suppose that $z_j\ne 0$, so $w_j\ne 0$. Then $\langle p(v_j),\varphi _B(z) \rangle = \langle p(v_j),\varphi _B(w) \rangle$ for each $v_j\in B_j$. This implies that

$$\begin{aligned} \alpha _j\langle v_j,z^*_j \rangle = \beta _j\langle v_j,w^*_j \rangle \quad \text{ for } \text{ all } v_j\in B_j, \end{aligned}$$

hence $\alpha _j z^*_j = \beta _j w^*_j$. It follows that $\alpha _j =\beta _j$ and $z^*_j=w^*_j$. Thus $z_j = \mu _j w_j$ for some $\mu _j>0$. As $\alpha _i=\beta _i$ for all $i\in \{1,\ldots ,r\}$, we know by Lemma 3.6 that $k_j(z_j,0) = k_j(w_j,0)$, hence $z_j =w_j$ by [1, Proposition 2.3.5]. So $z=w$, which shows that $\varphi _B$ is injective.

As $\varphi _B$ is injective and continuous on B, it follows from Brouwer’s domain invariance theorem that $\varphi _B(B)$ is an open subset of ${\textrm{int}}\,B^*_1$ by Lemma 3.5. Suppose, by way of contradiction, that $\varphi _B(B)\ne {\textrm{int}}\, B^*_1$. Then $\partial \varphi _B(B)\cap {\textrm{int}}\, B_1^*$ is nonempty, as otherwise $\varphi _B(B)$ is closed and open in ${\textrm{int}}\, B^*_1$, which would imply that ${\textrm{int}}\,B_1^*$ is the disjoint union of the nonempty open sets $\varphi _B(B)$ and its complement contradicting the connectedness of ${\textrm{int}}\, B^*_1$. So let $w\in \partial \varphi _B(B)\cap {\textrm{int}}\, B_1^*$ and $(z^n)$ be a sequence in B such that $\varphi _B(z^n)\rightarrow w$. As $\varphi _B$ is continuous on B, we have that $k_B(z^n,0)\rightarrow \infty$.

Using the product property, $k_B(z^n,0) = \max _i k_i(z^n_i,0)$, we may assume after taking subsequences that $\alpha _i^n = k_B(z^n,0) - k_{i}(z^n_i,0)\rightarrow \alpha _i\in [0,\infty ]$ and $z^n_i\rightarrow \zeta _i\in \overline{B_i}$ for all i. Let $I = \{i:\alpha _i<\infty \}$, and note that for each $i\in I$, $\zeta _i\in \partial B_i$, as $k_i(z_i^n,0)\rightarrow \infty$. Then

$$\begin{aligned} \varphi _B(z^n)= & {} \frac{1}{\sum _{i=1}^r e^{k_i(z^n_i,0)} + e^{-k_i(z^n_i,0)}} \left( \sum _{i=1}^r (e^{k_i(z^n_i,0)} - e^{-k_i(z^n_i,0)})p((z_i^n)^*)\right) \\= & {} \frac{1}{\sum _{i=1}^r e^{-\alpha ^n_i} + e^{-k_B(z^n,0) -k_i(z^n_i,0)}} \left( \sum _{i=1}^r (e^{-\alpha ^n_i} - e^{-k_B(z^n,0)-k_i(z^n_i,0)})p((z^n_i)^*)\right) . \end{aligned}$$

Letting $n\rightarrow \infty$, the righthand side converges to

$$\begin{aligned} \frac{1}{\sum _{i\in I} e^{-\alpha _i}} \left( \sum _{i\in I} e^{-\alpha _i}p(\zeta _i^*)\right) =w. \end{aligned}$$

But this implies that $w\in \partial B^*_1$, as ${\textrm{Re}}\langle \sum _{i\in I}p(\zeta _i), w\rangle = 1$ and $\sum _{i\in I} p(\zeta _i)\in \overline{B}$, where $p(\zeta _i) = (0,\ldots ,0,\zeta _i,0,\ldots ,0)$. This is impossible and hence $\varphi _B(B)={\textrm{int}}\, B^*_1$. $\square$

We now analyse $\varphi _B$ on $\partial {\overline{B}}^h$.

Lemma 3.8

The map $\varphi _B$ maps $\partial {\overline{B}}^h$ bijectively onto $\partial B_1^*$. Moreover, the part ${\mathcal {P}}_h$, where h is given by (3.2), is mapped onto the relative open boundary face

$$\begin{aligned} F(\{\xi _j\in \partial B_j:j\in J\}) = \left\{ \sum _{j\in J}\lambda _jp(\xi _j^*):\sum _{j\in J}\lambda _j =1 \text{ and } \lambda _j>0 \text{ for } \text{ all } j\in J\right\} . \end{aligned}$$

Proof

We know from Lemma 3.5 that $\varphi _B$ maps $\partial {\overline{B}}^h$ into $\partial B_1^*$. To show that it is onto we let $w\in \partial B_1^*$. As $B^*_1$ is the disjoint union of its relative open faces (see [52, Theorem 18.2]), there exist $J\subseteq \{1,\ldots ,r\}$, extreme points $p(\xi ^*_j)$ of $B_1^*$, and $0<\lambda _j\le 1$ for $j\in J$ with $\sum _{j\in J}\lambda _j=1$ such that $w=\sum _{j\in J}\lambda _j p(\xi ^*_j)$. Let $\mu _j = -\log \lambda _j$ and $\mu ^* = \min _{j\in J} \mu _j$. Now set $\alpha _j = \mu _j -\mu ^*$ for $j\in J$. Then $\alpha _j \ge 0$ for $j\in J$ and $\min _{j\in J} \alpha _j =0$.

Let $h\in \partial {\overline{B}}^h$ be given by $h(z) = \max _{j\in J} (h_{\xi _j}(z_j)-\alpha _j)$. Then

$$\begin{aligned} \varphi _B(h) = \frac{\sum _{j\in J} e^{-\alpha _j} p(\xi _j^*)}{\sum _{j\in J} e^{-\alpha _j}} = \frac{\sum _{j\in J} e^{-\mu _j} p(\xi _j^*)}{\sum _{j\in J} e^{-\mu _j} } = \frac{\sum _{j\in J} \lambda _j p(\xi ^*_j)}{\sum _{j\in J} \lambda _j}=w. \end{aligned}$$

To prove injectivity let $h,h'\in \partial {\overline{B}}^h$, where h is as in (3.2) and

$$\begin{aligned} h'(z) = \max _{j\in J'} (h_{\eta _j}(z_j)-\beta _j) \end{aligned}$$

(3.4)

for $z\in B$. Suppose that $\varphi _B(h)=\varphi _B(h')$, so

$$\begin{aligned} \varphi _B(h) = \frac{\sum _{j\in J}e^{-\alpha _j}p(\xi _j^*)}{\sum _{j\in J} e^{-\alpha _j}}= \frac{\sum _{j\in J'}e^{-\beta _j}p(\eta _j^*)}{\sum _{j\in J'} e^{-\beta _j} } =\varphi _B(h'). \end{aligned}$$

We have that $J=J'$. Indeed, if $k\in J$ and $k\not \in J'$, then

$$\begin{aligned} 0={\textrm{Re}}\langle p(\xi _k),\varphi _B(h')\rangle = {\textrm{Re}}\langle p(\xi _k),\varphi _B(h)\rangle >0, \end{aligned}$$

which is impossible. For the other case a contradiction can be derived in the same way.

Now suppose there exists $k\in J$ such that $\xi _k\ne \eta _k$. If

$$\begin{aligned} \frac{e^{-\alpha _k}}{\sum _{j\in J} e^{-\alpha _j}}\le \frac{e^{-\beta _k}}{\sum _{j\in J} e^{-\beta _j}}, \end{aligned}$$

then

$$\begin{aligned} {\textrm{Re}}\langle p(\eta _k),\varphi _B(h)\rangle= & {} \frac{e^{-\alpha _k}}{\sum _{j\in J} e^{-\alpha _j}}{\textrm{Re}}\langle \eta _k,\xi ^*_k\rangle < \frac{e^{-\alpha _k}}{\sum _{j\in J} e^{-\alpha _j}} \\ {}\le & {} \frac{e^{-\beta _k}}{\sum _{j\in J} e^{-\beta _j}} = {\textrm{Re}}\langle p(\eta _k),\varphi _B(h')\rangle , \end{aligned}$$

as $\overline{B_k}$ is smooth and strictly convex, which contradicts $\varphi _B(h) =\varphi _B(h')$. The other case goes in the same way. Thus, $J=J'$ and $\xi _j=\eta _j$ for all $j\in J$.

It follows that

$$\begin{aligned} \frac{e^{-\alpha _k}}{\sum _{j\in J} e^{-\alpha _j}} = {\textrm{Re}}\langle p(\xi _k),\varphi _B(h)\rangle = {\textrm{Re}}\langle p(\eta _k),\varphi _B(h')\rangle = \frac{e^{-\beta _k}}{\sum _{j\in J} e^{-\beta _j}} \end{aligned}$$

for all $k\in J$. To show that $\alpha _k=\beta _k$ for all $k\in J$ let $\nu :{\mathbb {R}}^J\rightarrow {\mathbb {R}}$ be given by $\nu (x) = \sum _{j\in J}e^{-x_j}$. Then for $x,y\in {\mathbb {R}}^J$ and $0<t<1$ we have that

$$\begin{aligned} \nu (tx+(1-t)y) \le \nu (x)^t\nu (y)^{1-t}, \end{aligned}$$

and we have equality if and only if there exists a constant c such that $x_k = y_k+c$ for all $k\in J$. So, if $x\ne y +(c,\ldots ,c)$ for all c, then $-\nabla \log \nu (x) \ne -\nabla \log \nu (y)$.

As $\min _{j\in J}\alpha _j =0=\min _{j\in J}\beta _j$, we can conclude that $\alpha _k=\beta _k$ for all $k\in J$. This shows that $h = h'$ and hence $\varphi _B$ is injective on $\partial {\overline{B}}^h$.

To complete the proof, note that $\varphi _B(h)$ is in the relative open boundary face $F(\{\xi _j\in \partial B_j:j\in J\})$ of $B_1^*$. Moreover, $h'$ given by (3.4) is in the same part as h if, and only if, $J=J'$ and $\xi _j=\eta _j$ for all $j\in J$ by [40, Propositions 2.8 and 2.9]. So, $\varphi _B(h')$ lies in $F(\{\xi _j\in \partial B_j:j\in J\})$ if and only if $h'$ lies in the same part as h. $\square$

3.3 Continuity and the Proof of Theorem 3.4

We now show that $\varphi _B$ is continuous on ${\overline{B}}^h$.

Proposition 3.9

The map $\varphi _B:{\overline{B}}^h\rightarrow B_1^*$ is continuous.

Proof

Clearly $\varphi _B$ is continuous on B. Suppose that $(z^n)$ is sequence in B converging to $h\in \partial {\overline{B}}^h$, where h is given by (3.2). To show that $\varphi _B(z^n)\rightarrow \varphi _B(h)$ we show that every subsequence of $(\varphi _B(z^n))$ has a subsequence converging to $\varphi _B(h)$. So, let $(\varphi _B(z^{n_k}))$ be a subsequence. We can take a further subsequence $(z^{n_{k,m}})$ such that

• (1)
  $$\begin{aligned} \beta ^m_j = k_B(z^{n_{k,m}},0) - k_j(z^{n_{k,m}}_j,0) \rightarrow \beta _j\in [0,\infty ]\quad \text{ for } \text{ all } j\in \{1,\ldots ,r\}. \end{aligned}$$

• (2) There exists $j_0$ such that $\beta ^m_{j_0} =0$ for all $m\ge 1$.

• (3) $(z^{n_{k,m}}_j)$ converges to $\eta _j\in \overline{B_j}$ and $h_{z^{n_{k,m}}_j}\rightarrow h_{\eta _j}$ for all $j\in \{1,\ldots ,r\}$.

Let $J'=\{j:\beta _j<\infty \}$. Then $h_{z^{n_{k,m}}}\rightarrow h'$, where $h'(z) = \max _{j\in J'} (h_{\eta _j}(z_j) - \beta _j)$ for $z\in B$, as

$$\begin{aligned}{} & {} \lim _{m\rightarrow \infty } k_B(z,z^{n_{k,m}}) - k_B(z^{n_{k,m}},0) \\{} & {} \quad = \lim _{m\rightarrow \infty } \max _j (k_j(z_j,z^{n_{k,m}}_j) - k_j(z^{n_{k,m}}_j,0) -\beta ^m_j) = \max _{j\in J'} (h_{\eta _j}(z_j) - \beta _j), \end{aligned}$$

by the product property of $k_B$.

As $h=h'$, we know by [40, Propositions 2.8 and 2.9] that $J=J'$, $\xi _j=\eta _j$ and $\alpha _j =\beta _j$ for all $j\in J$. We also know by Lemma 2.1 that $k_B(z^{n_{k,m}},0)\rightarrow \infty$, as h is a horofunction. So,

$$\begin{aligned} \varphi _B(z^{n_{k,m}})= & {} \frac{\sum _{i=1}^r (e^{-\beta _i^m} - e^{-k_B(z^{n_{k,m}},0) -k_i(z^{n_{k,m}}_i,0)})p((z^{n_{k,m}}_i)^*)}{\sum _{i=1}^r e^{-\beta _i^m} - e^{-k_B(z^{n_{k,m}},0) -k_i(z^{n_{k,m}}_i,0)}}\\\rightarrow & {} \frac{\sum _{j\in J} e^{-\beta _j}p(\eta _j^*)}{\sum _{j\in J} e^{-\beta _j}} = \varphi _B(h), \end{aligned}$$

which shows that $\varphi _B(z^n)\rightarrow \varphi _B(h)$.

We know from Lemma 3.5 that $\varphi _B(B)\subseteq {\textrm{int}}\, B_1^*$ and $\varphi _B(\partial {\overline{B}}^h)\subseteq \partial B_1^*$. So, to complete the proof it remains to show that if $(h_n)$ in $\partial {\overline{B}}^h$ converges to $h\in \partial {\overline{B}}^h$, where h is as in (3.2), then $\varphi _B(h_n)\rightarrow \varphi _B(h)$. For $n\ge 1$ let $h_n$ be given by

$$\begin{aligned} h_n(z) = \max _{j\in J_n} (h_{\eta _j^n}(z_j) - \beta _j^n)\qquad \text{ for }\, z\in B. \end{aligned}$$

Again we show that every subsequence of $(\varphi _B(h_n))$ has a convergent subsequence with limit $\varphi _B(h)$.

Let $(\varphi _B(h_{n_k}))$ be a subsequence. Taking a further subsequence we may assume that

• (1) There exists $J_0\subseteq \{1,\ldots ,r\}$ such that $J_{n_k} =J_0$ for all k.

• (2) There exists $j_0\in J_0$ such that $\beta ^{n_k}_{j_0}=0$ for all k.

• (3) $\beta _j^{n_k}\rightarrow \beta _j\in [0,\infty ]$ for all $j\in J_0$.

• (4) $\eta _j^{n_k}\rightarrow \eta _j$ for all $j\in J_0$.

Note that for each $j\in J_0$ we have that $h_{\eta _j^{n_k}}\rightarrow h_{\eta _j}$ in ${\overline{B}}_j^h$, as the identity map on $\overline{B_j}$, that is $\xi _j\in \overline{B_j}\rightarrow h_{\xi _j}\in {\overline{B}}_j^h$, is a homeomorphism by [5, Theorem 1.2].

Let $J'=\{j\in J_0:\beta _j<\infty \}$ and note that $j_0\in J'$. Then for each $z\in B$ we have that

$$\begin{aligned} \lim _{m\rightarrow \infty } h_{n_k}(z)= & {} \lim _{k\rightarrow \infty } \max _{j\in J_0} (h_{\eta _j^{n_k}}(z_j) -\beta _j^{n_k}) = \lim _{k\rightarrow \infty } \max _{j\in J'} (h_{\eta _j^{n_k}}(z_j) -\beta _j^{n_k}) \\ {}= & {} \max _{j\in J'} (h_{\eta _j}(z_j) -\beta _j). \end{aligned}$$

So, if we let $h'(z) = \max _{j\in J'} (h_{\eta _j}(z_j) -\beta _j)$ for $z\in B$, then $h'$ is a horofunction by Theorem 3.1 and $h_{n_k}\rightarrow h'$ in ${\overline{B}}^h$. As $h_n\rightarrow h$, we conclude that $h'=h$. This implies that $J'=J$ and $\eta _j=\xi _j$ and $\beta _j=\alpha _j$ for all $j\in J$, as otherwise $\delta (h,h') \ne 0$ by [40, Proposition 2.9 and Lemma 3.3]. This implies that $\beta _j^{k_m}\rightarrow \alpha _j$ and $\eta _j^{k_m}\rightarrow \xi _j$ for all $j\in J'$. Moreover, by definition $\beta _j^{n_k}\rightarrow \infty$ for all $j\in J_0{\setminus } J'$. Thus,

$$\begin{aligned} \varphi _B(h_{n_k}) = \frac{\sum _{j\in J_0} e^{-\beta _j^{n_k}} p((\eta _j^{n_k})^*) }{\sum _{j\in J_0}e^{-\beta _j^{n_k}}} \rightarrow \frac{\sum _{j\in J} e^{-\alpha _j} p(\xi _j^*) }{\sum _{j\in J}e^{-\alpha _j}} = \varphi _B(h), \end{aligned}$$

which completes the proof. $\square$

The proof of Theorem 3.4 is now straightforward.

Proof of Theorem 3.4

It follows from Lemmas 3.7 and 3.8 and Proposition 3.9 that $\varphi _B:{\overline{B}}^h\rightarrow B^*_1$ is a continuous bijection. As ${\overline{B}}^h$ is compact and $B^*_1$ is Hausdorff, we conclude that $\varphi _B$ is a homeomorphism. Moreover, $\varphi _B$ maps each part of $\partial {\overline{B}}^h$ onto the relative interior of a boundary face of $B_1^*$ by Lemma 3.8. $\square$

4 Euclidean Jordan Algebras with Spectral Norm

Every finite dimensional normed space $(V,\Vert \cdot \Vert )$ has a Finsler structure. Indeed, if we let

$$\begin{aligned} L(\gamma ) = \int _0^1 \Vert \gamma '(t)\Vert {\textrm{d}}t \end{aligned}$$

be the length of a piecewise $C^1$-smooth path $\gamma :[0,1]\rightarrow V$, then

$$\begin{aligned} \Vert x-y\Vert =\inf _\gamma L(\gamma ), \end{aligned}$$

where the infimum is taken over all $C^1$-smooth paths $\gamma :[0,1]\rightarrow V$ with $\gamma (0)=x$ and $\gamma (1)=y$. So, for normed spaces V the unit ball in the tangent space $T_bV$ is the same for all $b\in V$.

In this section we analyse the problem posed by Kapovich and Leeb [34, Question 6.18] concerning the existence of a natural homeomorphism between the horofunction compactification of a finite dimensional normed space V and the closed dual unit ball of V in the setting of Euclidean Jordan algebras equipped with the spectral norm. So we consider the Euclidean Jordan algebra not as inner-product space, but as an order-unit space, which makes it a finite dimensional (formally real) JB-algebra, see [4, Theorem 1.11]. We will give an explicit description of the horofunctions of these normed spaces and identify the parts and the detour distance. In our analysis we make frequent use of the theory of Jordan algebras and order-unit spaces. For the reader’s convenience we will recall some of the basic concepts. Throughout the paper we will follow the terminology used in [3, 4, 25].

4.1 Preliminaries

Order-unit spaces A cone $V_+$ in a real vector space V is a convex subset of V with $\lambda V_+\subseteq V_+$ for all $\lambda \ge 0$ and $V_+\cap -V_+=\{0\}$. The cone $V_+$ induces a partial ordering $\le$ on V by $x\le y$ if $y-x\in V_+$. We write $x<y$ if $x\le y$ and $x\ne y$. The cone $V_+$ is said to be Archimedean if for each $x\in V$ and $y\in V_+$ with $nx\le y$ for all $n\ge 1$ we have that $x\le 0$. An element u of $V_+$ is called an order-unit if for each $x\in V$ there exists $\lambda \ge 0$ such that $-\lambda u\le x\le \lambda u$. The triple $(V,V_+,u)$, where $V_+$ is an Archimedean cone and u is an order-unit, is called an order-unit space. An order-unit space admits a norm

$$\begin{aligned} \Vert x\Vert _u =\inf \{\lambda \ge 0:-\lambda u\le x\le \lambda u\}, \end{aligned}$$

which is called the order-unit norm, and we have that $-\Vert x\Vert _uu \le x\le \Vert x\Vert _uu$ for all $x\in V$. The cone $V_+$ is closed under the order-unit norm and $u\in {\textrm{int}}\, V_+$.

A linear functional $\varphi$ on an order-unit space is said to be positive if $\varphi (x) \ge 0$ for all $x\in V_+$. It is called a state if it is positive and $\varphi (u) =1$. The set of all states is denoted by S(V) and is called the state space, which is a convex set. In our case, the order-unit space is finite dimensional, hence S(V) is compact. The extreme points of S(V) are called the pure states.

The dual space $V^*$ of an order-unit space V is a base norm space, see [3, Theorem 1.19]. More specifically, $V^*$ is an ordered normed vector space with cone $V^*_+=\{\varphi \in V^*:\varphi \text{ is } \text{ positive }\}$, $V^*_+-V^*_+ = V^*$, and the unit ball of the norm of $V^*$ is given by

$$\begin{aligned} B_1^* = {\textrm{conv}} (S(V)\cup -S(V)). \end{aligned}$$

Jordan algebras Important examples of order-unit spaces come from Jordan algebras. A Jordan algebra (over ${\mathbb {R}}$) is a real vector space V equipped with a commutative bilinear product $\bullet$ that satisfies the identity

$$\begin{aligned} x^2\bullet (y\bullet x) = (x^2\bullet y)\bullet x\quad \text{ for } \text{ all } x,y\in V. \end{aligned}$$

A basic example is the space ${\textrm{Herm}}(n, {\mathbb {C}})$ consisting of $n\times n$ Hermitian matrices with Jordan product $A\bullet B = (AB+BA)/2$.

Throughout the paper we will assume that V has a unit, denoted u. For $x\in V$ we let $L_x$ be the linear map on V given by $L_xy =x\bullet y$. A finite dimensional Jordan algebra is said to be Euclidean if there exists an inner-product $(\cdot |\cdot )$ on V such that

$$\begin{aligned} (L_xy| z) = (y| L_xz)\quad \text{ for } \text{ all } x,y,z\in V. \end{aligned}$$

A Euclidean Jordan algebra has a cone $V_+=\{x^2:x\in V\}$. The interior of $V_+$ is a symmetric cone, i.e., it is self-dual and ${\textrm{Aut}}(V_+) =\{A\in {\textrm{GL}}(V):A(V_+) = V_+\}$ acts transitively on the interior of $V_+$. In fact, the Euclidean Jordan algebras are in one-to-one correspondence with the symmetric cones by the Koecher-Vinberg theorem, see for example [25].

The algebraic unit u of a Euclidean Jordan algebra is an order-unit for the cone $V_+$, so the triple $(V,V_+,u)$ is an order-unit space. We will consider the Euclidean Jordan algebras as an order-unit space equipped with the order-unit norm. These are precisely the finite dimensional formally real JB-algebras, see [4, Theorem 1.11]. In the analysis, however, the inner-product structure on V will be exploited to identify $V^*$ with V.

Throughout we will fix the rank of the Euclidean Jordan algebra V to be r. In a Euclidean Jordan algebra each x can be written in a unique way as $x= x^+-x^-$, where $x^+$ and $x^-$ are orthogonal element $x^+$ and $x^-$ in $V_+$, see [4, Proposition 1.28]. This is called the orthogonal decomposition of x.

Given x in a Euclidean Jordan algebra V, the spectrum of x is given by $\sigma (x)=\{ \lambda \in {\mathbb {R}}:\lambda u -x \text{ is } \text{ not } \text{ invertible }\}$, and we have that $V_+ =\{x\in V:\sigma (x)\subset [0,\infty )\}$. We write $\Lambda (x) =\inf \{\lambda :x\le \lambda u\}$ and note that $\Lambda (x)=\max \{\lambda :\lambda \in \sigma (x)\}$, so that

$$\begin{aligned} \Vert x\Vert _u= \max \{\Lambda (x),\Lambda (-x)\} = \max \{|\lambda |:\lambda \in \sigma (x)\} \end{aligned}$$

for all $x\in V$. We also note that

$$\begin{aligned} \Lambda (x+\mu u) = \Lambda (x)+\mu \end{aligned}$$

for all $x\in V$ and $\mu \in {\mathbb {R}}$. Moreover, if $x\le y$, then $\Lambda (x)\le \Lambda (y)$.

Recall that $p\in V$ is an idempotent if $p^2=p$. If, in addition, p is non-zero and cannot be written as the sum of two non-zero idempotents, then it is said to be a primitive idempotent. The set of all primitive idempotent is denoted ${\mathcal {J}}_1(V)$ and is known to be a compact set [30]. Two idempotents p and q are said to be orthogonal if $p\bullet q=0$, which is equivalent to $(p|q)=0$. According to the spectral theorem [25, Theorem III.1.2], each x has a spectral decomposition, $x = \sum _{i=1}^r \lambda _i p_i$, where each $p_i$ is a primitive idempotent, the $\lambda _i$’s are the eigenvalues of x (including multiplicities), and $p_1,\ldots ,p_r$ is a Jordan frame, i.e., the $p_i$’s are mutually orthogonal and $p_1+\cdots +p_r =u$.

Throughout the paper we will fix the inner-product on V to be

$$\begin{aligned} (x|y) = {\textrm{tr}}(x\bullet y), \end{aligned}$$

where ${\textrm{tr}}(x) = \sum _{i=1}^r \lambda _i$ and $x = \sum _{i=1}^r \lambda _ip_i$ is the spectral decomposition of x.

For $x\in V$ we denote the quadratic representation by $U_x:V\rightarrow V$, which is the linear map,

$$\begin{aligned} U_x y = 2 x\bullet (x\bullet y) - x^2\bullet y = 2L_x(L_x y) - L_{x^2} y. \end{aligned}$$

In case of a Euclidean Jordan algebra $U_x$ is self-adjoint, i.e. $(U_xy| z) = (y| U_xz)$.

We identify $V^*$ with V using the inner-product. So, $S(V) =\{w\in V_+:(u|w)=1\}$, which is a compact convex set, as V is finite dimensional. Moreover, the extreme points of S(V) are the primitive idempotents, see [25, Proposition IV.3.2]. The dual space $(V,\Vert \cdot \Vert _u^*)$ is a base norm space with norm,

$$\begin{aligned} \Vert z\Vert _u^* =\sup \{(x|z):x\in V \text{ with } \Vert x\Vert _u=1\}. \end{aligned}$$

If V is a Euclidean Jordan algebra, it is known that the (closed) boundary faces of the dual ball $B_1^*= {\textrm{conv}} (S(V)\cup -S(V))$ are precisely the sets of the form,

$$\begin{aligned} {\textrm{conv}}\, ( (U_p(V)\cap S(V))\cup (U_q(V)\cap -S(V))), \end{aligned}$$

(4.1)

where p and q are orthogonal idempotents not both zero, see [18, Theorem 4.4].

4.2 Summary of Results

To conveniently describe the horofunction compactification ${\overline{V}}^h$ of $(V,\Vert \cdot \Vert _u)$, where V is a Euclidean Jordan algebra, we need some additional notation. Throughout this section we will fix the basepoint $b\in V$ to be 0.

Let $p_1,\ldots , p_r$ be a Jordan frame in V. Given $I\subseteq \{1,\ldots , r\}$ nonempty, we write $p_I = \sum _{i\in I} p_i$ and we let $V(p_I) = U_{p_I}(V)$. For convenience we set $p_\emptyset =0$, so $V(p_\emptyset ) =U_0(V)=\{0\}$.

Recall that $V(p_I)$ is the Peirce 1-space of the idempotent $p_I$:

$$\begin{aligned} V(p_I)=\{x\in V:p_I\bullet x =x\}, \end{aligned}$$

which is a subalgebra, see [25, Theorem IV.1.1]. Given $z\in V(p_I)$, we write $\Lambda _{V(p_I)}(z)$ to denote the maximal eigenvalue of z in the subalgebra $V(p_I)$.

The following theorem characterises the horofunctions in ${\overline{V}}^h$.

Theorem 4.1

Let $p_1,\ldots ,p_r$ be a Jordan frame, $I,J\subseteq \{1,\ldots ,r\}$, with $I\cap J=\emptyset$ and $I\cup J$ nonempty, and $\alpha \in {\mathbb {R}}^{I\cup J}$ such that $\min \{\alpha _i :i\in I\cup J\}=0$. The function $h:V\rightarrow {\mathbb {R}}$, given by

$$\begin{aligned} h(x) = \max \left\{ \Lambda _{V(p_I)}\left( -U_{p_I}x - \sum _{i\in I} \alpha _ip_i\right) , \Lambda _{V(p_J)}\left( U_{p_J}x - \sum _{j\in J} \alpha _j p_j\right) \right\} \quad \text{ for } x\in V,\nonumber \\ \end{aligned}$$

(4.2)

is a horofunction, where we use the convention that if I or J is empty, the corresponding term is omitted from the maximum. Each horofunction in ${\overline{V}}^h$ is of the form (4.2) and a Busemann point.

To conveniently describe the parts and the detour distance (2.2) we introduce the following notation. Given orthogonal idempotents $p_I$ and $p_J$ we let $V(p_I,p_J)=V(p_I)+ V(p_J)$, which is a subalgebra of V with unit $p_{IJ}=p_I+p_J$. The subspace $V(p_I,p_J)$ can be equipped with the variation norm,

$$\begin{aligned} \Vert x\Vert _{{\textrm{var}}} = \Lambda _{V(p_I,p_J)}(x) + \Lambda _{V(p_I,p_J)}(-x)={\textrm{diam}}\, \sigma _{V(p_I,p_J)}(x), \end{aligned}$$

which is a semi-norm on $V(p_I,p_J)$. The variation norm is, however, a norm on the quotient space $V(p_I,p_J)/{\mathbb {R}}p_{IJ}$.

Theorem 4.2

Given horofunctions h and $h'$, where

$$\begin{aligned} h(x) = \max \left\{ \Lambda _{V(p_I)}\left( -U_{p_I}x - \sum _{i\in I} \alpha _ip_i\right) , \Lambda _{V(p_J)}\left( U_{p_J}x - \sum _{j\in J} \alpha _j p_j\right) \right\} \nonumber \\ \end{aligned}$$

(4.3)

and

$$\begin{aligned} h'(x) = \max \left\{ \Lambda _{V(q_{I'})}\left( -U_{q_{I'}}x - \sum _{i\in I'} \beta _iq_i\right) , \Lambda _{V(q_{J'})}\left( U_{q_{J'}}x - \sum _{j\in J'} \beta _j q_j\right) \right\} ,\nonumber \\ \end{aligned}$$

(4.4)

we have that

• (i) h and $h'$ are in the same part if and only if $p_I=q_{I'}$ and $p_J = q_{J'}$.

• (ii) If h and $h'$ are in the same part, then $\delta (h,h') = \Vert a -b\Vert _{{\textrm{var}}}$, where $a= \sum _{i\in I} \alpha _ip_i + \sum _{j\in J} \alpha _j p_j$ and $b = \sum _{i\in I'} \beta _iq_i + \sum _{j\in J'} \beta _j q_j$ in $V(p_I,p_J)$.

• (iii) The part $({\mathcal {P}}_h,\delta )$ is isometric to $(V(p_I,p_J)/{\mathbb {R}}p_{IJ},\Vert \cdot \Vert _{{\textrm{var}}})$.

Remark 4.3

A basic example is $({\mathbb {R}}^n,\Vert \cdot \Vert _\infty )$, where $\Vert z\Vert _\infty = \max _i |z_i|$, which is an associative Euclidean Jordan algebra. In that case every horofunction is a Busemann points and of the form

$$\begin{aligned} h(x) = \max \{ \max _{i\in I} ( -x_i -\alpha _i),\max _{j\in J} (x_j -\alpha _i)\}, \end{aligned}$$

where $I,J\subseteq \{1,\ldots ,n\}$ are disjoint, $I\cup J$ is nonempty and $\alpha \in {\mathbb {R}}^{I\cup J}$ with $\min _{k\in I\cup J}\alpha _k =0$, (see [22, Theorem 5.2] and [40]). Moreover, $({\mathcal {P}}_h,\delta )$ is isometric to $({\mathbb {R}}^{I\cup J}/{\mathbb {R}}{\textbf{1}},\Vert \cdot \Vert _{{\textrm{var}}})$, where ${\textbf{1}}=(1,\ldots ,1)\in {\mathbb {R}}^{I\cup J}$.

We will show that the following map is a homeomorphism from ${\overline{V}}^h$ onto $B^*_1$. Let $\varphi :{\overline{V}}^h\rightarrow B^*_1$ be given by

$$\begin{aligned} \varphi (x) = \frac{e^x - e^{-x}}{(e^x+e^{-x}|u)} = \frac{1}{\sum _{i=1}^r e^{\lambda _i} +e^{-\lambda _i}}\left( \sum _{i=1}^r (e^{\lambda _i} -e^{-\lambda _i})p_i\right) \end{aligned}$$

(4.5)

for $x = \sum _{i=1}^r \lambda _ip_i\in V$, and

$$\begin{aligned} \varphi (h) = \frac{1}{\sum _{i\in I} e^{-\alpha _i} +\sum _{j\in J} e^{-\alpha _j}}\left( \sum _{i\in I} e^{-\alpha _i}p_i -\sum _{j\in J} e^{-\alpha _j}p_j\right) \end{aligned}$$

(4.6)

for $h\in \partial {\overline{V}}^h$ given by (4.2).

We should note that $\varphi$ is well defined. To verify this assume that the horofunction h given by (4.2) is represented as

$$\begin{aligned} h(x) = \max \left\{ \Lambda _{V(q_{I'})}\left( -U_{q_{I'}}x - \sum _{i\in I'} \beta _iq_i\right) , \Lambda _{V(q_{J'})}\left( U_{q_{J'}}x - \sum _{j\in J'} \beta _j q_j\right) \right\} \end{aligned}$$

for $x\in V$. Then it follows from Theorem 4.2 that $p_I=q_{I'}$ and $p_J = q_{J'}$. Moreover, as $\delta (h,h) = 0$, we have that $a= \sum _{i\in I} \alpha _ip_i + \sum _{j\in J} \alpha _j p_j= \sum _{i\in I'} \beta _iq_i + \sum _{j\in J'} \beta _j q_j=b$, as $\min \{\alpha _i :I\cup J\} = 0 =\min \{\beta _i:I\cup J\}$. This implies that $U_{p_I} a = U_{q_{I'}}b$ and $U_{p_J} a = U_{q_{J'}}b$, so that

$$\begin{aligned} \sum _{i\in I} \alpha _ip_i = \sum _{i\in I'} \beta _iq_i \quad \text{ and } \quad \sum _{j\in J} \alpha _j p_j = \sum _{j\in J'} \beta _j q_j. \end{aligned}$$

Using the map $v\in V\mapsto e^{-v}$ we deduce that $\sum _{i\in I} e^{-\alpha _i}p_i +(u-p_I) = \sum _{i\in I'} e^{-\beta _i}q_i +(u-q_{I'})$, and hence $\sum _{i\in I} e^{-\alpha _i}p_i = \sum _{i\in I'}e^{-\beta _i}q_i$. Likewise $\sum _{j\in J} e^{-\alpha _j}p_j = \sum _{j\in J'}e^{-\beta _j}q_j$. We also find that

so $\varphi (h)$ is well defined.

We will also show that $\varphi$ maps each part of the horofunction boundary onto the relative interior of a boundary face of the dual unit ball. Recall that the relative interior of a face F of $B^*_1$ is the interior of F as a subset of the affine span of F.

Theorem 4.4

Given a Euclidean Jordan algebra $(V,\Vert \cdot \Vert _u)$, the map $\varphi :{\overline{V}}^h\rightarrow B^*_1$ is a homeomorphism. Moreover, the part ${\mathcal {P}}_h$, with h given by (4.2), is mapped onto the relative interior of the closed boundary face

$$\begin{aligned} {\textrm{conv}}\, ( U_{p_I}(V)\cap S(V))\cup (U_{p_J}(V)\cap -S(V))). \end{aligned}$$

4.3 Horofunctions

In this subsection we will prove Theorem 4.1. We first make some preliminary observations. Note that $x\le \lambda u$ if and only if $0\le \lambda u -x$, which by the Hahn–Banach separation theorem is equivalent to $(\lambda u -x|w)\ge 0$ for all $w\in S(V)$. As the state space is compact, we have for each $x\in V$ that

$$\begin{aligned} \Lambda (x) = \max _{w\in S(V)} (x|w). \end{aligned}$$

(4.7)

As $\Vert \cdot \Vert _u$ is the JB-algebra norm, $\Vert x\bullet y\Vert _u \le \Vert x\Vert _u\Vert y\Vert _u$, see [4, Theorem 1.11]. It follows that if $x^n\rightarrow x$ and $y^n\rightarrow y$ in $(V,\Vert \cdot \Vert _u)$, then $x^n\bullet y^n\rightarrow x\bullet y$. Thus, we have the following lemma.

Lemma 4.5

If $x^n\rightarrow x$ and $y^n\rightarrow y$ in $(V,\Vert \cdot \Vert _u)$, then $U_{x^n} y^n \rightarrow U_x y$.

We will also use the following technical lemma several times.

Lemma 4.6

For $n\ge 1$, let $p^n_1,\ldots ,p^n_r$ be a Jordan frame in V and $I\subseteq \{1,\ldots ,r\}$ nonempty. Suppose that

• (i) $p_i^n\rightarrow p_i$ for all $i\in I$.

• (ii) $x^n \in V(p_I^n)$ with $x^n\rightarrow x\in V(p_I)$.

• (iii) $\beta _i^n\ge 0$ with $\beta _i^n\rightarrow \beta _i\in [0,\infty ]$ for all $i\in I$.

If $I'=\{i\in I:\beta _i<\infty \}$ is nonempty, then

$$\begin{aligned} \lim _{n\rightarrow \infty }\Lambda _{V(p_I^n)}\left( x^n - \sum _{i\in I}\beta _i^np_i^n\right) =\Lambda _{V(p_{I'})}\left( U_{p_{I'}}x-\sum _{i\in I'}\beta _ip_i\right) . \end{aligned}$$

Proof

We will show that every subsequence of $(\Lambda _{V(p_I^n)}(x^n -\sum _{i\in I}\beta _i^np_i^n))$ has a convergent subsequence with limit $\Lambda _{V(p_{I'})}(U_{p_{I'}}x-\sum _{i\in I'}\beta _ip_i)$. So let $(\Lambda _{V(p_I^{n_k})}(x^{n_k} - \sum _{i\in I}\beta _i^{n_k}p_i^{n_k}))$ be a subsequence. By (4.7) there exists $d^{n_k}\in S(V(p_I^{n_k}))$ with

By taking subsequences we may assume that $d^{n_k}\rightarrow d\in S(V(p_I))$.

Using the Peirce decomposition with respect to the Jordan frame $p^{n_k}_i$, $i\in I$, in $V(p_I^{n_k})$, we can write

$$\begin{aligned} d^{n_k} =\sum _{i\in I} \mu _i^{n_k} p^{n_k}_i +\sum _{i<j\in I} d_{ij}^{n_k}. \end{aligned}$$

Note that as $d^{n_k}\ge 0$, we have that $\mu _i^{n_k} =(d^{n_k}|p^{n_k}_i) \ge 0$ for all $i\in I$.

We claim that for each $i\in I\setminus I'$ we have that $\mu _i^{n_k}\rightarrow 0$. Indeed, as $I'$ is nonempty, there exist $l\in I'$ and a constant $C>0$ such that

for all k, since $(x^{n_k}|p^{n_k}_l) \le \Vert x^{n_k}\Vert _u$. Moreover,

$$\begin{aligned} \left( x^{n_k} -\sum _{i\in I} \beta _i^{n_k}p_i^{n_k}\bigg {|}d^{n_k}\right)= & {} \left( x^{n_k}|d^{m_k}\right) -\sum _{i\in I} \beta _i^{n_k}\mu _i^{n_k} \\ {}\le & {} \Vert x^{n_k}\Vert _u - \sum _{i\in I'} \beta _i^{n_k}\mu _i^{n_k}- \sum _{i\in I\setminus I'} \beta _i^{n_k}\mu _i^{n_k}. \end{aligned}$$

As $\beta _i^{n_k},\mu _i^{n_k}\ge 0$ for all $i\in I$ and $\beta _i^{n_k}\rightarrow \infty$ for all $i\in I{\setminus } I'$, we conclude from the previous two inequalities that $\mu _i^{n_k}\rightarrow 0$ for all $i\in I\setminus I'$.

Using the Peirce decomposition with respect to the Jordan frame $p_i$, $i\in I$, we write

$$\begin{aligned} d = \sum _{i\in I}\mu _ip_i +\sum _{i<j\in I} d_{ij}. \end{aligned}$$

We now show that

$$\begin{aligned} d=\sum _{i\in I'} \mu _{i} p_i +\sum _{i<j\in I'} d_{ij}, \end{aligned}$$

(4.8)

and hence $d\in V(p_{I'})$. Note that

$$\begin{aligned} \mu _i -\mu _i^{n_k}=(d|p_i)-(d^{n_k}|p^{n_k}_i) = (d-d^{n_k}|p_i)+(d^{n_k}|p_i-p_i^{n_k})\rightarrow 0. \end{aligned}$$

We conclude that $\mu _i^{n_k} \rightarrow \mu _i$ for all $i\in I$, and hence $(d|p_j)=\mu _j=0$ for all $j\in I{\setminus } I'$. This implies by [25, III, Exercise 3] that $d\bullet p_j=0$ for all $j\in I{\setminus } I'$. So,

$$\begin{aligned} 0 =d\bullet p_j = \frac{1}{2}\left( \sum _{l<j} d_{lj} + \sum _{j<m} d_{jm}\right) , \end{aligned}$$

which shows that $d_{lj}=0 =d_{jm}$ for all $l<j<m$, as they are all orthogonal. This gives (4.8).

Next we show that $\lim _{k\rightarrow \infty } \Lambda _{V(p_I^{n_k})}(x^{n_k} - \sum _{i\in I}\beta _i^{n_k}p_i^{n_k}) = (U_{p_{I'}}x - \sum _{i\in I'}\beta _ip_i|d)$. First note that

as $\beta _i^{n_k},\mu _i^{n_k}\ge 0$ for all i and k. This implies that

As $U_{p_{I'}}d =d$ and $U_{p_{I'}}$ is self-adjoint, we find that

so that

(4.9)

Now let $p^{n_k}_{I'} = \sum _{i\in I'} p_i^{n_k}$. As $p^{n_k}_{I'}\rightarrow p_{I'}$, it follows from Lemma 4.5 that $U_{p^{n_k}_{I'}}d\rightarrow U_{p_{I'}}d = d$. This implies that

for all k large, as $(U_{p^{n_k}_{I'}}d|p^{n_k}_{I})\rightarrow (U_{p_{I'}}d|p_{I})= (d|U_{p_{I'}}p_{I}) =(d|p_{I'}) =(d|p_{I})=1$. Moreover,

This shows that $(U_{p_{I'}}x -\sum _{i\in I'} \beta _ip_i|d)\le \liminf _{k\rightarrow \infty } \Lambda _{V(p_I^{n_k})}(x^{n_k} - \sum _{i\in I}\beta _i^{n_k}p_i^{n_k})$. From (4.9) we conclude that

(4.10)

To complete the proof we show that

(4.11)

As $(d|p_{I'})=(d|p_{I})=1$, we know that $d\in S(V_{p_{I'}})$. So, we get from (4.7) that

On the other hand, if $w\in S(V(p_{I'}))$ is such that

then by definition of $d^{n_k}$ we get for all k large that

as $(U_{p^{n_k}_{I'}}w|p^{n_k}_I)\rightarrow (U_{p_{I'}}w|p_I) = (w|p_{I'})=1$. This implies that

and hence (4.11) holds by (4.10). $\square$

To prove that all horofunctions in ${\overline{V}}^h$ are of the form (4.2), we first establish the following proposition using the previous lemma.

Proposition 4.7

Let $(y^n)$ be a sequence in V, with $y^n =\sum _{i=1}^r \lambda _i^n p^n_i$. Suppose that $h_{y^n}\rightarrow h\in \partial {\overline{V}}^h$ and $(y^n)$ satisfies the following properties:

• (1) There exists $1\le s\le r$ such that $|\lambda ^n_s|=r^n$ for all n, where $r^n =\Vert y^n\Vert _u$.

• (2) $p_k^n\rightarrow p_k$ for all $1\le k\le r$.

• (3) There exist $I,J\subseteq \{1,\ldots ,r\}$ disjoint with $I\cup J$ nonempty, and $\alpha \in {\mathbb {R}}^{I\cup J}$ with $\min \{\alpha _i:i\in I\cup J\}=0$ such that $r^n - \lambda _i^n\rightarrow \alpha _i$ for all $i\in I$, $r^n+\lambda _j^n\rightarrow \alpha _j$ for all $j\in J$, and $r^n- |\lambda _k^n|\rightarrow \infty$ for all $k\not \in I\cup J$.

Then h satisfies (4.2).

Proof

Take $x\in V$ fixed. Note that for all $n\ge 1$,

$$\begin{aligned} \Vert x-y^n\Vert _u -\Vert y^n\Vert _u= & {} \max \{\Lambda (x -y^n),\Lambda (-x+y^n)\} -r^n \\= & {} \max \{ \Lambda (x -y^n -r^nu),\Lambda (-x+y^n -r^nu)\}. \end{aligned}$$

As h is a horofunction, $\Vert y^n\Vert _u = r^n\rightarrow \infty$ by Lemma 2.1. Thus, $\lambda ^n_i\rightarrow \infty$ for all $i\in I$ and $\lambda _j^n\rightarrow -\infty$ for all $j\in J$. Now suppose that J is nonempty. Then $r^n+\lambda _k^n\ge r^n-|\lambda _k^n|\rightarrow \infty$ for all $k\not \in J$. As

$$\begin{aligned} \Lambda (x -y^n -r^nu) = \Lambda (x - \sum _{j\in J} (r^n+\lambda _j^n)p^n_j - \sum _{k\not \in J} (r^n+\lambda _k^n)p^n_k), \end{aligned}$$

it follows that

$$\begin{aligned} \lim _{n\rightarrow \infty } \Lambda (x -y^n -r^nu) = \Lambda _{V(p_J)}(U_{p_J}x -\sum _{j\in J}\alpha _jp_j) \end{aligned}$$

by Lemma 4.6. Likewise, if I is nonempty, then

$$\begin{aligned} \lim _{n\rightarrow \infty } \Lambda (-x +y^n -r^nu) = \Lambda _{V(p_I)}(-U_{p_I}x -\sum _{i\in I}\alpha _ip_i) \end{aligned}$$

by Lemma 4.6. We conclude that if I and J are both nonempty, then

$$\begin{aligned} h(x)= & {} \lim _{n\rightarrow \infty } \Vert x-y^n\Vert _u -\Vert y^n\Vert _u\\= & {} \lim _{n\rightarrow \infty } \max \{\Lambda ( -x +y^n-r^nu),\Lambda ( x -y^n-r^nu) \} \\= & {} \max \{\Lambda _{V(p_I)}(-U_{p_I}x -\sum _{i\in I}\alpha _ip_i), \Lambda _{V(p_J)}(U_{p_J}x -\sum _{j\in J}\alpha _jp_j)\}. \end{aligned}$$

To complete the proof it remains to show that $\lim _{n\rightarrow \infty } \Vert x-y^n\Vert _u -\Vert y^n\Vert _u = \lim _{n\rightarrow \infty } \Lambda ( -x +y^n-r^nu)$ if J is empty, and $\lim _{n\rightarrow \infty } \Vert x-y^n\Vert _u -\Vert y^n\Vert _u = \lim _{n\rightarrow \infty } \Lambda ( x -y^n-r^nu)$ if I is empty. Suppose that I is empty, so J is nonempty. Then for each $i\in \{1,\ldots ,r\}$ we have that $r^n-\lambda _i^n\rightarrow \infty$. Note that

$$\begin{aligned} -x+y^n-r^nu= & {} -x -\sum _i (r^n-\lambda _i^n)p_i^n\le -x -\min _i (r^n-\lambda _i^n)u\\ {}\le & {} (\Vert x\Vert _u-\min _i (r^n-\lambda _i^n))u. \end{aligned}$$

Thus, $\Lambda (-x+y^n-r^nu) \le \Lambda ((\Vert x\Vert _u-\min _i (r^n-\lambda _i^n))u) = \Vert x\Vert _u-\min _i (r^n-\lambda _i^n)$ for all n, hence $\Lambda (-x+y^n-r^nu)\rightarrow -\infty$. As

$$\begin{aligned} \max \{\Lambda (x-y^n-r^nu),\Lambda (-x+y^n-r^nu)\} = \big \Vert x-y^n\big \Vert _u -\big \Vert y^n\big \Vert _u\ge -\Vert x\Vert _u, \end{aligned}$$

we conclude that $\Vert x-y^n\Vert _u -|y^n\Vert _u = \Lambda (x-y^n-r^nu)$ for all n sufficiently large, hence

$$\begin{aligned} h(x) = \lim _{n\rightarrow \infty } \Lambda (x-y^n-r^nu) = \Lambda _{V(p_J)}(U_{p_J}x -\sum _{j\in J}\alpha _jp_j). \end{aligned}$$

The argument for the case where J is empty goes in the same way. $\square$

The following corollary shows that each horofunction is of the form (4.2).

Corollary 4.8

If h is a horofunction in ${\overline{V}}^h$, then there exist a Jordan frame $p_1,\ldots ,p_r$ in V, disjoint subsets $I,J\subseteq \{1,\ldots ,r\}$, with $I\cup J$ nonempty, and $\alpha \in {\mathbb {R}}^{I\cup J}$ with $\min \{\alpha _i :i\in I\cup J\}=0$, such that $h:V\rightarrow {\mathbb {R}}$ satisfies (4.2) for all $x\in V$.

Proof

Suppose that $(y^n)$ is a sequence in V with $h_{y^n}\rightarrow h$ in ${\overline{V}}^h$. Then for each $x\in V$ we have that

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert x-y^n\Vert _u -\Vert y^n\Vert _u = h(x) \end{aligned}$$

and $\Vert y^n\Vert _u\rightarrow \infty$ by Lemma 2.1.

To show that the limit is equal to (4.2) it suffices to show that we can take a subsequences of $(y^n)$ that satisfies the conditions in Proposition 4.7. First we note that by the spectral theorem [25, Theorem III.1.2], there exist for each $n\ge 1$ a Jordan frame $p^n_1,\ldots ,p^n_r$ in V and $\lambda ^n_1,\ldots ,\lambda ^n_r\in {\mathbb {R}}$ such that

$$\begin{aligned} y^n = \lambda ^n_1p^n_1+\cdots +\lambda ^n_r p^n_r, \end{aligned}$$

where r is the rank of V. Denote $r^n=\Vert y^n\Vert _u = \max _i |\lambda ^n_i|$.

Now by taking subsequences we may assume that there exist $I_+\subseteq \{1,\ldots , r\}$ and $1\le s\le r$ such that for each $n\ge 1$ we have $r^n = |\lambda ^n_s|$ and

$$\begin{aligned} \lambda _i^n>0 \text{ for } \text{ all } i\in I_+ \quad \text{ and } \lambda ^n_i\le 0 \text{ for } \text{ all } i\not \in I_+. \end{aligned}$$

Now for each $i\in \{1,\ldots ,r\}$ and $n\ge 1$ define

$$\begin{aligned} \alpha _i^n = \left[ \begin{array}{cl} r^n - \lambda _i^n &{} \text{ for } i\in I_+\\ r^n +\lambda ^n_i &{} \text{ for } i\not \in I_+. \end{array}\right. \end{aligned}$$

Note that $\alpha ^n_i \in [0,\infty )$ for all i. Again by taking subsequences we may assume that $\alpha _i^n\rightarrow \alpha _i\in [0,\infty ]$ as $n\rightarrow \infty$, for all i. Recall that $\alpha ^n_s = 0$ for all n, so $\alpha _s=0$. Furthermore, we may assume that $p^n_i\rightarrow p_i$ in ${\mathcal {J}}_1(V)$ for all i, as it is a compact set [30]. Note that $p_1,\ldots ,p_r$ is a Jordan frame in V.

Now let

$$\begin{aligned} I =\{i:\alpha _i<\infty \text{ and } i\in I_+\}\quad \text{ and } \quad J=\{j:\alpha _j<\infty \text{ and } j\not \in I_+\}. \end{aligned}$$

So, $I\cap J$ is empty, $s\in I\cup J$ and $\min \{\alpha _i:i\in I\cup J\} =\alpha _s=0$. Then the subsequence of $(y^n)$ satisfies the conditions in Proposition 4.7, hence h is a horofunction of the form (4.2). $\square$

The next proposition shows that each function of the form (4.2) can be realised as a horofunction, and is a Busemann point.

Proposition 4.9

Let $p_1,\ldots ,p_r$ be a Jordan frame in V. Suppose that $I,J\subseteq \{1,\ldots ,r\}$ with $I\cap J=\emptyset$ and $I\cup J$ nonempty, and $\alpha \in {\mathbb {R}}^{I\cup J}$ with $\min \{\alpha _i :i\in I\cup J\}=0$. If for $n\ge 1$ we let $y^n = \lambda _1^np_1+\cdots +\lambda _r^n p_r$, where

$$\begin{aligned} \lambda _i^n =\left[ \begin{array}{ll} n -\alpha _i &{} \quad \text{ if }\; i\in I\\ -n+\alpha _i&{} \quad \text{ if }\;i\in J\\ 0&{} \text{ otherwise, } \end{array}\right. \end{aligned}$$

then $(y^n)$ is an almost geodesic sequence and $h_{y^n}\rightarrow h$, where h satisfies (4.2) for all $x\in V$. In particular, h is a Busemann point in ${\overline{V}}^h$.

Proof

Let $k\ge \max \{ \alpha _i:i\in I\cup J\}$ and note that for $n\ge k$ we have that $r^n = \Vert y^n\Vert _u= n$, as $\min \{\alpha _i :i\in I\cup J\}=0$. The sequence $(y^n)$, where $n\ge k$, satisfies the conditions in Proposition 4.7. Indeed, for $n\ge k$ we have that $r^n -\lambda _i^n = \alpha _i$ for all $i\in I$, $r^n+\lambda _i^n =\alpha _i$ for all $i\in J$, and $r^n -\lambda _i^n= n$ otherwise. Also for s with $\alpha _s = 0$, we have that $|\lambda _s^n| = n = \Vert y^n\Vert _u$.

Finally to see that $(h_{y^n})$ converges, we note that if we define $z = \sum _{i\in I} -\alpha _ip_i + \sum _{j\in J} \alpha _jp_j$ and $w= \sum _{i\in I} p_i -\sum _{j\in J}p_j$, then $y^n= nw+z$, which lies on the straight-line $t\mapsto tw+z$. Hence $(y^n)$ is an almost geodesic sequence, so

$$\begin{aligned} h(x) = \lim _{n\rightarrow \infty } \Vert x-y^n\Vert _u -\Vert y^n\Vert _u \end{aligned}$$

exists for all $x\in V$. Thus, we can apply Proposition 4.7 and conclude that h satisfies (4.2), and h is a Busemann point in the horofunction boundary. $\square$

Combining the results so far we now prove Theorem 4.1.

Proof of Theorem 4.1

Corollary 4.8 shows that each horofunction in ${\overline{V}}^h$ is of the form (4.2). It follows from Proposition 4.9 that any function of the form (4.2) is a horofunction and by the second part of that proposition each horofunction is a Busemann point. $\square$

4.4 Parts and the Detour Metric

In this subsection we will identify the parts in the horofunction boundary of ${\overline{V}}^h$, derive a formula for the detour distance (2.2), and establish Theorem 4.2. We begin by proving the following proposition.

Proposition 4.10

If

$$\begin{aligned} h(x) = \max \left\{ \Lambda _{V(p_{I})}\left( -U_{p_{I}}x - \sum _{i\in I} \alpha _ip_i\right) , \Lambda _{V{(p_{J})}}\left( U_{p_{J}}x - \sum _{j\in J} \alpha _j p_j\right) \right\} ,\nonumber \\ \end{aligned}$$

(4.12)

and

$$\begin{aligned} h'(x) = \max \left\{ \Lambda _{V(q_{I'})}\left( -U_{q_{I'}}x - \sum _{i\in I'} \beta _iq_i\right) , \Lambda _{V(q_{J'})}\left( U_{q_{J'}}x - \sum _{j\in J'} \beta _j q_j\right) \right\} \nonumber \\ \end{aligned}$$

(4.13)

are horofunctions with $p_{I} = q_{I'}$ and $p_{J}=q_{J'}$, then h and $h'$ are in the same part and

$$\begin{aligned} \delta (h,h') = \Vert a-b \Vert _{{\textrm{var}}} = \Lambda _{V(p_I,p_J)}(a-b) + \Lambda _{V(p_I,p_J)}(b-a), \end{aligned}$$

where $a =\sum _{i\in I} \alpha _ip_i + \sum _{j\in J} \alpha _jp_j$ and $b = \sum _{i\in I'} \beta _iq_i + \sum _{j\in J'} \beta _j q_j$ in $V(p_I,p_J)=V(p_I)+V(p_J)$.

Proof

As in Proposition 4.9, for $n\ge 1$ let $y^n = \lambda _1^np_1+\cdots +\lambda _r^n p_r$, where

$$\begin{aligned} \lambda _i^n =\left[ \begin{array}{rl} n -\alpha _i &{} \text{ if } i\in I\\ -n+\alpha _i &{} \text{ if } i\in J\\ 0&{} \text{ otherwise, } \end{array}\right. \end{aligned}$$

and let $w^n = \mu _1^nq_1+\cdots +\mu _r^n q_r$, where

$$\begin{aligned} \mu _i^n =\left[ \begin{array}{rl} n -\beta _i &{} \text{ if } i\in I'\\ -n+\beta _i &{} \text{ if } i\in J'\\ 0&{} \text{ otherwise. } \end{array}\right. \end{aligned}$$

By Proposition 4.9 we know that $(y^n)$ and $(w^n)$ are almost geodesic sequences with $h_{y^n}\rightarrow h$ and $h_{w^n}\rightarrow h'$. Note that

$$\begin{aligned} U_{p_I} w^m= U_{q_{I'}} w^m = \sum _{i\in I'}\mu _i^m U_{q_{I'}}q_i = \sum _{i\in I'}\mu _i^m q_i \end{aligned}$$

for all m, so

$$\begin{aligned}{} & {} \Lambda _{V\left( p_I\right) }\left( -U_{p_I}w^m -\sum _{i\in I}\alpha _ip_i +\Vert w^m\Vert _u p_I\right) \\ {}{} & {} \quad =\Lambda _{V\left( p_I\right) }\left( -U_{q_{I'}}w^m -\sum _{i\in I} \alpha _ip_i +\Vert w^m\Vert _uq_{I'}\right) \\ {}{} & {} \quad =\Lambda _{V\left( p_I\right) } \left( \sum _{i\in I'} \left( \Vert w^m\Vert _u -\mu _i^m\right) q_i -\sum _{i\in I} \alpha _ip_i \right) . \end{aligned}$$

Thus,

$$\begin{aligned}{} & {} \lim _{m\rightarrow \infty } \Lambda _{V\left( p_I\right) }\left( -U_{p_I}w^m -\sum _{i\in I}\alpha _ip_i +\Vert w^m\Vert _u p_I\right) \\{} & {} \quad = \lim _{m\rightarrow \infty } \Lambda _{V\left( p_I\right) } \left( \sum _{i\in I'} \left( \Vert w^m\Vert _u -\mu _i^m\right) q_i -\sum _{i\in I} \alpha _ip_i \right) \\{} & {} \quad = \Lambda _{V\left( p_I\right) } \left( \sum _{i\in I'} \beta _iq_i -\sum _{i\in I} \alpha _ip_i \right) \\{} & {} \quad = \Lambda _{V\left( p_I\right) }\left( b-a\right) . \end{aligned}$$

In the same way it can be shown that

$$\begin{aligned}{} & {} \lim _{m\rightarrow \infty } \Lambda _{V\left( p_J\right) }\left( U_{p_J}w^m -\sum _{j\in J}\alpha _j p_i +\Vert w^m\Vert _u p_J\right) \\{} & {} \quad = \Lambda _{V\left( p_J\right) } \left( \sum _{j\in J'} \beta _jq_j -\sum _{j\in J} \alpha _jp_j\right) = \Lambda _{V\left( p_J\right) }\left( b-a\right) . \end{aligned}$$

So, it follows from (2.3) that

$$\begin{aligned} H\left( h,h'\right)= & {} \lim _{m\rightarrow \infty } \Vert w^m\Vert _u \\{} & {} + \max \left\{ \Lambda _{V\left( p_I\right) }\left( -U_{p_I}w^m - \sum _{i\in I} \alpha _ip_i\right) , \Lambda _{V\left( p_J\right) }\left( U_{p_J}w^m - \sum _{j\in J} \alpha _j p_j\right) \right\} \\= & {} \lim _{m\rightarrow \infty } \max \left\{ \Lambda _{V\left( p_I\right) }\left( -U_{p_I}w^m - \sum _{i\in I} \alpha _ip_i + \Vert w^m\Vert _u p_I\right) , \Lambda _{V\left( p_J\right) }\left( U_{p_J}w^m \right. \right. \\{} & {} \left. \left. - \sum _{j\in J} \alpha _j p_j + \Vert w^m\Vert _u p_J\right) \right\} \\= & {} \max \left\{ \Lambda _{V\left( p_I\right) } \left( \sum _{i\in I'} \beta _iq_i -\sum _{i\in I} \alpha _ip_i \right) , \Lambda _{V\left( p_J\right) } \left( \sum _{j\in J'} \beta _jq_j -\sum _{j\in J} \alpha _jp_j\right) \right\} \\= & {} \Lambda _{V\left( p_I,p_J\right) }\left( b-a\right) . \end{aligned}$$

Interchanging the roles of h and $h'$ gives $H(h',h) = \Lambda _{V(p_I,p_J)}(a-b)$, hence $\delta (h,h') = \Vert a-b\Vert _{{\textrm{var}}}$. $\square$

To show that h and $h'$ are in different part if $p_I\ne q_{I'}$ or $p_J\ne q_{J'}$, we need the following lemma.

Lemma 4.11

If p and q are idempotents in V with $p\nleq q$, then $U_pq<p$.

Proof

We have that $U_pq\le U_p u = p$. In fact, $U_p q < p$. Indeed, if $U_p q= p$, then

$$\begin{aligned} p = U_p u = U_p (u-q) +U_p q = U_p( u-q) +p, \end{aligned}$$

and hence $U_p (u-q) =0$. This implies that $p +(u-q) \le u$ by [29, Lemma 4.2.2], so that $p\le q$. This is impossible, as $p\nleq q$, and hence $U_p q < p$. $\square$

Proposition 4.12

If h and $h'$ are horofunctions given by (4.12) and (4.13), respectively, and $p_{I} \ne q_{I'}$ or $p_{J}\ne q_{J'}$, then

$$\begin{aligned} \delta (h,h') = \infty . \end{aligned}$$

Proof

Suppose that $p_I\ne q_{I'}$. Then $p_I\nleq q_{I'}$ or $q_{I'}\nleq p_I$. Without loss of generality assume that $p_I\nleq q_{I'}$. Let $(y^n)$ in $V(p_I)$ and $(w^n)$ in $V(q_{I'})$ be as in Proposition 4.9, so $h_{y^n}\rightarrow h$ and $h_{w^m}\rightarrow h'$. To prove the statement in this case, we use (2.3) and show that

$$\begin{aligned} H(h',h) = \lim _{m\rightarrow \infty } \Vert w^m\Vert _u + h(w^m) = \infty . \end{aligned}$$

(4.14)

Note that

$$\begin{aligned} \Vert w^m\Vert _u +h\left( w^m\right)\ge & {} \Vert w^m\Vert _u + \Lambda _{V\left( p_I\right) }\left( -U_{p_I}w^m -\sum _{i\in I} \alpha _i p_i \right) \\= & {} \Lambda _{V\left( p_I\right) }\left( -U_{p_I}w^m -\sum _{i\in I} \alpha _i p_i +\Vert w^m\Vert _u p_I\right) . \end{aligned}$$

As $w^m\le \Vert w^m\Vert _u q_{I'}$ for all m, we have that $U_{p_I}w^m\le \Vert w^m\Vert _uU_{p_I}q_{I'}$ for all m. Thus,

$$\begin{aligned} -U_{p_I}w^m -\sum _{i\in I} \alpha _i p_i +\Vert w^m\Vert _u p_I\ge & {} -\Vert w^m\Vert _uU_{p_I}q_{I'} -\sum _{i\in I} \alpha _i p_i +\Vert w^m\Vert _u p_I\\= & {} \Vert w^m\Vert _u (p_I -U_{p_I}q_{I'}) - \sum _{i\in I} \alpha _i p_i \end{aligned}$$

for all m.

We know from Lemma 4.11 that $p_I-U_{p_I}q_{I'}>0$. As $p_I-U_{p_I}q_{I'}\in V(p_I)$ we also have that $p_I-U_{p_I}q_{I'} = \sum _{j=1}^s \gamma _j r_j$, where $\gamma _j>0$ for all j and the $r_j$’s are orthogonal idempotents in $V(p_I)$. It now follows that for all m,

The right-hand side goes to $\infty$ as $m\rightarrow \infty$, and hence (4.14) holds.

For the case $p_J\ne q_{J'}$ a similar argument can be used. $\square$

We now prove Theorem 4.2.

Proof

Parts (i) and (ii) follow directly from Propositions 4.10 and 4.12. Clearly the map $\rho :{\mathcal {P}}_h\rightarrow V(p_I,p_J)/{\mathbb {R}}p_{IJ}$ given by $\rho (h') = [b]$, where

$$\begin{aligned} h'(x) = \max \left\{ \Lambda _{V(q_{I'})}\left( -U_{q_{I'}}x - \sum _{i\in I'} \beta _iq_i\right) , \Lambda _{V(q_{J'})}\left( U_{q_{J'}}x - \sum _{j\in J'} \beta _j q_j\right) \right\} , \end{aligned}$$

and $b = \sum _{i\in I'} \beta _iq_i + \sum _{j\in J'} \beta _j q_j\in V(p_I,p_J)$ with $\min _{i\in I\cup J} \beta _i=0$, is a bijection. So, by Proposition 4.10, $\rho$ is an isometry from $({\mathcal {P}}_h,\delta )$ onto $(V(p_I,p_J)/{\mathbb {R}}p_{IJ},\Vert \cdot \Vert _{{\textrm{var}}} )$. $\square$

4.5 The Homeomorphism onto the Dual Unit Ball

In this subsection we prove Theorem 4.4. To start we prove a lemma that will be useful in the sequel.

Lemma 4.13

If $q\le p$ are idempotents in V and $z\in V(p)$, then $\Lambda _{V(q)}(U_qz)\le \Lambda _{V(p)}(z)$.

Proof

If $\lambda = \Lambda _{V(p)}(z)$, then $0\le \lambda p-z$, so that $0\le \lambda U_qp -U_qz$. As $q= U_q q \le U_qp\le U_qu =q^2=q$, we find that $0\le \lambda U_qp -U_qz =\lambda q -U_qz$, hence $\Lambda _{V(q)}(U_qz)\le \lambda$. $\square$

We will show that $\varphi$ given by (4.5) and (4.6) is a continuous bijection from ${\overline{V}}^h$ onto $B_1^*$. As ${\overline{V}}^h$ is compact and $B_1^*$ is Hausdorff, we can then conclude that $\varphi$ is a homeomorphism. We begin by showing that $\varphi$ maps V into the interior of $B_1^*$.

Lemma 4.14

For each $x\in V$ we have that $\varphi (x) \in {\textrm{int}}\,B^*_1$.

Proof

For $x\in V$ there exists $y\in V$ with $\Vert y\Vert _u= 1$, such that

$$\begin{aligned} \Vert \varphi (x)\Vert _u^* = \sup _{w\in V:\Vert w\Vert _u \le 1} |(w|\varphi (x))| = (y|\varphi (x)), \end{aligned}$$

where $(v|w) ={\textrm{tr}}(v\bullet w)$. So, if x has spectral decomposition $x=\sum _{i=1}^r \lambda _ip_i$, then we can consider the Peirce decomposition of y,

$$\begin{aligned} y = \sum _{i=1}^r \mu _i p_i+\sum _{i<j} y_{ij}, \end{aligned}$$

to find that

as $\mu _i =(y|p_i)\le (u|p_i) =1$ and $\mu _i =(y|p_i)\ge (-u|p_i) =-1$. $\square$

Lemma 4.15

The map $\varphi$ is injective on V.

Proof

Suppose that $x,y\in V$ with $x=\sum _{i=1}^r \sigma _i p_i$ and $y=\sum _{i=1}^r \tau _i q_i$, where $\sigma _1\le \ldots \le \sigma _r$ and $\tau _1\le \ldots \le \tau _r$, satisfy $\varphi (x) =\varphi (y)$. Then $\varphi (x) =\sum _{i=1}^r \alpha _i p_i =\sum _{i=1}^r \beta _i q_i=\varphi (y)$. where

$$\begin{aligned} \alpha _j =\frac{e^{\sigma _j}-e^{-\sigma _j}}{\sum _{i=1}^r e^{\sigma _i}+e^{-\sigma _i}} \quad \text{ and } \quad \beta _j =\frac{e^{\tau _j}-e^{-\tau _j}}{\sum _{i=1}^r e^{\tau _i}+e^{-\tau _i}}\qquad \text{ for } \text{ all }\, j. \end{aligned}$$

As $\alpha _1 \le \ldots \le \alpha _r$ and $\beta _1\le \ldots \le \beta _r$, it follows from the spectral theorem (version 2) [25, Theorem III.1.2] that $\alpha _j = \beta _j$ for all j. Lemma 3.6 now implies that $\sigma = (\sigma _1,\ldots ,\sigma _r) = (\tau _1,\ldots ,\tau _r) =\tau$, as

$$\begin{aligned} (\alpha _1,\ldots ,\alpha _r) = \nabla \log \mu (\sigma )\quad \text{ and } \quad (\beta _1,\ldots ,\beta _r) = \nabla \log \mu (\tau ). \end{aligned}$$

Note that $\alpha _i =\alpha _j$ if and only if $\sigma _i=\sigma _j$, and $\beta _i =\beta _j$ if and only if $\tau _i=\tau _j$, as $\nabla \log \mu (x)$ is injective. It now follows from the spectral theorem (version 1) [25, Theorem III.1.1] that $x=y$. $\square$

Lemma 4.16

The map $\varphi$ maps V onto ${\textrm{int}}\,B_1^*$.

Proof

As $\varphi$ is continuous on V and $\varphi (V)\subseteq {\textrm{int}}\,B_1^*$, it follows from Brouwer’s domain invariance theorem that $\varphi (V)$ is open in ${\textrm{int}}\,B_1^*$. Suppose, for the sake of contradiction, that $\varphi (V)\ne {\textrm{int}}\,B_1^*$. So, we can find a $z\in \partial \varphi (V)\cap {\textrm{int}}\,B_1^*$. Let $(y^n)$ in V be such that $\varphi (y^n)\rightarrow z$ and write $y^n = \sum _{i=1}^r \lambda _i^np_i^n$. As $\varphi$ is continuous on V, we may assume that $r^n =\Vert y^n\Vert _u\rightarrow \infty$. Furthermore, after taking a subsequence, we may assume that $(y^n)$ satisfies the conditions in Proposition 4.7. So, using the notation as in Proposition 4.7, we get that

$$\begin{aligned} \varphi (y^n) = \frac{\sum _{i=1}^r(e^{\lambda ^n_i}-e^{-\lambda ^n_i})p_i^n}{\sum _{i=1}^r e^{\lambda ^n_i}+e^{-\lambda ^n_i}} = \frac{\sum _{i=1}^r(e^{-r^n+\lambda ^n_i}-e^{-r^n-\lambda ^n_i})p_i^n}{\sum _{i=1}^r e^{-r^n+\lambda ^n_i}+e^{-r^n-\lambda ^n_i}}. \end{aligned}$$

The right-hand side converges to

$$\begin{aligned} \frac{1}{\sum _{i\in I} e^{-\alpha _i}+\sum _{j\in J} e^{-\alpha _j}} \left( \sum _{i\in I}e^{-\alpha _i}p_i-\sum _{j\in J}e^{-\alpha _j}p_j\right) =z. \end{aligned}$$

But this implies that $z\in \partial B_1^*$, which is impossible. Indeed, if we let $p_I =\sum _{i\in I} p_i$ and $p_J=\sum _{j\in J} p_j$, then $1 \ge \Vert z\Vert _u^*\ge ( z|p_I-p_J) = 1$, as $-u\le p_I -p_J\le u$. $\square$

For simplicity we denote the (closed) boundary faces of $B_1^*$ by

$$\begin{aligned} F_{p,q} = {\textrm{conv}}\,( (U_p(V)\cap S(V))\cup (U_q(V)\cap -S(V))) \end{aligned}$$

where p and q are orthogonal idempotents in V not both zero, see [18, Theorem 4.4].

Lemma 4.17

If h is a horofunction given by (4.2), then $\varphi$ maps ${\mathcal {P}}_h$ into ${\textrm{relint}} \,F_{p_I,p_J}$.

Proof

Clearly, $\varphi (h)\in F_{p_I,p_J}$ if h is given by (4.2). So, $\varphi$ maps ${\mathcal {P}}_h$ into $F_{p_I,p_J}$ by Theorem 4.2(i). To show that $\varphi$ maps ${\mathcal {P}}_h$ into ${\textrm{relint}} \,F_{p_I,p_J}$, it suffices to show that $\varphi (h)\in {\textrm{relint}} \,F_{p_I,p_J}$.

To do this we first consider $w= (|I|+|J|)^{-1}(p_I-p_J)\in F_{p_I,q_J}$ and show that $w\in {\textrm{relint}} F_{p_i,q_J}$. Let $c\in F_{p_I,p_J}$ be arbitrary. Note that we can write $c= \sum _{i\in I'} \lambda _i q_i -\sum _{j\in J'} \lambda _jq_j$, where $\sum _{i\in I'} q_i =p_I$, $\sum _{j\in J'} q_j =p_J$, and $\sum _{i\in I'} \lambda _i + \sum _{j\in J'} \lambda _j =1$ with $0\le \lambda _i,\lambda _j\le 1$ for all i and j. We see that $w+\varepsilon (w-c) = (1+\varepsilon )w -\varepsilon c\in F_{p_I,p_J}$ for all $\varepsilon >0$ small, so $w\in {\textrm{relint}}\, F_{p_I,p_J}$ by [52, Theorem 6.4].

To complete the proof we argue by contradiction. So suppose that $\varphi (h)\not \in {\textrm{relint}} F_{p_I,p_J}$. Then $\varphi (h)$ is in the (relative) boundary of $F_{p_I,p_J}$, hence

$$\begin{aligned} z_\varepsilon = (1+\varepsilon ) \varphi (h) -\varepsilon w\not \in F_{p_I,p_J} \end{aligned}$$

for all $\varepsilon >0$, as $w\in {\textrm{relint}} F_{p_I,p_J}$ and $F_{p_I,p_J}$ is convex. However, for each $i\in I$ we have that the coefficient of $p_i$ in $z_\varepsilon$,

$$\begin{aligned} \frac{(1+\varepsilon )e^{-\alpha _i} }{\sum _{i\in I} e^{-\alpha _i}+\sum _{j\in J}e^{-\alpha _j}} - \frac{\varepsilon }{|I|+|J|}, \end{aligned}$$

is strictly positive for all $\varepsilon >0$ sufficiently small. Likewise, for each $j\in J$ we have that the coefficient of $-p_j$ in $z_\varepsilon$,

$$\begin{aligned} \frac{(1+\varepsilon )e^{-\alpha _j} }{\sum _{i\in I} e^{-\alpha _i}+\sum _{j\in J}e^{-\alpha _j}} - \frac{\varepsilon }{|I|+|J|}, \end{aligned}$$

is strictly positive for all $\varepsilon >0$ sufficiently small. This implies that $z_\varepsilon \in F_{p_I,p_J}$ for all $\varepsilon >0$ small, which is impossible. This completes the proof. $\square$

Using the previous results we now show that $\varphi$ is injective on ${\overline{V}}^h$.

Corollary 4.18

The map $\varphi :{\overline{V}}^h\rightarrow B_1^*$ is injective.

Proof

We already saw in Lemmas 4.14 and 4.15 that $\varphi$ maps V into ${\textrm{int}}\, B_1^*$ and is injective on V. So by the previous lemma, it suffices to show that if $\varphi (h)=\varphi (h')$ for horofunctions $h\sim h'$, then $h =h'$. Let h be given by (4.2) and suppose that $h'$ is given by

$$\begin{aligned} h'(x) = \max \left\{ \Lambda _{V(q_{I'})}\left( -U_{q_{I'}}x - \sum _{i\in I'} \beta _iq_i\right) , \Lambda _{V(q_{J'})}\left( U_{q_{J'}}x - \sum _{j\in J'} \beta _j q_j\right) \right\} . \end{aligned}$$

Then

$$\begin{aligned} \frac{\sum _{i\in I} e^{-\alpha _i}p_i -\sum _{j\in J} e^{-\alpha _j}p_j}{\sum _{i\in I} e^{-\alpha _i} +\sum _{j\in J} e^{-\alpha _j}} = \frac{\sum _{i\in I'} e^{-\beta _i}q_i -\sum _{j\in J'} e^{-\beta _j}q_j}{\sum _{i\in I'} e^{-\beta _i} +\sum _{j\in J'} e^{-\beta _j}}. \end{aligned}$$

As $\min _k \alpha _k =0 =\min _k\beta _k$, it follows from the spectral theorem [25, Theorem III.1.2] that

$$\begin{aligned} \frac{1}{\sum _{i\in I} e^{-\alpha _i} +\sum _{j\in J} e^{-\alpha _j}} =\Vert \varphi (h)\Vert _u=\Vert \varphi (h')\Vert _u = \frac{1}{\sum _{i\in I'} e^{-\beta _i} +\sum _{j\in J'}e^{-\beta _j}}, \end{aligned}$$

so that

$$\begin{aligned} \sum _{i\in I} e^{-\alpha _i}p_i -\sum _{j\in J} e^{-\alpha _j}p_i = \sum _{i\in I'} e^{-\beta _i}q_i -\sum _{j\in J'} e^{-\beta _j}q_j. \end{aligned}$$

As each $x\in V$ can be written in a unique way as $x= x^+-x^-$, where $x^+$ and $x^-$ are orthogonal element $x^+$ and $x^-$ in $V_+$, see [4, Proposition 1.28], we find that $\sum _{i\in I} e^{-\alpha _i}p_i = \sum _{i\in I'} e^{-\beta _i}q_i$ and $\sum _{j\in J} e^{-\alpha _j}p_i = \sum _{j\in J'} e^{-\beta _j}q_j$. This implies that

$$\begin{aligned} \sum _{i\in I} \alpha _ip_i= & {} -\log \left( \sum _{i\in I} e^{-\alpha _i}p_i +\left( u-p_I\right) \right) =-\log \left( \sum _{i\in I'} e^{-\beta _i}q_i+\left( u-q_{I'}\right) \right) \\ {}= & {} \sum _{i\in I'} \beta _iq_i \end{aligned}$$

and

$$\begin{aligned} \sum _{j\in J} \alpha _jp_j= & {} -\log \left( \sum _{j\in J} e^{-\alpha _j}p_i +\left( u-p_J\right) \right) =-\log \left( \sum _{j\in J'} e^{-\beta _j}q_j +\left( u-q_{J'}\right) \right) \\ {}= & {} \sum _{j\in J'}\beta _jq_j, \end{aligned}$$

and hence $h=h'$. $\square$

The next result shows that $\varphi$ is continuous on $\partial {\overline{V}}^h$.

Theorem 4.19

The map $\varphi :{\overline{V}}^h\rightarrow B_1^*$ is continuous.

Proof

Clearly $\varphi$ is continuous on V. Suppose $(y^n)$ is a sequence in V with $h_{y^n}\rightarrow h\in \partial {\overline{V}}^h$. We wish to show that $\varphi (y^n)\rightarrow \varphi (h)$. Let $(\varphi (y^{n_k}))$ be a subsequence. We will show that it has a subsequence which converges to $\varphi (h)$.

As h is a horofunction, we know that $r^n=\Vert y^{n_k}\Vert _u\rightarrow \infty$ by Lemma 2.1. For each k there exists a Jordan frame $q_1^{n_k},\ldots ,q_r^{n_k}$ in V and $\lambda _1^{n_k},\ldots ,\lambda _r^{n_k}\in {\mathbb {R}}$ such that

$$\begin{aligned} y^{n_k} = \sum _{i=1}^r \lambda _i^{n_k}q_i^{n_k}. \end{aligned}$$

By taking a subsequence we may assume that there exist $I_+\subseteq \{1,\ldots ,r\}$ and $1\le s\le r$ such that for each k, $r^{n_k} =\Vert y^{n_k}\Vert _u =|\lambda ^{n_k}_s|$, and $\lambda _i^{n_k}>0$ if and only if $i\in I_+$.

For each k, let $\beta ^{n_k}_i = r^{n_k} -\lambda _i^{n_k}$ for $i \in I_+$, and $\beta ^{n_k}_i = r^{n_k} +\lambda _i^{n_k}$ for $i \not \in I_+$. Note that $\beta ^{n_k}\ge 0$ for all i and k, and $\beta _s^{n_k}=0$ for all k. By taking a further subsequence we may assume that $\beta ^{n_k}_i\rightarrow \beta _i\in [0,\infty ]$ and $q^{n_k}_i\rightarrow q_i$ for all i. Let $I'=\{i\in I_+:\beta _i<\infty \}$ and $J'=\{j\not \in I_+:\beta _j<\infty \}$. Note that $s\in I'\cup J'$ and we can apply Proposition 4.7 to conclude that $h_{y^{n_k}}\rightarrow h'\in \partial {\overline{V}}^h$, where

$$\begin{aligned} h'\left( x\right) = \max \left\{ \Lambda _{V\left( q_{I'}\right) }\left( -U_{q_{I'}}x - \sum _{i\in I'}\beta _iq_i\right) ,\Lambda _{V\left( q_{J'}\right) }\left( U_{q_{J'}}x - \sum _{j\in J'}\beta _jq_j\right) \right\} . \end{aligned}$$

As $h_{y^{n_k}}\rightarrow h$, we know that $h=h'$ and hence $\delta (h,h')=0$. This implies that $p_I=q_{I'}$ and $p_J=q_{J'}$ by Theorem 4.2. Moreover,

$$\begin{aligned} \sum _{i\in I}\alpha _ip_i +\sum _{j\in J}\alpha _jp_j = \sum _{i\in I'}\beta _iq_i+\sum _{j\in J'}\beta _jq_j. \end{aligned}$$

It follows that

$$\begin{aligned} \sum _{i\in I}\alpha _ip_i= & {} U_{p_I}\left( \sum _{i\in I}\alpha _ip_i +\sum _{j\in J}\alpha _jp_j\right) = U_{q_{I'}}\left( \sum _{i\in I'}\beta _iq_i+\sum _{j\in J'}\beta _jq_j\right) \\ {}= & {} \sum _{i\in I'}\beta _iq_i \end{aligned}$$

and

$$\begin{aligned} \sum _{j\in J}\alpha _jp_j= & {} U_{p_J}\left( \sum _{i\in I}\alpha _ip_i +\sum _{j\in J}\alpha _jp_j\right) = U_{q_{J'}}\left( \sum _{i\in I'}\beta _iq_i+\sum _{j\in J'}\beta _jq_j\right) = \sum _{j\in J'}\beta _jq_j, \end{aligned}$$

so that $\sum _{i\in I}e^{\alpha _i} p_i = \sum _{i\in I'}e^{\beta _i}q_i$ and $\sum _{j\in J}e^{\alpha _j} p_j = \sum _{j\in J'}e^{\beta _j}q_j$. We conclude that

$$\begin{aligned} \lim _{k\rightarrow \infty } \varphi \left( y^{n_k}\right)= & {} \lim _{k\rightarrow \infty } \frac{\sum _{i=1}^r \left( e^{-r^{n_k}+\lambda _i^{n_k}} -e^{-r^{n_k}-\lambda _i^{n_k}}\right) q_i^{n_k}}{\sum _{i=1}^r \left( e^{-r^{n_k}+\lambda _i^{n_k}} +e^{-r^{n_k}-\lambda _i^{n_k}}\right) } \\ {}= & {} \frac{\sum _{i\in I'} e^{-\beta _i}q_i - \sum _{j\in J'}e^{-\beta _j}q_j}{\sum _{i\in I'} e^{-\beta _i} + \sum _{j\in J'}e^{-\beta _j}} =\varphi \left( h\right) . \end{aligned}$$

From Lemmas 4.14 and 4.17 we know that $\varphi$ maps V into ${\textrm{int}}\, B_1^*$ and $\partial {\overline{V}}^h$ into $\partial B_1^*$. So to complete the proof it remains to show that if $(h_n)$ in $\partial {\overline{V}}^h$ converges to $h\in \partial {\overline{V}}^h$, then $\varphi (h_n)\rightarrow \varphi (h)$. Suppose h is given by (4.2) and for each n the horofunction $h_n$ is given by

$$\begin{aligned} h_n(x)= & {} \max \left\{ \Lambda _{V(q^n_{I_n})}\left( -U_{q^n_{I_n}}x - \sum _{i\in I_n} \beta ^n_iq^n_i\right) , \Lambda _{V(q^n_{J_n})}\left( U_{q^n_{J_n}}x - \sum _{j\in J_n} \beta ^n_j q^n_j\right) \right\} \nonumber \\ {}{} & {} \quad \text{ for } \, x\in V, \end{aligned}$$

(4.15)

where $I_n,J_n\subseteq \{1,\ldots ,r\}$ are disjoint, $I_n\cup J_n$ is nonempty, and $\min \{\beta ^n_k:k\in I_n\cup J_n\} =0$.

To prove the assertion we show that each subsequence of $(\varphi (h_n))$ has a convergent subsequence with limit $\varphi (h)$. Let $(\varphi (h_{n_k}))$ be a subsequence. By taking subsequences we may assume that

• (1) There exist $I_0,J_0\subseteq \{1,\ldots ,r\}$ disjoint with $I_0\cup J_0$ nonempty, such that $I_{n_k} = I_0$ and $J_{n_k} = J_0$ for all k.

• (2) $\beta ^{n_k}_i\rightarrow \beta _i\in [0,\infty ]$ and $q^{n_k}_i\rightarrow q_i$ for all $i\in I_0\cup J_0$.

• (3) There exists $i^*\in I_0\cup J_0$ such that $\beta ^{n_k}_{i^*}=0$ for all k.

Let $I' = \{i\in I_0:\beta _i<\infty \}$ and $J'=\{j\in J_0:\beta _j<\infty \}$, and note that $i^*\in I'\cup J'$.

Using Lemma 4.6 we now show that $h_{n_k}\rightarrow h'$, where

$$\begin{aligned} h'(x)=\max \left\{ \Lambda _{V(q_{I'})}\left( -U_{q_{I'}}x - \sum _{i\in I'} \beta _iq_i\right) , \Lambda _{V(q_{J'})}\left( U_{q_{J'}}x - \sum _{j\in J'} \beta _j q_j\right) \right\} .\nonumber \\ \end{aligned}$$

(4.16)

Note that if $I'$ is nonempty, then by Lemma 4.6 we have that

$$\begin{aligned} \lim _{k\rightarrow \infty } \Lambda _{V(q^{n_k}_{I_0})}\left( -U_{q^{n_k}_{I_0}}x - \sum _{i\in I_0} \beta ^{n_k}_iq^{n_k}_i\right) = \Lambda _{V(q_{I'})}\left( -U_{q_{I'}}x - \sum _{i\in I'} \beta _iq_i\right) , \end{aligned}$$

as $U_{q^{n_k}_{I_0}}x \rightarrow U_{q_{I_0}}x$ by Lemma 4.5 and $U_{q_{I'}}(U_{q_{I_0}}x) = U_{q_{I'}}x$ by [4, Proposition 2.26]. Likewise if $J'$ is nonempty, we have that

$$\begin{aligned} \lim _{k\rightarrow \infty } \Lambda _{V(q^{n_k}_{J_0})}\left( U_{q^{n_k}_{J_0}}x - \sum _{j\in J_0} \beta ^{n_k}_jq^{n_k}_j\right) = \Lambda _{V(q_{J'})}\left( U_{q_{J'}}x - \sum _{j\in J'} \beta _jq_j\right) . \end{aligned}$$

Thus, if $I'$ and $J'$ are both nonempty (4.16) holds.

Now suppose that $I'$ is empty, so $J'$ is nonempty. As $-x\le \Vert x\Vert _u u$, we get that

$$\begin{aligned} -U_{q_{I_0}^{n_k}}x\le \Vert x\Vert _u U_{q_{I_0}^{n_k}}u = \Vert x\Vert _u q_{I_0}^{n_k}. \end{aligned}$$

This implies that $-U_{q_{I_0}^{n_k}}x - \sum _{i\in I_0}\beta _i^{n_k}q_i^{n_k}\le \sum _{i\in I_0} (\Vert x\Vert _u -\beta _i^{n_k})q_i^{n_k}$, hence

$$\begin{aligned} \Lambda _{V(q^{n_k}_{I_0})}\left( -U_{q^{n_k}_{I_0}}x - \sum _{i\in I_0} \beta ^{n_k}_iq^{n_k}_i\right) \le \max _{i\in I_0}(\Vert x\Vert _u -\beta _i^{n_k})\rightarrow -\infty . \end{aligned}$$

On the other hand, $h_{n_k}(x)\ge -\Vert x\Vert _u$ for all k. Thus, for all k sufficiently large, we have that

$$\begin{aligned} h_{n_k}(x) = \Lambda _{V(q^{n_k}_{J_0})}\left( U_{q^{n_k}_{J_0}}x - \sum _{j\in J_0} \beta _j^{n_k}q_j^{n_k}\right) , \end{aligned}$$

which implies that (4.16) holds if $I'$ is empty. In the same way it can be shown that (4.16) holds if $J'$ is empty.

As $h_n\rightarrow h$, we know that $h'=h$, so $\delta (h,h')=0$. It follows from Theorem 4.2 that $p_I=q_{I'}$, $p_J=q_{J'}$, and $\sum _{i\in I}\alpha _ip_i +\sum _{j\in J}\alpha _jp_j = \sum _{i\in I'}\beta _iq_i+\sum _{j\in J'}\beta _jq_j$. This implies that

$$\begin{aligned} \sum _{i\in I}\alpha _ip_i = \sum _{i\in I'}\beta _iq_i\quad \text{ and } \quad \sum _{j\in J}\alpha _jp_j = \sum _{j\in J'}\beta _jq_j, \end{aligned}$$

so that $\sum _{i\in I}e^{-\alpha _i} p_i = \sum _{i\in I'}e^{-\beta _i}q_i$ and $\sum _{j\in J}e^{-\alpha _j} p_j = \sum _{j\in J'}e^{-\beta _j}q_j$. Thus,

$$\begin{aligned} \lim _{k\rightarrow \infty } \varphi (h_{n_k})= & {} \lim _{k\rightarrow \infty } \frac{\sum _{i\in I_0} e^{-\beta _i^{n_k}}q^{n_k}_i -\sum _{j\in J_0} e^{-\beta _j^{n_k}}q^{n_k}_j}{\sum _{i\in I_0} e^{-\beta _i^{n_k}}+\sum _{j\in J_0} e^{-\beta _j^{n_k}}} \\= & {} \frac{\sum _{i\in I'} e^{-\beta _i}q_i - \sum _{j\in J'}e^{-\beta _j}q_j}{\sum _{i\in I'} e^{-\beta _i} + \sum _{j\in J'}e^{-\beta _j}} =\varphi (h), \end{aligned}$$

which completes the proof. $\square$

Theorem 4.20

The map $\varphi :{\overline{V}}^h\rightarrow B_1^*$ is onto.

Proof

From Lemma 4.16 we know that $\varphi (V) = {\textrm{int}}\, B_1^*$. Let $z\in \partial B_1^*$. As $B_1^*$ is the disjoint union of the relative interiors of its faces, see [52, Theorem 18.2], we know that there exist orthogonal idempotents $p_I$ and $p_J$ such that $z\in {\textrm{relint}} F_{p_I,p_J}$. Thus, we can write

$$\begin{aligned} z =\sum _{i\in I} \lambda _ip_i-\sum _{j\in J}\lambda _jp_j, \end{aligned}$$

where $p_I =\sum _{i\in I}p_i$, $q_J =\sum _{j\in J}q_j$, $0<\lambda _k\le 1$ for all $k\in I\cup J$, and $\sum _{k\in I\cup J}\lambda _k =1$.

Define $\mu _k = -\log \lambda _k$ for $k\in I\cup J$. So, $\mu _k\ge 0$. Let $\mu ^*=\min \{\mu _k:k\in I\cup J\}$ and set $\alpha _k = \mu _k-\mu ^*\ge 0$. Note that $\min \{\alpha _k:k\in I\cup J\}=0$.

Then h, given by

$$\begin{aligned} h(x) = \max \left\{ \Lambda _{V(p_I)}\left( -U_{p_I}x - \sum _{i\in I} \alpha _ip_i\right) , \Lambda _{V(p_J)}\left( U_{p_J}x - \sum _{j\in J} \alpha _j p_j\right) \right\} \end{aligned}$$

for $x\in V$, is a horofunction by Proposition 4.9. Moreover,

$$\begin{aligned}{} & {} \frac{1}{\sum _{i\in I} e^{-\mu _i} +\sum _{j\in J} e^{-\mu _j}}\left( \sum _{i\in I} e^{-\mu _i}p_i -\sum _{j\in J} e^{-\mu _j}p_j\right) \\{} & {} \quad = \frac{1}{\sum _{i\in I} \lambda _i +\sum _{j\in J} \lambda _j}\left( \sum _{i\in I} \lambda _ip_i -\sum _{j\in J} \lambda _jp_j\right) , \end{aligned}$$

hence $\varphi (h)=z$, which completes the proof. $\square$

The proof of Theorem 4.4 is now straightforward.

Proof of Theorem 4.4

It follows from Theorems 4.19 and 4.20 and Corollary 4.18 that $\varphi :{\overline{V}}^h\rightarrow B_1^*$ is a continuous bijection. As ${\overline{V}}^h$ is compact and $B_1^*$ is Hausdorff, we conclude that $\varphi$ is a homeomorphism. It follows from Lemma 4.17 that $\varphi$ maps each part onto the relative interior of a boundary face of $B_1^*$. $\square$

Remark 4.21

It is interesting to note that a similar idea can be used to show that the horofunction compactification of a finite dimensional normed space $(V,\Vert \cdot \Vert )$ with a smooth and strictly convex norm is homeomorphic to the closed dual unit ball. Indeed, in that case the horofunctions are given by $h:z\mapsto -x^*(z)$, where $x^*\in V^*$ has norm 1, see for example [23, Lemma 5.3]. Moreover, for $(y^n)$ in V we have that $h_{y^n}\rightarrow h$ if and only if $y^n/\Vert y^n\Vert \rightarrow x$ and $\Vert y^n\Vert \rightarrow \infty$.

In this case we define a map $\psi :{\overline{V}}^h\rightarrow B_1^*$ as follows. For $x\in V$ with $x\ne 0$, let

$$\begin{aligned} \psi (x) = -\left( \frac{e^{\Vert x\Vert } - e^{-\Vert x\Vert }}{e^{\Vert x\Vert }+e^{-\Vert x\Vert }}\right) x^*, \end{aligned}$$

where $x^* \in V^*$ is the unique functional with $x^*(x) =\Vert x\Vert$ and $\Vert x^*\Vert =1$, and let $\psi (0)=0$. For $h\in \partial {\overline{V}}^h$ with $h:z\mapsto -x^*(z)$ let

$$\begin{aligned} \psi (h) = -x^*. \end{aligned}$$

It is straightforward to check that $\psi$ is a bijection from ${\overline{V}}^h$ onto $B_1^*$, and $\psi$ is continuous on ${\textrm{int}}\,B_1^*$. To show continuity on $\partial {\overline{V}}^h$, we assume, by way of contradiction, that $(h_n)$ is a sequence of horofunctions with $h_n\rightarrow h$ and $h_n (z) = -x^*_n(z)$ for all $z\in V$, and there exists a neighbourhood U of $\psi (h)$ in $B_1^*$ such that $\psi (h_n)\not \in U$ for all n. Then, for each $z^*\in \partial B_1^*$ with $z^*\not \in U$ we have that $z^*(x)<1$. So, by compactness, $\delta = \max \{ 1 - z^*(x):z^*\in \partial B_1^*{\setminus } U\}>0$. It now follows that

$$\begin{aligned} h_n(x) - h(x) = -x^*_n(x) +x^*(x) = 1- x^*_n(x)\ge \delta >0 \end{aligned}$$

for all n, which contradicts $h_n\rightarrow h$. This shows that $\psi$ is a continuous bijection, and hence a homeomorphism, as ${\overline{V}}^h$ is compact and $B_1^*$ is Hausdorff.

More generally, one can consider product spaces $V=\prod _{i=1}^r V_i$ with norm $\Vert x\Vert _V = \max _{i=1}^r \Vert v_i\Vert _i$, where each $(V_i,\Vert \cdot \Vert _i)$ is a finite dimensional normed space with a smooth and strictly convex norm. In that case we have by [40, Theorem 2.10] that the horofunctions of V are given by

$$\begin{aligned} h(v) = \max _{j\in J} (h_{\xi _j^*}(v_j)-\alpha _j), \end{aligned}$$

(4.17)

where $J\subseteq \{1,\ldots ,r\}$ nonempty, $\min _{j\in J}\alpha _j =0$, $\xi _j^*\in V_j^*$ with $\Vert \xi _j^*\Vert =1$, and $h_{\xi _j^*}(v_j) = -\xi _j^*(v_j)$. One can use similar ideas as the ones in Sect. 3 to show that the horofunction compactification is homeomorphic to the closed unit dual ball of V. Indeed, one can define a map $\varphi _V:{\overline{V}}^h\rightarrow B_1^*$ by

$$\begin{aligned} \varphi _V(v) = \frac{1}{\sum _{i=1}^r e^{\Vert v_i\Vert _i} + e^{-\Vert v_i\Vert _i}} \left( \sum _{i=1}^r (e^{\Vert v_i\Vert _i} - e^{-\Vert v_i\Vert _i})p(v^*_i)\right) \qquad \text{ for } \, v\in V\setminus \{0\} \end{aligned}$$

and $\varphi _V(0)=0$. Here $p(v^*_i) = (0,\ldots ,0,v^*_i,0,\ldots ,0)$ and $v^*_i$ is the unique functional such that $v^*_i(v_i) =\Vert v_i\Vert _i$ and $\Vert v^*_i\Vert _i=1$ if $v_i\ne 0$, and we set $p(v_i^*)=0$, if $v_i=0$. For a horofunction h given by (4.17) we define

$$\begin{aligned} \varphi _V(h) = \frac{1}{\sum _{j\in J} e^{-\alpha _j}}\left( \sum _{j\in J} e^{-\alpha _j}p(\xi _j^*)\right) . \end{aligned}$$

Following the same line of reasoning as in Sect. 3 one can prove that $\varphi _V$ is a homeomorphism.

Remark 4.22

The connection between the geometry of the horofunction compactification and the dual unit ball seems hard to establish for general finite dimensional normed spaces, and might not even hold. For the normed spaces discussed in this paper and in [32, 33] all horofunctions are Busemann points, but there are normed spaces with horofunctions that are not Busemann, see [54]. It could well be the case that the horofunction compactification of these spaces is not naturally homeomorphic to the closed dual unit ball, but no counter example is known at present.

5 Symmetric Cones with the Hilbert Distance

In this section we study the global topology and geometry of the horofunction compactification of symmetric cones under the Hilbert distance. Recall that the Hilbert distance is defined as follows. Let A be a real finite dimensional affine space. Consider a bounded, open, convex set $\Omega \subseteq A$. For $x,y\in \Omega$, let $\ell _{xy}$ be the straight-line through x and y in A, and denote the points of intersection of $\ell _{xy}$ and $\partial \Omega$ by $x'$ and $y'$, where x is between $x'$ and y, and y is between x and $y'$. On $\Omega$ the Hilbert distance is then defined by

$$\begin{aligned} \rho _H(x,y) = \log \Big (\frac{|x'-y|}{|x'-x|} \frac{|y'-x|}{|y'-y|}\Big ) \end{aligned}$$

(5.1)

for all $x\ne y$ in $\Omega$, and $\rho _H(x,x) =0$ for all $x\in \Omega$. The metric space $(\Omega , \rho _H)$ is called the Hilbert geometry on $\Omega$.

These metric spaces generalise Klein’s model of hyperbolic space and have a Finsler structure, see [48, 49]. In our analysis we will work with Birkhoff’s version of the Hilbert metric, which is defined on a cone in an order-unit space in terms of its partial ordering. This provides a convenient way to work with the Hilbert distance and its Finsler structure. In the next subsection we will recall the basic concepts involved in our analysis. Throughout we will follow the terminology used in [42, Chap. 2], which contains a detailed discussion of Hilbert geometries and some of their applications. We refer the reader to [49] for a comprehensive account of the theory of Hilbert geometries.

5.1 Preliminaries and Finsler Structure

Let $(V,V_+,u)$ be a finite dimensional order-unit space. So, $V_+$ is a closed cone in V with $u\in {\textrm{int}}\,V_+$. Recall that the cone $V_+$ induces a partial ordering on V by $x\le y$ if $y-x\in V_+$, see Sect. 4.1. For $x\in V$ and $y\in V_+$, we say that y dominates x if there exist $\alpha ,\beta \in {\mathbb {R}}$ such that $\alpha y\le x\le \beta y$. In that case, we write

$$\begin{aligned} M(x/y) =\inf \{\beta \in {\mathbb {R}}:x\le \beta y\} \quad \text{ and } \quad m(x/y) = \sup \{\alpha \in {\mathbb {R}}:\alpha y\le x\}. \end{aligned}$$

By the Hahn–Banach theorem, $x\le y$ if and only if $\psi (x)\le \psi (y)$ for all $\psi \in V^*_+=\{\varphi \in V^*:\varphi \text{ positive }\}$, which is equivalent to $\psi (x)\le \psi (y)$ for all $\psi \in S(V)$. Using this fact we see that for each $x\in V$ and $y\in {\textrm{int}}\,V_+$,

$$\begin{aligned} M(x/y) = \sup _{\psi \in S(V)}\frac{\psi (x)}{\psi (y)}\quad \text{ and } \quad m(x/y) =\inf _{\psi \in S(V)}\frac{\psi (x)}{\psi (y)}. \end{aligned}$$

We also note that if $A\in {\textrm{GL}}(V)$ is a linear automorphism of $V_+$, i.e., $A(V_+)= V_+$, then $x\le \beta y$ if, and only if, $Ax\le \beta Ay$. It follows that $M(Ax/Ay)=M(x/y)$ and $m(x/y)= m(Ax/Ay)$.

If $w\in {\textrm{int}}\, V_+$, then w dominates each $x\in V$, and we define

$$\begin{aligned} |x|_w = M(x/w)- m(x/w). \end{aligned}$$

One can verify that $|\cdot |_w$ is a semi-norm on V, see [42, Lemma A.1.1], and a genuine norm on the quotient space $V/{\mathbb {R}}w$, as $|x|_w=0$ if and only if $x=\lambda w$ for some $\lambda \in {\mathbb {R}}$.

Clearly, if $x,y\in V$ are such that $y=0$ and y dominates x, then $x=0$, as $V_+$ is a cone. On the other hand, if $y\in V_+{\setminus }\{0\}$, and y dominates x, then $M(x/y)\ge m(x/y)$. The domination relation yields an equivalence relation on $V_+$ by $x\sim y$ if y dominates x and x dominates y. The equivalence classes are called the parts of $V_+$. As $V_+$ is closed, one can check that $\{0\}$ and ${\textrm{int}}\,V_+$ are parts of $V_+$. The parts of a finite dimensional cone are closely related to its faces. Indeed, if $V_+$ is the cone of a finite dimensional order-unit space, then it can be shown that the parts correspond to the relative interiors of the faces of $V_+$, see [42, Lemma 1.2.2]. Recall that a face of a convex set $S\subseteq V$ is a subset F of S with the property that if $x,y\in S$ and $\lambda x+ (1-\lambda )y\in F$ for some $0<\lambda <1$, then $x,y\in F$.

It is easy to verify that if $x,y\in V_+{\setminus }\{0\}$, then $x\sim y$ if, and only if, there exist $0<\alpha \le \beta$ such that $\alpha y\le x\le \beta y$. Furthermore, if $x\sim y$, then

$$\begin{aligned} m(x/y) = \sup \{\alpha >0:y\le \alpha ^{-1}x\}= M(y/x)^{-1}. \end{aligned}$$

(5.2)

Birkhoff’s version of the Hilbert distance on $V_+$ is defined as follows:

$$\begin{aligned} d_H(x,y) = \log \Big (\frac{M(x/y)}{m(x/y)}\Big ) = \log M(x/y) +\log M(y/x) \end{aligned}$$

(5.3)

for all $x\sim y$ with $y\ne 0$, $d_H(0,0)=0$, and $d_H(x,y)=\infty$ otherwise.

Note that $d_H(\lambda x,\mu y) = d_H(x,y)$ for all $x,y\in V_+$ and $\lambda ,\mu >0$, so $d_H$ is not a distance on $V_+$. It is, however, a distance between pairs of rays in each part of $V_+$. In particular, if $\varphi :V\rightarrow {\mathbb {R}}$ is a linear functional such that $\varphi (x)>0$ for all $x\in V_+{\setminus }\{0\}$, then $d_H$ is a distance on

$$\begin{aligned} \Omega _V=\{x\in {\textrm{int}}\, V_+:\varphi (x)=1\}, \end{aligned}$$

which is a (relatively) open, bounded, convex set, see [42, Lemma 1.2.4]. Moreover, the following holds, see [42, Proposition 2.1.1 and Theorem 2.1.2].

Theorem 5.1

$(\Omega _V,d_H)$ is a metric space and $d_H=\rho _H$ on $\Omega _V$.

It is worth noting that any Hilbert geometry can be realised as $(\Omega _V,d_H)$ for some order-unit space V and strictly positive linear functional $\varphi$.

A Hilbert geometry $(\Omega _V,d_H)$ has a Finsler structure, see [48]. Indeed, if one defines the length of a piecewise $C^1$-smooth path $\gamma :[0,1]\rightarrow \Omega _V$ by

$$\begin{aligned} L(\gamma ) = \int ^1_0 |\gamma '(t)|_{\gamma (t)}{\textrm{d}}t, \end{aligned}$$

then $d_H(x,y) = \inf _\gamma L(\gamma )$, where the infimum is taken over all piecewise $C^1$-smooth paths in $\Omega _V$ with $\gamma (0)=x$ and $\gamma (1)=y$.

It should be noted that in the case of Hilbert geometries the unit ball $\{x\in V/{\mathbb {R}}w:|x|_w\le 1\}$ in the tangent space at $w\in \Omega _V$ may have a different facial structure for different w. This phenomenon appears frequently in the case where $\Omega _V$ is a polytope, but does not appear in the Hilbert geometries considered here.

Let $(V,V_+,u)$ be an order-unit space, where V is a Euclidean Jordan algebra of rank r, $V_+$ is the cone of squares, and u is the algebraic unit. So, ${\textrm{int}}\, V_+$ is a symmetric cone and ${\textrm{Isom}}(\Omega _V)$ acts transitively on $\Omega _V$.

Throughout we will take $\varphi :V\rightarrow {\mathbb {R}}$ with $\varphi (x) = \frac{1}{r}{\textrm{tr}}(x)$, which is a state, and

$$\begin{aligned} \Omega _V=\{x\in {\textrm{int}}\, V_+:\varphi (x)=1\}=\{x\in {\textrm{int}}\, V_+:{\textrm{tr}}(x) = r\}. \end{aligned}$$

We shall call $(\Omega _V,d_H)$ a symmetric Hilbert geometry. A prime example is

$$\begin{aligned} \Omega _V =\{ A\in {\textrm{Herm}}(n, {\mathbb {C}}):{\textrm{tr}}(A) = n \text{ and } A \text{ positive } \text{ definite }\}. \end{aligned}$$

These spaces are important examples of noncompact type symmetric spaces with an invariant Finsler metric, see [50]. In particular, the example above corresponds to the symmetric space ${\textrm{SL}}(n,{\mathbb {C}})/{\textrm{SU}}(n)$.

In a symmetric Hilbert geometry the distance can be expressed in terms of the spectrum. Indeed, we know that for $x\in V$ invertible, the quadratic representation $U_x:V\rightarrow V$ is a linear automorphism of $V_+$, see [25, Proposition III.2.2]. Moreover, $U_x^{-1} = U_{x^{-1}}$ and $U_{x^{-1/2}}x=u$. Furthermore, for $x\in V$ we have that

$$\begin{aligned} M(x/u) =\inf \{\lambda :x\le \lambda u\} = \max \sigma (x)\quad \text{ and } \quad m(x/u) =\sup \{\lambda :\lambda u\le x\} = \min \sigma (x), \end{aligned}$$

so that $|x|_u = \max \sigma (x) - \min \sigma (x)$. Also for $x,y\in {\textrm{int}}\, V_+$ we have that

$$\begin{aligned} \log M(x/y) = \max \sigma (\log U_{y^{-1/2}}x) \quad \text{ and } \quad \log M(y/x) = -\min \sigma (\log U_{y^{-1/2}}x). \end{aligned}$$

It follows that

$$\begin{aligned} d_H(x,y)= & {} \log M(x/y)+\log M(y/x) = |\log U_{y^{-1/2}}x|_u \\ {}= & {} {\mathrm {diam\,}}\sigma (\log U_{y^-1/2}x)\quad \text{ for } \text{ all } x,y\in {\textrm{int}}\, V_+. \end{aligned}$$

Moreover, for each $w\in \Omega _V$ we have that

$$\begin{aligned} |x|_w= & {} M(x/w)- m(x/w) = M(U_{w^{-1/2}}x/u)-m(U_{w^{-1/2}}x/u)\\ {}= & {} |U_{w^{-1/2}}x|_u\quad \text{ for } \text{ all } x\in V, \end{aligned}$$

which shows that the facial structure of the unit ball in each tangent space is identical, as $U_{w^{-1/2}}$ is an invertible linear map.

5.2 Horofunctions of Symmetric Hilbert Geometries

The main objective is to show for symmetric Hilbert geometries $(\Omega _V,d_H)$ that there exists a natural homeomorphism between ${{\overline{\Omega }}}_V^h$ and the closed dual unit ball of the Finsler metric $|\cdot |_u$ in the tangent space $V/{\mathbb {R}}u$ at the unit u. To describe the homeomorphism, we recall the description of the horofunction compactification of symmetric Hilbert geometries given in [44, Theorem 5.6].

Theorem 5.2

The horofunctions of a symmetric Hilbert geometry $(\Omega _V,d_H)$ are precisely the functions $h:\Omega _V\rightarrow {\mathbb {R}}$ of the form

$$\begin{aligned} h(x) = \log M(y/x)+\log M(z/x^{-1})\quad \text{ for } x\in \Omega _V, \end{aligned}$$

(5.4)

where $y,z\in \partial V_+$ are such that $\Vert y\Vert _u=\Vert z\Vert _u =1$ and $(y|z)=0$.

It follows from the proof of [44, Theorem 5.6] that all horofunctions are in fact Busemann points. Indeed, if y and z have spectral decompositions

$$\begin{aligned} y = \sum _{i \in I} \lambda _ip_i\quad \text{ and } \quad z =\sum _{j\in J}\mu _jp_j, \end{aligned}$$

where $I,J\subset \{1,\ldots ,r\}$ are nonempty and disjoint, and $p_1,\ldots ,p_r$ is a Jordan frame, then the sequence $(y_n)\in {\textrm{int}}\, V_+$ given by

$$\begin{aligned} y_n = \sum _{i \in I } \lambda _ip_i+\sum _{j\in J}\frac{1}{n^2\mu _j}p_j + \sum _{k\not \in I\cup J} \frac{1}{n}p_k \end{aligned}$$

has the property that $y_n\rightarrow y$, $y_n^{-1}/\Vert y_n^{-1}\Vert _u\rightarrow z$ and $h_{y_n}\rightarrow h$, where h is as in (5.4). Note that if we let $v_n = y_n/\varphi (y_n)\in \Omega _V$, then $h_{v_n}(z)=h_{y_n}(z)$ for all $z\in \Omega _V$, so $h_{v_n}\rightarrow h$.

Also note that for $n,m\ge 1$,

$$\begin{aligned} U_{y_n^{-1/2}}y_m = \sum _{i \in I } p_i+\sum _{j\in J}\frac{n^2}{m^2}p_j + \sum _{k\not \in I\cup J} \frac{n}{m}p_k. \end{aligned}$$

This implies that for each $n\ge m\ge 1$,

$$\begin{aligned} M(y_m/y_n) = M(U_{y_n^{-1/2}}y_m/u) = \Vert U_{y_n^{-1/2}}y_m\Vert _u = n^2/m^2, \end{aligned}$$

so that $\log M(y_m/y_n) = 2\log n -2\log m$. Moreover, $\log M(y_n/y_m) =\log 1 =0$ for all $n\ge m\ge 1$. It follows that

$$\begin{aligned} d_H(v_n,v_m)+d_H(v_m,v_1)= d_H(y_n,y_m)+d_H(y_m,y_1) = d_H(y_n,y_1) = d_H(v_n,v_1) \end{aligned}$$

for all $n\ge m\ge 1$. Thus, $(v_n)$ is an almost geodesic sequence in $\Omega _V$, and hence each horofunction in ${{\overline{\Omega }}}_V^h$ is a Busemann point.

To identify the parts and describe the detour distance (2.2) we need the following general lemma.

Lemma 5.3

Let $(V,V_+,u)$ be a finite dimensional order-unit space. If $v\in \partial V_+{\setminus }\{0\}$ and $w_n\in {\textrm{int}}\, V_+$ with $w_{n+1}\le w_n$ for all $n\ge 1$ and $w_n\rightarrow w\in \partial V_+{\setminus }\{0\}$, then

$$\begin{aligned} \lim _{n\rightarrow \infty } M(v/w_n) = \left[ \begin{array}{ll} M(v/w) &{} \text{ if } \,w \text{ dominates } v\\ \infty &{} \text{ otherwise. } \end{array}\right. \end{aligned}$$

Proof

Set $\lambda _n = M(v/w_n)$ for $n\ge 1$. Then for $n\ge m\ge 1$ we have that $0\le \lambda _nw_n -v \le \lambda _nw_m-v$. This implies that $\lambda _m\le \lambda _n$ for all $m\le n$, hence $(\lambda _n)$ is monotonically increasing.

Now suppose that $\lambda = M(v/w)<\infty$, i.e., w dominates v. Then $0\le \lambda w-v\le \lambda w_n -v$, hence $\lambda _n\le \lambda$ for all n. This implies that $\lambda _n\rightarrow \lambda ^*\le \lambda <\infty$. As $0\le \lambda _n w_n -v$ for all n and $V_+$ is closed, we know that $\lim _{n\rightarrow \infty } \lambda _nw_n -v = \lambda ^* w-v\in V_+$. So $\lambda ^*\ge \lambda$, hence $\lambda ^*= \lambda$. We conclude that if w dominates v, then $\lim _{n\rightarrow \infty } M(v/w_n) = M(v/w)$.

On the other hand, if w does not dominate v, then

$$\begin{aligned} \lambda w-v\not \in V_+\quad \text{ for } \text{ all } \lambda \ge 0. \end{aligned}$$

(5.5)

Assume, by way of contradiction, that $(\lambda _n)$ is bounded. Then $\lambda _n\rightarrow \lambda ^*<\infty$, since $(\lambda _n)$ is increasing, and $\lambda _nw_n -v \rightarrow \lambda ^* w -v\in V_+$, as $V_+$ is closed. This contradicts (5.5), and hence $\lambda _n = M(v/w_n)\rightarrow \infty$, if w does not dominate v. $\square$

Before we identify the parts in $\partial {\overline{\Omega }}^h_V$ and the detour distance, it is useful to recall the following fact:

$$\begin{aligned} M(x/y) = M(y^{-1}/x^{-1}) \quad \text{ for } \text{ all } x,y\in {\textrm{int}}\,V_+, \end{aligned}$$

if ${\textrm{int}}\, V_+$ is a symmetric cone, see [45, Sect. 2.4].

Proposition 5.4

Let $(\Omega _V,d_H)$ be a symmetric Hilbert geometry and $h,h'\in \partial {\overline{\Omega }}^h_V$ with

$$\begin{aligned} h(x) = \log M(y/x)+\log M(z/x^{-1}) \quad \text{ and } \quad h'(x) = \log M(y'/x)+\log M(z'/x^{-1}) \end{aligned}$$

for $x\in \Omega _V$. The following assertions hold:

• (i) h and $h'$ are in the same part if and only if $y\sim y'$ and $z\sim z'$.

• (ii) If h and $h'$ are in the same part, then $\delta (h,h') = d_H(y,y') +d_H(z,z')$.

Proof

Consider the spectral decompositions: $y = \sum _{i\in I}\lambda _ip_i$, $z = \sum _{j\in J}\mu _jp_j$, $y' = \sum _{i\in I'}\alpha _iq_i$, and $z' = \sum _{j\in J'}\beta _jq_j$. Set

$$\begin{aligned} y_n= & {} \sum _{i\in I}\lambda _ip_i + \sum _{j\in J}\frac{1}{n^2\mu _j}p_j + \sum _{k\not \in I\cup J} \frac{1}{n} p_k \quad \text{ and } \\ w_n= & {} \sum _{i\in I'}\alpha _iq_i + \sum _{j\in J'}\frac{1}{n^2\beta _j}q_j + \sum _{k\not \in I'\cup J'} \frac{1}{n} q_k. \end{aligned}$$

Then $h_{y_n}\rightarrow h$ and $h_{w_n}\rightarrow h'$ by the proof of [44, Theorem 5.6].

For all $n\ge 1$ large we have that $\Vert w_n\Vert _u =\Vert y'\Vert _u=1$, so that

$$\begin{aligned} d_H(w_n,u) = \log M(w_n/u) +\log M(u/w_n) = \log \Vert w_n\Vert _u+\log M(w_n^{-1}/u) = \log \Vert w_n^{-1}\Vert _u. \end{aligned}$$

Now set $v_n = w^{-1}_n/\Vert w_n^{-1}\Vert _u$ and note that by (2.3),

$$\begin{aligned} H(h',h)= & {} \lim _{n\rightarrow \infty } d_H(w_n,u) +h(w_n)\\= & {} \lim _{n\rightarrow \infty } \log \Vert w_n^{-1}\Vert _u + \log M(y/w_n) +\log M(z/w_n^{-1})\\= & {} \lim _{n\rightarrow \infty } \log M(y/w_n) +\log M(z/v_n^{-1}). \end{aligned}$$

Clearly $w_{n+1}\le w_n$ and $w_n\rightarrow y'$. Also,

$$\begin{aligned} w_n^{-1} = \sum _{i\in I'} \alpha _i^{-1}q_i +\sum _{j\in J'} n^2\beta _j q_j +\sum _{k\not \in I'\cup J'} nq_k. \end{aligned}$$

So, for all $n\ge 1$ large, we have that $\Vert w_n^{-1}\Vert _u =n^2$, as $\max _{j\in J}\beta _j = \Vert z'\Vert _u =1$. It follows that

$$\begin{aligned} v_n = \sum _{i\in I'} \frac{1}{n^2\alpha _i}q_i +\sum _{j\in J'}\beta _j q_j +\sum _{k\not \in I'\cup J'} \frac{1}{n}q_k \end{aligned}$$

for all $n\ge 1$ large. So, $v_{n+1}\le v_n$ for all $n\ge 1$ large and $v_n\rightarrow z'$. It now follows from Lemma 5.3 that $H(h',h)=\infty$ if $y'$ does not dominate y, or, $z'$ does not dominate z. Moreover, if $y'$ dominates y, and, $z'$ dominates z, then $H(h',h) = \log M(y/y') +\log M(z/z')$.

Interchanging the roles between h and $h'$ we find that $H(h,h') = \infty$ if y does not dominate $y'$, or, z does not dominate $z'$, and $H(h,h') = \log M(y'/y) +\log M(z'/z)$, otherwise. Thus, $\delta (h,h') = d_H(y,y')+d_H(z,z')$ if and only if $y\sim y'$ and $z\sim z'$, and $\delta (h,h')=\infty$ otherwise. $\square$

The characterisation of the parts and the detour distance is a more explicit description of the general one one given in [43, Theorem 4.9] in the case of symmetric Hilbert geometries.

5.3 The Homeomorphism

Let us now define a map $\varphi _H:{\overline{\Omega }}_V^h\rightarrow B^*_1$, where $B^*_1$ is the unit ball of the dual norm of $|\cdot |_u$ on $V/{\mathbb {R}}u$. For $x\in \Omega _V$ let

$$\begin{aligned} \varphi _H(x) = \frac{x}{{\textrm{tr}}(x)}-\frac{x^{-1}}{{\textrm{tr}}(x^{-1})}, \end{aligned}$$

and for $h\in \partial {\overline{\Omega }}_V^h$ given by (5.4) let

$$\begin{aligned} \varphi _H(h) = \frac{y}{{\textrm{tr}}(y)}-\frac{z}{{\textrm{tr}}(z)}. \end{aligned}$$

We note that $\varphi _H(h)$ is well-defined by Proposition 5.4.

We will prove the following theorem in the sequel.

Theorem 5.5

If $(\Omega _V,d_H)$ is a symmetric Hilbert geometry, then the map $\varphi _H:{\overline{\Omega }}_V^h\rightarrow B^*_1$ is a homeomorphism which maps each part of $\partial {\overline{\Omega }}_V^h$ onto the relative interior of a boundary face of $B^*_1$.

We first analyse the dual unit ball $B_1^*$ of $|\cdot |_u$ and its facial structure. The following fact, which can be found in [45, Sect. 2.2], will be useful.

Lemma 5.6

Given an order-unit space $(V,V_+,u)$, the norm $|\cdot |_u$ on $V/{\mathbb {R}}u$ coincides with the quotient norm of $2\Vert \cdot \Vert _u$ on $V/{\mathbb {R}}u$.

Recall that in a Euclidean Jordan algebra V each x has a unique orthogonal decomposition $x= x^+-x^-$, where $x^+$ and $x^-$ are orthogonal elements in $V_+$, see [4, Proposition 1.28]. Let

$$\begin{aligned} {\mathbb {R}}u^\perp =\{x\in V:(u|x) = 0\} = \{x\in V:{\textrm{tr}}(x^+) = {\textrm{tr}}(x^-)\}. \end{aligned}$$

It follows from Lemma 5.6 that

$$\begin{aligned} (V/{\mathbb {R}}u,|\cdot |_u)^* = ({\mathbb {R}}u^\perp ,\frac{1}{2}\Vert \cdot \Vert _u^*). \end{aligned}$$

So the dual unit ball $B_1^*$ in ${\mathbb {R}}u^\perp$ is given by

$$\begin{aligned} B_1^* =2{\textrm{conv}}(S(V)\cup -S(V))\cap {\mathbb {R}}u^\perp , \end{aligned}$$

see [3, Theorem 1.19], and its (closed) boundary faces are precisely the nonempty sets of the form

$$\begin{aligned} A_{p,q} = 2{\textrm{conv}}\, ( (U_p(V)\cap S(V))\cup (U_q(V)\cap -S(V)))\cap {\mathbb {R}}u^\perp , \end{aligned}$$

where p and q are orthogonal idempotents, see [18, Theorem 4.4].

To prove Theorem 5.5 we collect a number of preliminary results.

Lemma 5.7

For each $x\in \Omega _V$ we have that $\varphi _H(x)\in {\textrm{int}}\, B_1^*$, and for each $h\in \partial {\overline{\Omega }}_V^h$ we have that $\varphi _H(h)\in \partial B_1^*$.

Proof

Let $x = \sum _{i=1}^r \lambda _i p_i\in \Omega _V$, so $\lambda _i>0$ for all i. Note that $(u|\varphi _H(x)) = 1 -1 =0$ and hence $\varphi _H(x) \in {\mathbb {R}}u^\perp$. Given $-u\le z\le u$, we have the Peirce decomposition of z with respect to the frame $p_1,\ldots ,p_r$,

$$\begin{aligned} z = \sum _{i=1}^r \sigma _i p_i +\sum _{i<j}z_{ij}, \end{aligned}$$

with $-1 =-(u|p_i)\le \sigma _i = (z|p_i) \le (u|p_i) = 1$. As this is an orthogonal decomposition we have that

$$\begin{aligned} (z|\varphi _H(x))= & {} \sum _{i=1}^r\sigma _i\left( \frac{\lambda _i}{\sum _{j=1}^r \lambda _j} - \frac{\lambda ^{-1}_i}{\sum _{j=1}^r \lambda ^{-1}_j}\right) < \sum _{i=1}^r\left( \frac{\lambda _i}{\sum _{j=1}^r \lambda _j}\right) \\{} & {} + \sum _{i=1}^r\left( \frac{\lambda ^{-1}_i}{\sum _{j=1}^r \lambda ^{-1}_j}\right) = 2. \end{aligned}$$

This implies that $\frac{1}{2}\Vert \varphi _H(x)\Vert _u^* = \frac{1}{2}\sup _{-u\le z\le u} (z|\varphi _H(x))<1$, hence $\varphi _H(x)\in {\textrm{int}}\, B_1^*$.

To prove the second assertion let h be a horofunction given by $h(x) =\log M(y/x)+\log M(z/x^{-1})$, where $\Vert y\Vert _u=\Vert z\Vert _u =1$ and $(y|z)=0$. Write $y=\sum _{i\in I}\alpha _iq_i$ and $z=\sum _{j\in J}\beta _jq_j$. If we now let $q_I = \sum _{i\in I}q_i$ and $q_J = \sum _{j\in J}q_j$, then $-u\le q_I-q_J\le u$ and

$$\begin{aligned} \Vert \varphi _H(h)\Vert _u^*\ge \frac{1}{2}(q_I-q_J|\varphi _H(h)) = (1+1)/2=1. \end{aligned}$$

Moreover, for each $-u\le w\le u$ we have that

$$\begin{aligned} |(w|\varphi _H(h))| \le |(w|y/{\textrm{tr}}(y))|+|(w|z/{\textrm{tr}}(z))|\le (u|y/{\textrm{tr}}(y))+(u|z/{\textrm{tr}}(z)) =2. \end{aligned}$$

Combining the inequalities shows that $\varphi _H(h)\in \partial B_1^*$. $\square$

To prove injectivity of $\varphi _H$ on $\Omega _V$ we need the following lemma, which has a proof similar to the one of Lemma 3.6 given in [32, Sect. 4].

Lemma 5.8

Let $\mu _i:{\mathbb {R}}^r\rightarrow {\mathbb {R}}$, for $i=1,2$, be given by $\mu _1(x) = \sum _{i=1}^r e^{x_i}$ and $\mu _2(x) = \sum _{i=1}^r e^{-x_i}$ for $x\in {\mathbb {R}}^r$, and let $g:x\mapsto \log \mu _1(x)+\log \mu _2(x)$. If $x,y\in {\mathbb {R}}^r$ are such that $y\ne x+c(1,\ldots ,1)$ for all $c\in {\mathbb {R}}$, then $\nabla g(x)\ne \nabla g(y)$.

Lemma 5.9

The map $\varphi _H$ is injective on $\Omega _V$.

Proof

Suppose that $\varphi _H(x)=\varphi _H(y)$, where $x=\sum _{i=1}^r \lambda _i p_i$ and $y=\sum _{i=1}^r \mu _i q_i$ in $\Omega _V$. Note that $0<\lambda _i,\mu _i$ for all i and $(x|u)= {\textrm{tr}}(x) =r = {\textrm{tr}}(y)= (y|u)$. After possibly relabelling we can write

$$\begin{aligned} \varphi _H(x) = \sum _{i=1}^r \left( \frac{\lambda _i}{\sum _{j=1}^r \lambda _j} - \frac{\lambda ^{-1}_i}{\sum _{j=1}^r \lambda ^{-1}_j} \right) p_i =\sum _{i=1}^r \alpha _i p_i \end{aligned}$$

and

$$\begin{aligned} \varphi _H(y) = \sum _{i=1}^r \left( \frac{\mu _i}{\sum _{j=1}^r \mu _j} - \frac{\mu ^{-1}_i}{\sum _{j=1}^r \mu ^{-1}_j} \right) q_i =\sum _{i=1}^r \beta _i q_i, \end{aligned}$$

where $\alpha _1\le \ldots \le \alpha _r$ and $\beta _1\le \ldots \le \beta _r$. By the spectral theorem (version 2) [25] we conclude that $\alpha _i =\beta _i$ for all i.

Consider the injective map ${\textrm{Log}}:{\textrm{int}}\,{\mathbb {R}}_+^r\rightarrow {\mathbb {R}}^r$ given by ${\textrm{Log}}(\gamma )= (\log \gamma _1,\ldots ,\log \gamma _r)$. Let $\Delta = \{\gamma \in {\textrm{int}}\,{\mathbb {R}}^r_+:\sum _{i=1}^r \gamma _i=r\}$. The map $(\nabla g)\circ {\textrm{Log}}$ is injective on $\Delta$ by Lemma 5.8 and

$$\begin{aligned} \nabla g({\textrm{Log}}(\gamma )) = \left( \frac{\gamma _1}{\sum _{i=1}^r \gamma _i} - \frac{\gamma ^{-1}_1}{\sum _{i=1}^r \gamma ^{-1}_i}, \ldots , \frac{\gamma _r}{\sum _{i=1}^r \gamma _i} - \frac{\gamma ^{-1}_r}{\sum _{i=1}^r \gamma ^{-1}_i}\right) . \end{aligned}$$

Writing $\lambda =(\lambda _1,\ldots ,\lambda _r)$ and $\mu = (\mu _1,\ldots ,\mu _r)$, we have that $\lambda ,\mu \in \Delta$ and

$$\begin{aligned} \nabla g({\textrm{Log}}(\lambda )) =(\alpha _1,\ldots ,\alpha _r) = (\beta _1,\ldots ,\beta _r) = \nabla g({\textrm{Log}}(\mu )), \end{aligned}$$

so that $\lambda = \mu$.

As $(\nabla g)\circ {\textrm{Log}}$ is injective on $\Delta$, we also know that $\alpha _k=\alpha _{k+1}$ if and only if $\lambda _k =\lambda _{k+1}$. Likewise, $\beta _k=\beta _{k+1}$ if and only if $\mu _k =\mu _{k+1}$. From the spectral theorem (version 1) [25] we now conclude that $x=y$. $\square$

In the next couple of lemmas we show that $\varphi _H$ is onto.

Lemma 5.10

The map $\varphi _H$ maps $\Omega _V$ onto ${\textrm{int}}\, B_1^*$.

Proof

Note that $\Omega _V$ is an open set of the affine space $\{x\in V:{\textrm{tr}}(x) =r\}$, which has dimension $\dim V-1$. Also $B_1^*\subset {\mathbb {R}}u^\perp$ has dimension $\dim V - 1$. As $\varphi _H$ is a continuous injection from $\Omega _V$ into ${\textrm{int}}\, B_1^*$ by Lemmas 5.7 and 5.9, we know that $\varphi _H(\Omega _V)$ is a open subset of ${\textrm{int}}\, B_1^*$ by Brouwer’s invariance of domain theorem. We now argue by contradiction. So, suppose that $\varphi _V(\Omega _V)\ne {\textrm{int}}\, B_1^*$. There then exists a $w\in \partial \varphi _H(\Omega _V)\cap {\textrm{int}}\, B_1^*$. Let $(v_n)$ in $\Omega _V$ be such that $\varphi _H(v_n)\rightarrow w$.

As $\varphi _H$ is continuous on $\Omega _V$, we may assume that $d_H(v_n,u)\rightarrow \infty$. After taking a subsequence, we may also assume that $v_n\rightarrow v\in \partial \Omega _V$. Now let $y_n = v_n/\Vert v_n\Vert _u$ and set $y = v/\Vert v\Vert _u$. Furthermore, let $z_n = y_n^{-1}/\Vert y_n^{-1}\Vert _u$. After taking subsequences we may assume that $z_n\rightarrow z\in \partial V_+$ and $y_n\rightarrow y\in \partial V_+$, so $\Vert y\Vert _u=\Vert z\Vert _u=1$. As $y_n\bullet z_n = u/\Vert y_n^{-1}\Vert _u \rightarrow 0$, we find that $y\bullet z = 0$, which implies that $(y|z) = 0$.

Using the spectral decomposition we write $y_n =\sum _{i=1}^r \lambda _i^np_i^n$ and $y =\sum _{i\in I}\lambda _ip_i$, where $\lambda _i>0$ for all $i\in I$. Likewise, we let $z_n = \sum _{i=1}^r \mu _i^np_i^n$ and $z =\sum _{j\in J}\mu _jp_j$ with $\mu _j>0$ for all $j\in J$. Note that $\mu _i^n = (\lambda _i^n)^{-1}/\Vert y^{-1}_n\Vert _u$.

Then

$$\begin{aligned} \varphi _h(v_n)= & {} \frac{\sum _{i=1}^r \lambda _i^np_i^n}{\sum _{k=1}^r \lambda _k^n} - \frac{\sum _{i=1}^r(\lambda _i^n)^{-1}p_i^n}{\sum _{k=1}^r (\lambda _k^n)^{-1}} = \frac{\sum _{i=1}^r \lambda _i^np_i^n}{\sum _{k=1}^r \lambda _k^n} - \frac{\sum _{i=1}^r \mu _i^np_i^n}{\sum _{k=1}^r \mu ^n_k}\rightarrow \frac{\sum _{i\in I} \lambda _ip_i}{\sum _{k\in I} \lambda _k} \\{} & {} - \frac{\sum _{j\in J} \mu _jp_j}{\sum _{k\in J} \mu _j} = w. \end{aligned}$$

Now let $w^* = \sum _{i\in I}p_i - \sum _{j\in J}p_j$ and note that $-u\le w^*\le u$, as $(y|z)=0$. We find that

$$\begin{aligned} \frac{1}{2}\Vert w\Vert ^*_u \ge \frac{1}{2}(w|w^*) = (1+1)/2 =1, \end{aligned}$$

hence $w\in \partial B_1^*$, which is a contradiction. $\square$

Lemma 5.11

The map $\varphi _H$ maps $\partial {\overline{\Omega }}_V^h$ onto $\partial B_1^*$.

Proof

We know from Lemma 5.7 that $\varphi _H$ maps $\partial {\overline{\Omega }}_V^h$ into $\partial B_1^*$. To prove that it is onto let $w\in \partial B_1^*$. Then there exists a face, say

$$\begin{aligned} A_{p,q} = 2{\textrm{conv}}\, ( (U_p(V)\cap S(V))\cup (U_q(V)\cap -S(V)))\cap {\mathbb {R}}u^\perp \end{aligned}$$

where p and q are orthogonal idempotents, such that w is in the relative interior of $A_{p,q}$, as $B_1^*$ is the disjoint union of the relative interiors of its faces [52, Theorem 18.2]. So,

$$\begin{aligned} w= \sum _{i\in I} \alpha _i p_i -\sum _{j\in J}\beta _j q_j, \end{aligned}$$

where $\alpha _i>0$ for all $i\in I$, $\beta _j>0$ for all $j\in J$, and $\sum _{i\in I}\alpha _i +\sum _{j\in J} \beta _j =2$. Moreover, $\sum _{i\in I}p_i = p$ and $\sum _{j\in J}q_j =q$.

As $w\in {\mathbb {R}}u^\perp$, we have that $0=(u|w) = \sum _{i\in I}\alpha _i -\sum _{j\in J} \beta _j$, hence $\sum _{i\in I}\alpha _i =\sum _{j\in J} \beta _j =1$.

Put $\alpha ^* =\max _{i\in I} \alpha _i$ and $\beta ^*=\max _{j\in J} \beta _j$. Furthermore, for $i\in I$ set $\lambda _i = \alpha _i/\alpha ^*$ and for $j\in J$ set $\mu _j= \beta _j/\beta ^*$. Then

$$\begin{aligned} w = \left( \frac{\sum _{i\in I} \alpha _i p_i}{\sum _{k\in I} \alpha _k}\right) - \left( \frac{\sum _{j\in J} \beta _jq_j}{\sum _{k\in J} \beta _k}\right) = \left( \frac{\sum _{i\in I} \lambda _i p_i}{\sum _{k\in I} \lambda _k}\right) - \left( \frac{\sum _{j\in J} \mu _jq_j}{\sum _{k\in J} \mu _k}\right) . \end{aligned}$$

Note that $0<\lambda _i\le 1$ for all $i\in I$ and $\max _{i\in I}\lambda _i =1$. Likewise, $0<\mu _j\le 1$ for all $j\in J$ and $\max _{j\in J}\beta _j =1$.

Now let $y = \sum _{i\in I}\lambda _i p_i$ and $z=\sum _{j\in J} \mu _jq_j$. Then $\Vert y\Vert _ u =\Vert z\Vert _u =1$ and $(y|z) = 0$. Furthermore, if we let $h:\Omega _V\rightarrow {\mathbb {R}}$ be given by

$$\begin{aligned} h(x) = \log M(y/x) +\log M(z/x^{-1}) \end{aligned}$$

for $x\in \Omega _V$, then h is a horofunction by Theorem 5.2 and $\varphi _H(h)=w$, which completes the proof. $\square$

We already saw in Lemma 5.10 that $\varphi _H$ is injective on $\Omega _V$. The next lemma shows that $\varphi _H$ is injective on ${\overline{\Omega }}_V^h$.

Lemma 5.12

The map $\varphi _H:{\overline{\Omega }}_V^h\rightarrow B_1^*$ is injective.

Proof

We know from Lemma 5.10 that $\varphi _H$ is injective on $\Omega _V$. So, it remains to show that if $h,h'\in \partial {\overline{\Omega }}_V^h$ and $\varphi _H(h)=\varphi _H(h')$, then $h=h'$.

Suppose $h(x) =\log M(y/x) +\log M(z/x^{-1})$ and $h'(x)= \log M(y'/x)+\log M(z'/x^{-1})$ for all $x\in \Omega _V$. Then

$$\begin{aligned} \varphi _H(h) = \frac{y}{{\textrm{tr}}(y)} - \frac{z}{{\textrm{tr}}(z)} = \frac{y'}{{\textrm{tr}}(y')} - \frac{z'}{{\textrm{tr}}(z')} =\varphi _H(h'). \end{aligned}$$

Using the fact that the orthogonal decomposition of an element in V is unique, see [4, Proposition 1.26], we conclude that

$$\begin{aligned} \frac{y}{{\textrm{tr}}(y)} = \frac{y'}{{\textrm{tr}}(y')} \quad \text{ and } \quad \frac{z}{{\textrm{tr}}(z)} = \frac{z'}{{\textrm{tr}}(z')}. \end{aligned}$$

As $\Vert y\Vert _u=\Vert y'\Vert _u=1$, we get that ${\textrm{tr}}(y)={\textrm{tr}}(y')$, and hence $y=y'$. Likewise, $\Vert z\Vert _u=\Vert z'\Vert _u=1$ implies that $z=z'$, hence $h=h'$. $\square$

5.4 Proof of Theorem 5.5

Before we prove Theorem 5.5, we recall a fact from Jordan theory. For $x,z\in V$ let $[x,z]=\{y\in V:x\le y\le z\}$ be the order-interval. Given $y\in V_+$ we write

$$\begin{aligned} {\textrm{face}}(y) = \{x\in V_+:x\le \lambda y \text{ for } \text{ some } \lambda \ge 0\}. \end{aligned}$$

In a Euclidean Jordan algebra V every idempotent p satisfies

$$\begin{aligned} {\textrm{face}}(p) \cap [0,u] =[0,p], \end{aligned}$$

see [4, Lemma 1.39]. Also note that $y\sim y'$ if and only if ${\textrm{face}}(y) ={\textrm{face}}(y')$.

Proof of Theorem 5.5

We know from the results in the previous subsection that $\varphi _H:{\overline{\Omega }}_V^h\rightarrow B_1^*$ is a bijection, which is continuous on $\Omega _V$.

To prove continuity of $\varphi _H$ on the whole of ${\overline{\Omega }}_V^h$ we first show that if $(v_n)$ in $\Omega _V$ is such that $h_{v_n}\rightarrow h\in \partial {\overline{\Omega }}_V^h$, then $\varphi _H(v_n)\rightarrow \varphi _H(h)$. Let $h(x) = \log M(y/x)+\log M(z/x^{-1})$ for $x\in \Omega _V$, where $\Vert y\Vert _u=\Vert z\Vert _u =1$ and $(y|z)=0$. For $n\ge 1$ let $y_n =v_n/\Vert v_n\Vert _u$ and note that $\varphi _H(v_n) = \varphi _H(y_n)$ for all n. Let $w_k = \varphi _H(v_{n_k})$, $k\ge 1$ be a subsequence of $(\varphi _H(v_n))$. We need to show that $(w_k)$ has a subsequence that converges to $\varphi _H(h)$.

As h is a horofunction and $(\Omega _V,d_H)$ is a proper metric space, $d_H(v_n,u)=d_H(y_n,u)\rightarrow \infty$ by Lemma 2.1. It follows that $(y_{n_k})$ has a subsequence $(y_{k_m})$ with $y_{k_m}\rightarrow y'\in \partial V_+$ and $z_{k_m} = y^{-1}_{k_m}/\Vert y^{-1}_{k_m}\Vert _u \rightarrow z'\in V_+$. Note that as $y\in \partial V_+$, we have that $\Vert y_{k_m}^{-1}\Vert _u\rightarrow \infty$. This implies that

$$\begin{aligned} y'\bullet z' =\lim _{m\rightarrow \infty } y_{k_m}\bullet \frac{y_{k_m}^{-1}}{\Vert y_{k_m}^{-1}\Vert _u} = \lim _{m\rightarrow \infty } \frac{u}{\Vert y_{k_m}^{-1}\Vert _u}=0, \end{aligned}$$

hence $(y'|z')=0$ (see [25, III, Exercise 3.3]) and $z'\in \partial V_+$. For $x\in \Omega _V$,

$$\begin{aligned} \lim _{m\rightarrow \infty } h_{y_{k_m}}(x)= & {} \lim _{m\rightarrow \infty } \log M(y_{k_m}/x) +\log M(x/y_{k_m}) - \log M(y_{k_m}/u)- \log M(u/y_{k_m})\\= & {} \lim _{m\rightarrow \infty }\log M(y_{k_m}/x) +\log M(y^{-1}_{k_m}/x^{-1}) - \log \Vert y^{-1}_{k_m}\Vert _u\\= & {} \lim _{m\rightarrow \infty }\log M(y_{k_m}/x) +\log M(z_{k_m}/x^{-1}) \\= & {} \log M(y'/x)+\log M(z'/x^{-1}). \end{aligned}$$

So, if we let $h'(x)= \log M(y'/x)+\log M(z'/x^{-1})$, then $h'$ is a horofunction by Theorem 5.2 and $h_{y_{k_m}}\rightarrow h'$. As $h=h'$, we know that $\delta (h,h') = d_H(y,y') +d_H(z,z') =0$, hence $y=y'$ and $z=z'$. It follows that

$$\begin{aligned} \varphi _H(v_{k_m})=\varphi _H(y_{k_m}) = \frac{y_{k_m}}{{\textrm{tr}}(y_{k_m})} - \frac{y^{-1}_{k_m}}{{\textrm{tr}}(y^{-1}_{k_m})}\rightarrow \frac{y}{{\textrm{tr}}(y)} -\frac{z}{{\textrm{tr}}(z)}= \varphi _H(h). \end{aligned}$$

Recall that $\varphi _H$ maps $\Omega _V$ into ${\textrm{int}}\, B_1^*$ and $\varphi _H$ maps $\partial {\overline{\Omega }}_V^h$ into $\partial B_1^*$ by Lemma 5.7. So, to prove the continuity of $\varphi _H$ it remains to show that if $(h_n)$ is a sequence in $\partial {\overline{\Omega }}_V^h$ converging to $h\in \partial {\overline{\Omega }}_V^h$, then $\varphi _H(h_n)\rightarrow \varphi _H(h)$.

Let $(\varphi _H(h_{n_k}))$ be a subsequence of $(\varphi _H(h_n))$. We show that it has a subsequence $(\varphi _H(h_{k_m}))$ converging to $\varphi _H(h)$. We know there exists $v_m,w_m\in \partial V_+$, with $\Vert v_m\Vert _u=\Vert w_m\Vert _u =1$ and $(v_m|w_m) =0$ such that

$$\begin{aligned} h_{k_m}(x) = \log M(v_m/x)+\log M(w_m/x^{-1}) \end{aligned}$$

for $x\in \Omega _V$. By taking a further subsequence we may assume that $v_m\rightarrow v\in \partial V_+$ and $w_m\rightarrow w\in \partial V_+$. Then $\Vert v\Vert _u=\Vert w\Vert _u=1$ and $(v|w)=0$. Moreover,

$$\begin{aligned} \log M(v_m/x)\rightarrow \log M(v/x)\quad \text{ and } \quad \log M(w_m/x^{-1})\rightarrow \log M(w/x^{-1}) \end{aligned}$$

for each $x\in \Omega _V$, as $y\mapsto M(y/x)$ is continuous on V, see [44, Lemma 2.2]. Thus, $h_{k_m}\rightarrow h^*\in \partial {\overline{\Omega }}_V^h$, where

$$\begin{aligned} h^*(x) = \log M(v/x)+\log M(w/x^{-1}), \end{aligned}$$

by Theorem 5.2. As $h_n\rightarrow h$, we have that $h=h^*$. This implies that $y=v$ and $z=w$ by Proposition 5.4. Thus, $v_m\rightarrow y$ and $w_m\rightarrow z$, hence

$$\begin{aligned} \varphi _H(h_{k_m}) = \frac{v_m}{{\textrm{tr}}(v_m)} -\frac{w_m}{{\textrm{tr}}(w_m)}\rightarrow \frac{y}{{\textrm{tr}}(y)} -\frac{z}{{\textrm{tr}}(z)}=\varphi _H(h). \end{aligned}$$

This completes the proof of the continuity of $\varphi _H$.

As $\varphi _H:{\overline{\Omega }}_V^h\rightarrow B_1^*$ is a continuous bijection, ${\overline{\Omega }}_V^h$ is compact, and $B_1^*$ is Hausdorff, we conclude that $\varphi _H$ is a homeomorphism.

To prove the second assertion let $h(x) =\log M(y/x)+\log M(z/x^{-1})$ be a horofunction, where $y =\sum _{i\in I}\lambda _i p_i$ and $z=\sum _{j\in J}\mu _j p_j$ with $\lambda _i,\mu _j>0$ for all $i\in I$ and $j\in J$. Let $p_I=\sum _{i\in I} p_i$ and $p_J=\sum _{j\in J}p_j$. As $\varphi _H$ is surjective, it suffices to show that $\varphi _H$ maps ${\mathcal {P}}_h$ into the relative interior of

$$\begin{aligned} A_{p_I,p_J} = 2{\textrm{conv}}\, ( (U_{p_I}(V)\cap S(V))\cup (U_{p_J}(V)\cap -S(V)))\cap {\mathbb {R}}u^\perp . \end{aligned}$$

So, let $h'\in {\mathcal {P}}_h$ where $h'(x) = \log M(y'/x)+\log M(z'/x^{-1})$ for $x\in \Omega _V$. Then $p_I\sim y\sim y'$ and $p_J\sim z\sim z'$. Using the spectral decomposition write $y'=\sum _{i\in I'} \alpha _iq_i$ and $z'=\sum _{j\in J'} \beta _jq_j$, where $\alpha _i>0$ for all $i\in I'$ and $\beta _j>0$ for all $j\in J'$. Now let $q_{I'} = \sum _{i\in I'} q_i$ and $q_{J'} = \sum _{j\in J'} q_j$. It follows that $p_I\sim q_{I'}$ and $p_J \sim q_{J'}$. So, ${\textrm{face}}(p_I)={\textrm{face}}(q_{I'})$ and ${\textrm{face}}(p_J) ={\textrm{face}}(q_{J'})$. As ${\textrm{face}}(p_I) \cap [0,u] = [0,p_I]$ and ${\textrm{face}}(q_{I'}) \cap [0,u] = [0,q_{I'}]$ by [4, Lemma 1.39], we conclude that $p_I =q_{I'}$. In the same way we get that $p_J = q_{J'}$. As $\alpha _i>0$ for all $i\in I'$ and $\beta _j>0$ for all $j\in J'$, we have that

$$\begin{aligned} \varphi _H(h')= \frac{y'}{{\textrm{tr}}(y')} - \frac{z'}{{\textrm{tr}}(z')} \end{aligned}$$

is in the relative interior of $A_{q_{I'},q_{J'}} = A_{p_I,p_J}$. $\square$

6 Final Remarks

Besides the problem posed by Kapovich and Leeb [34, Question 6.18] for finite dimensional normed spaces the results in this paper show that there should be milage in analysing the following problem.

Problem 6.1

Suppose $X=G/K$ is a noncompact type symmetric space with a G-invariant Finsler metric. When does there exist a homeomorphism between the horofunction compactification of X with basepoint b under the Finsler distance, and the closed dual unit ball $B_1^*$ of the Finsler metric in the tangent space at b, which maps each part in the horofunction boundary onto the relative interior of a boundary face of $B_1^*$?

In a sequel to this paper [41] the first author has shown for various classes of noncompact type symmetric spaces X with invariant Finsler distances coming from symmetric cones that the exponential map ${\textrm{exp}}_b$ from the tangent space $T_b$ at the basepoint b onto X extends as a homeomorphism between the horofunction compactification of $T_b$ as a normed space under the Finsler metric and the horofunction compactification of X under the Finsler distance. Moreover, the extension of the exponential map preserves the parts in the horofunction boundaries. In particular, this is true for symmetric Hilbert geometries $(\Omega _V,d_H)$ and the normed spaces $(V/{\mathbb {R}}u,|\cdot |_u)$. It would be interesting to know if this is true for all noncompact type symmetric spaces with invariant Finsler metrics.