Standard Subspaces of Hilbert Spaces of Holomorphic Functions on Tube Domains

Abstract

In this article we study standard subspaces of Hilbert spaces of vector-valued holomorphic functions on tube domains $E + i C^0$, where $C \subseteq E$ is a pointed generating cone invariant under $e^{{{\mathbb {R}}}h}$ for some endomorphism $h \in \mathop {\mathrm{End}}\nolimits (E)$, diagonalizable with the eigenvalues $1,0,-1$ (generalizing a Lorentz boost). This data specifies a wedge domain $W(E,C,h) \subseteq E$ and one of our main results exhibits corresponding standard subspaces as being generated using test functions on these domains. We also investigate aspects of reflection positivity for the triple $(E,C,e^{\pi i h})$ and the support properties of distributions on E, arising as Fourier transforms of operator-valued measures defining the Hilbert spaces $\mathcal {H}$. For the imaginary part of these distributions, we find similarities to the well known Huygens’ principle, relating to wedge duality in the Minkowski context. Interesting examples are the Riesz distributions associated to euclidean Jordan algebras.

Appendices

A Standard subspaces

In this appendix we collect some facts about standard subspaces $\mathtt{V}\subseteq \mathcal {H}$. In particular we describe the connection to antiunitary representations of the multiplicative group ${{\mathbb {R}}}^\times$, and the connection to KMS conditions and modular objects. Most of the material in this section is standard and well known. We refer to [Lo08] for the basic theory of standard subspaces, other references are [NÓ17, NÓ19]. Proofs are sometimes included for the sake of clarity of exposition.

1.1 A.1 Standard subspaces and antiunitary representations

Definition A.1

A closed real subspace $\mathtt{V}$ of a complex Hilbert space $\mathcal {H}$ is called standard if

$$\begin{aligned} \mathtt{V}\cap i \mathtt{V}= \{0\}\quad \text{ and } \quad \mathcal {H}= \overline{\mathtt{V}+ i \mathtt{V}}. \end{aligned}$$

(A.1)

If $\mathtt{V}\subseteq \mathcal {H}$ is a standard subspace, then

$$\begin{aligned} T_\mathtt {V}\, :\,\mathcal {D}(T_\mathtt {V}) := \mathtt {V}+ i \mathtt {V}\rightarrow \mathcal {H}, \quad x + i y \mapsto x- iy \end{aligned}$$

(A.2)

defines a closed operator with $\mathtt{V}= \mathop {\mathrm{Fix}}\nolimits (T_\mathtt{V})$. It is called the Tomita operator of $\mathtt{V}$. Its polar decomposition can be written as $T_\mathtt{V}= J_\mathtt{V}\Delta _\mathtt{V}^{1/2}$, where $J_\mathtt{V}$ is a conjugation (an antiunitary involution) and $\Delta _\mathtt{V}$ is a positive selfadjoint operator such that the modular relation

$$\begin{aligned} J_\mathtt{V}\Delta _\mathtt{V}J_\mathtt{V}= \Delta _\mathtt{V}^{-1} \end{aligned}$$

(A.3)

holds. We call $(\Delta _\mathtt{V}, J_\mathtt{V})$ the pair of modular objects associated to $\mathtt{V}$.

Denote the inner product on $\mathcal {H}$ by $\langle \cdot ,\cdot \rangle$ and let $\omega (u,v)={\text {Im}} \langle u,v \rangle$. Then $\omega$ is a symplectic form on $\mathcal {H}$. For a real subspace $\mathtt{W}\subset \mathcal {H}$ let

$$\begin{aligned} \mathtt {W}^\prime =\{v\in \mathcal {H}\,:\,(\forall w\in \mathtt {W})\, \omega (w,v)=0\}. \end{aligned}$$

Then $\mathtt{W}^\prime$ is also a real subspace and $\mathtt{W}^{\prime \prime } = {\overline{\mathtt{W}}}$, the closure of $\mathtt{W}$.

In the following lemma we collect several properties of standard subspaces that will be used in this article:

Lemma A.2

Let $\mathtt{V},\mathtt{V}_1,\mathtt{V}_2$ be standard subspaces. Then the following assertions hold:

• (i) $\mathtt{V}_1\subseteq \mathtt{V}_2$ implies $\mathtt{V}_2^\prime \subseteq \mathtt{V}_1^\prime$.

• (ii) $\langle \xi ,J_\mathtt{V}\xi \rangle \ge 0$ for all $\xi \in \mathtt{V}$.

• (iii) $\mathtt{V}$ is standard if and only if $\mathtt{V}^\prime$ is standard.

• (iv) $J_\mathtt{V}= J_{\mathtt{V}^\prime }$ and $\Delta _{\mathtt{V}^\prime } = \Delta _{\mathtt{V}}^{-1}$.

• (v) $\mathtt{V}=\mathtt{V}^\prime$ if and only if $\Delta _\mathtt{V}=\mathbf {1}$.

• (vi) $J_\mathtt{V}\mathtt{V}=\mathtt{V}^\prime$.

• (vii) $(\mathtt{V}^\prime )^\prime = \mathtt{V}$.

Proof

(i) is obvious.

(ii) Let $\xi \in \mathtt{V}$. Then $\xi = T_\mathtt{V}(\xi ) = J_\mathtt{V}\Delta _\mathtt{V}^{1/2} \xi$ implies that $\Delta ^{1/2}_\mathtt{V}\xi = J_\mathtt{V}\xi$. As $\Delta ^{1/2}_\mathtt{V}$ is positive selfadjoint, it follows that $\langle \xi ,J_\mathtt{V}\xi \rangle \ge 0$.

(iii) follows from [Lo08, § 3.1] and (iv) is contained in [Lo08, Prop. 3.3].

(v) follows from (iii), the fact that the pair $(\Delta _\mathtt{V}, J_\mathtt{V})$ determines $\mathtt{V}$ and the observation that $\Delta _\mathtt{V}= \Delta _\mathtt{V}^{-1}$ is equivalent to $\Delta _\mathtt{V}= \mathbf {1}$.

(vi) As $\langle \xi ,J_\mathtt{V}\xi \rangle$ is real by (ii), it follows that $J_\mathtt{V}\mathtt{V}\subseteq \mathtt{V}^\prime$. Applying this argument to $\mathtt{V}'$ and using (iii), we also obtain $J_\mathtt{V}\mathtt{V}' \subseteq \mathtt{V}$, so that (v) follows from the fact that $J_\mathtt{V}$ is an involution.

(vii) follows from (iv) which entails $J_{\mathtt{V}''} = J_\mathtt{V}$ and $\Delta _{\mathtt{V}''} = \Delta _\mathtt{V}$. $\quad \square$

We have already seen that every standard subspace $\mathtt{V}$ determines a pair $(\Delta _\mathtt{V}, J_\mathtt{V})$ of modular objects and that $\mathtt{V}$ can be recovered from this pair by $\mathtt{V}= \mathop {\mathrm{Fix}}\nolimits (J_\mathtt{V}\Delta _\mathtt{V}^{1/2})$. This observation can be used to obtain a representation theoretic parametrization of the set of standard subspaces of $\mathcal {H}$ (cf. [BGL02, NÓ17]): Each standard subspace $\mathtt{V}$ specifies a homomorphism $U^\mathtt{V}: {{\mathbb {R}}}^\times \rightarrow \mathop {\mathrm{AU}}\nolimits (\mathcal {H})$ by

$$\begin{aligned} U^\mathtt{V}(e^t) := \Delta _\mathtt{V}^{-it/2\pi } =e^{it H_\mathtt{V}}, \quad U^\mathtt{V}(-1) := J_\mathtt{V}, \quad \text {where } H_\mathtt{V}=-\frac{1}{2\pi } \log \Delta _\mathtt{V}. \end{aligned}$$

(A.4)

Theorem A.3

The map $\mathtt{V}\mapsto U^\mathtt{V}$ defines a bijection between standard subspaces and antiunitary representations of the graded group $({{\mathbb {R}}}^\times , \varepsilon _{{{\mathbb {R}}}^\times })$. The inverse is given by assigning to the antiunitary representation $U \,:\, {{\mathbb {R}}}^\times \rightarrow \mathop {\mathrm {AU}}\nolimits (\mathcal {H})$ the operators

$$\begin{aligned} H =- i \frac{d}{dt}\Big |_{t = 0} U(e^t), \quad \Delta := e^{-2\pi H}, \quad \text {and}\quad J := U(-1). \end{aligned}$$

Lemma A.4

Let $\mathtt{V}$ be a standard subspace. Then the following assertions hold:

• (a) $U^\mathtt{V}(e^t)\mathtt{V}=\mathtt{V}$ for all $t\in {{\mathbb {R}}}$.

• (b) $U^{\mathtt{V}^\prime }(r)=U^{\mathtt{V}}(r^{-1})$ for $r \in {{\mathbb {R}}}^\times$.

• (c) $\mathtt{V}\cap \mathtt{V}^\prime =\mathcal {H}^{U^\mathtt{V}}$.

Proof

(a) Let $\xi \in \mathtt{V}$ and $t\in {{\mathbb {R}}}$. Then

$$\begin{aligned} T_\mathtt{V}(U^\mathtt{V}(e^t)\xi )= J_\mathtt{V}\Delta _\mathtt{V}^{\frac{1}{2}-it/2\pi }\xi = \Delta _\mathtt{V}^{-it/2\pi }(J_\mathtt{V}\Delta _\mathtt{V}^{1/2}\xi )= U^\mathtt{V}(e^t)T_\mathtt{V}\xi = U^{\mathtt{V}}(e^t)\xi . \end{aligned}$$

(b), (c) follow from [NÓ17, Lemma 3.7]. $\quad \square$

Definition A.5

Let $\mathtt{V}\subseteq \mathcal {H}$ be a real subspace and J be a conjugation on $\mathcal {H}$. We say that $\mathtt{V}$ is J-positive if $\langle \xi , J \xi \rangle \ge 0$ for $\xi \in \mathtt{V}$.

Recall that a conjugation on $\mathcal {H}$ is an antiunitary involution. The following lemma explores the question when the positivity of a conjugation J on a real subspace $\mathtt{V}$ implies that $\mathtt{V}$ is standard with $J=J_\mathtt{V}$.

Lemma A.6

For a closed real subspace $\mathtt{V}\subseteq \mathcal {H}$ and a conjugation J, the following assertions hold:

• (i) If $\mathtt{V}$ is J-positive, then $J\mathtt{V}\subseteq \mathtt{V}^\prime$.

• (ii) If $\mathtt{V}+ i \mathtt{V}$ is dense in $\mathcal {H}$ and $J\mathtt{V}\subseteq \mathtt{V}^\prime$, then $\mathtt{V}\cap i\mathtt{V}=\{0\}$.

• (iii) Assume that $\mathtt{V}$ is standard. Then the following are equivalent

  • (a) $J=J_\mathtt{V}$.

  • (b) $\mathtt{V}^\prime$ is J-positive and $J\mathtt{V}\subseteq \mathtt{V}^\prime$.

  • (c) $\mathtt{V}$ and $\mathtt{V}^\prime$ are both J-positive.

Proof

(i) The form $\beta (\xi ,\eta ) := \langle J\xi , \eta \rangle$ on $\mathcal {H}$ is complex bilinear and symmetric. That $\mathtt{V}$ is J-positive implies that $\beta$ is real on all pairs $(\xi ,\xi )$, $\xi \in \mathtt{V}$, hence by polarization also on $\mathtt{V}\times \mathtt{V}$. This means that $J\mathtt{V}\subseteq \mathtt{V}'$.

(ii) The subspace $\mathtt{V}_0 := \mathtt{V}\cap i \mathtt{V}$ of $\mathcal {H}$ is complex and satisfies $J\mathtt{V}_0 \subseteq \mathtt{V}'$. Since $J\mathtt{V}_0$ is also a complex subspace, it follows that $J\mathtt{V}_0$ is orthogonal to the total subset $\mathtt{V}$, hence trivial.

(iii) That (a) implies (b),(c) follows from Lemma A.2(ii),(iv),(vi). Further, (b) implies $J\mathtt{V}' \supseteq JJ\mathtt{V}= \mathtt{V}$, so that the J-positivity of $\mathtt{V}'$ implies by [Lo08, Prop. 3.9] that $J = J_{\mathtt{V}'} = J_\mathtt{V}$, hence (a). If (c) holds, then (i) shows that the J-positivity of $\mathtt{V}$ implies $J\mathtt{V}\subseteq \mathtt{V}'$. Hence (c) implies (b). This proves (iii). $\quad \square$

Proposition A.7

(Reflection positivity and standard subspaces) Let $\mathtt{V}\subseteq \mathcal {H}$ be a standard subspace with modular objects $(\Delta , J)$. Then the following assertions hold:

• (i) $(\mathcal {E},\mathcal {E}_+, \theta ) := (\mathcal {H}^{{\mathbb {R}}}, \mathtt{V},J_\mathtt{V})$ is a real reflection positive Hilbert space.⁴

• (ii) The map $\Delta ^{1/4} \, :\,\mathtt {V}\rightarrow \mathcal {H}^J$ extends to an isometric isomorphism ${{\widehat{\mathtt {V}}}} \rightarrow \mathcal {H}^J$, where ${{\widehat{\mathtt {V}}}}$ is the completion of $\mathtt{V}$ with respect to scalar product $\langle v,w \rangle _J := \langle v, J w \rangle$ for $v,w \in \mathtt{V}$.

Proof

(i) follows directly from $\langle v, J v \rangle = \langle v, \Delta ^{1/2} v \rangle = \Vert \Delta ^{1/4} v\Vert ^2.$

(ii) Next we note that $\mathtt{V}\subseteq \mathcal {D}(\Delta ^{1/2}) \subseteq \mathcal {D}(\Delta ^{1/4})$ implies that $\Delta ^{1/4}$ is defined on $\mathtt{V}$. For $v \in \mathtt{V}$, we have

$$\begin{aligned} J \Delta ^{1/4}v = \Delta ^{-1/4} Jv = \Delta ^{-1/4} \Delta ^{1/2}v = \Delta ^{1/4}v, \end{aligned}$$

so that $\Delta ^{1/4}\mathtt{V}\subseteq \mathcal {H}^J$. Using the spectral decomposition of $\Delta$ it follows easily that $\Delta ^{1/4}\mathtt{V}$ is dense in $\mathcal {H}^J$. This implies (ii). $\quad \square$

The following simple observation is taken from [MN21]. It slightly extends [Lo08, Prop. 3.10].

Proposition A.8

Suppose that $\mathtt{V}_1 \subseteq \mathtt{V}_2$ are standard subspaces of $\mathcal {H}$. If

• (a) $\Delta _{\mathtt{V}_2}^{it} \mathtt{V}_1 = \mathtt{V}_1$ for $t \in {{\mathbb {R}}}$, or

• (b) $\Delta _{\mathtt{V}_1}^{it} \mathtt{V}_2 = \mathtt{V}_2$ for $t \in {{\mathbb {R}}}$,

then $\mathtt{V}_1 = \mathtt{V}_2$.

Proof

That (a) implies $\mathtt{V}_1 = \mathtt{V}_2$ follows from [Lo08, Prop. 3.10]. From (b) we obtain by dualization $\mathtt{V}_2^\prime \subseteq \mathtt{V}_1^\prime$ with $\Delta _{\mathtt{V}_1^\prime }^{it} \mathtt{V}_2^\prime = \mathtt{V}_2'$ for $t \in {{\mathbb {R}}}$, so that we obtain $\mathtt{V}_1^\prime = \mathtt{V}_2^\prime$ with (a), hence $\mathtt{V}_1 = \mathtt{V}_2$ also holds in this case. $\quad \square$

1.2 A.2 Standard subspaces and the KMS condition

As mentioned above, the bijection in Theorem A.3 is closely related to the characterization of standard subspaces and their modular objects in terms of a KMS condition ([Lo08, NÓ19]).

Definition A.9

Let V be a real vector space and $\mathop {\mathrm{Bil}}\nolimits (V)$ be the space of real bilinear maps $V \times V \rightarrow {{\mathbb {C}}}$. A function $\psi \,:\, {{\mathbb {R}}}\rightarrow \mathop {\mathrm {Bil}}\nolimits (V)$ is said to be positive definite if the kernel $\psi (t-s)(v,w)$ on the product set ${{\mathbb {R}}}\times V$ is positive definite.

We say that a positive definite function $\psi \,:\, {{\mathbb {R}}}\rightarrow \mathop {\mathrm {Bil}}\nolimits (V)$ satisfies the KMS condition for $\beta > 0$ if $\psi$ extends to a function $\overline{\mathcal {S}_\beta } \rightarrow \mathop {\mathrm{Bil}}\nolimits (V)$ which is pointwise continuous and pointwise holomorphic on the interior $\mathcal {S}_\beta$, and satisfies

$$\begin{aligned} \psi (i \beta +t) = \overline{\psi (t)}\quad \text{ for } \quad t \in {{\mathbb {R}}}. \end{aligned}$$

(A.5)

In a similar fashion as Lemma A.6(iv) characterizes the conjugation $J_\mathtt{V}$ of a standard subspace $\mathtt{V}$ in terms of the J-positivity of $\mathtt{V}$ and $\mathtt{V}'$, the following proposition characterizes the corresponding modular group in terms of a KMS condition.

Proposition A.10

Let $\mathtt{V}\subseteq \mathcal {H}$ be a standard subspace and $U \,:\, {{\mathbb {R}}}\rightarrow \mathrm {U{}}(\mathcal {H})$ be a continuous unitary one-parameter group. Then $U(t) = \Delta _\mathtt{V}^{-it/2\pi }$ holds for all $t \in {{\mathbb {R}}}$ if and only if the positive definite function

$$\begin{aligned} \varphi \,:\, {{\mathbb {R}}}\rightarrow \mathop {\mathrm {Bil}}\nolimits (\mathtt {V}),\quad \varphi (t)(\xi ,\eta ) := \langle \xi , U(t) \eta \rangle \end{aligned}$$

satisfies the KMS condition for $\beta = 2\pi$.

Proof

(see also [NÓ19, Thm. 2.6]). In [Lo08, Prop. 3.7], this characterization is stated for the function $\langle U(t)\xi , \eta \rangle$, but this should be $\langle \xi , U(t) \eta \rangle$ if the scalar product is linear in the second argument. $\quad \square$

1.3 A.3 Hardy space and graph realizations

Let $\Delta > 0$ be a positive selfadjoint operator on $\mathcal {H}$. Then $\mathcal {D}(\Delta ^{1/2})$ is a dense subspace of $\mathcal {H}$, and the map

$$\begin{aligned} \Psi : \mathcal {D}(\Delta ^{1/2}) \rightarrow \Gamma (\Delta ^{1/2}), \quad \xi \mapsto (\xi , \Delta ^{1/2}\xi ) \end{aligned}$$

is a complex linear bijection onto the closed graph of the selfadjoint operator $\Delta ^{1/2}$ in the Hilbert space $\mathcal {H}\oplus \mathcal {H}$. We thus obtain on $\mathcal {D}(\Delta ^{1/2})$ the structure of a complex Hilbert space for which $\Psi$ is unitary.

The operator $\Delta$ defines a unitary one-parameter group $(\Delta ^{it})_{t \in {{\mathbb {R}}}}$, and we consider the $\mathcal {H}$-valued Hardy space

$$\begin{aligned}&H^2(\mathcal {S}_\pi ,\mathcal {H})^{\Delta } \\ {}&:=\Big \{ f \in \mathrm {Hol}(\mathcal {S}_\pi ,\mathcal {H}) \,:\, (\forall z \in \mathcal {S}_\pi )(\forall t \in {{\mathbb {R}}}) \ f(z + t) = \Delta ^{-it/2\pi } f(z), \quad \sup _{0< y< \pi } \Vert f(iy)\Vert < \infty \Big \} \end{aligned}$$

of equivariant bounded holomorphic maps $\mathcal {S}_\pi \rightarrow \mathcal {H}$. For $\Delta ^{-it/2\pi } = e^{itH}$, i.e., $H = -\frac{1}{2\pi } \log \Delta$, and the spectral measure $P_H$ of H, we have

$$\begin{aligned} \Vert \Delta ^{y/2\pi }\xi \Vert ^2 = \Vert e^{-yH}\xi \Vert ^2 = \int _{{\mathbb {R}}}e^{-2\lambda y}\, dP_H^\xi (\lambda ), \end{aligned}$$

so that the Monotone Convergence Theorem implies for $f \in H^2(\mathcal {S}_\pi ,\mathcal {H})^{\Delta }$ and $\xi := f(\pi i/2)$

$$\begin{aligned} \int _{{\mathbb {R}}}e^{\pm \lambda \pi }\, dP_H^\xi (\lambda ) < \infty , \quad \text{ so } \text{ that } \quad \xi \in \mathcal {D}(\Delta ^{\pm 1/4}). \end{aligned}$$

Thus [NÓ18, Lemma A.2.5] implies that f extends to a continuous function on $\overline{\mathcal {S}_\pi }$, also denoted f. It satisfies

$$\begin{aligned} \sup _{0< y < \pi } \Vert f(iy)\Vert = \max (\Vert f(0)\Vert , \Vert f(\pi i)\Vert ). \end{aligned}$$

In particular, the map

$$\begin{aligned} \Phi : H^2(S_\pi , \mathcal {H})^{\Delta } \rightarrow \mathcal {H}\oplus \mathcal {H}, \quad \Phi (f) := (f(0), f(\pi i)) \end{aligned}$$

is defined. To identify the range of $\Phi$, we use [NÓ18, Lemma A.2.5] to see that $\xi = f(0)$ for some $f \in H^2(S_\pi , \mathcal {H})^{\Delta _V}$ if and only if $\xi \in \mathcal {D}(\Delta ^{1/2})$. Then $f(\pi i) = \Delta ^{1/2} \xi$, and we conclude that

$$\begin{aligned} \Phi \big (H^2(S_\pi , \mathcal {H})^{\Delta }\big ) = \Gamma (\Delta ^{1/2}) \end{aligned}$$

(cf. [LLQR18, Prop. 3.4]). As $\Phi$ is injective with closed range, it is an isomorphism of Banach spaces but not necessarily isometric.

[LLQR18, Prop. 3.2] also contains observations which are very similar to the following lemma.

Lemma A.11

If J is a conjugation on $\mathcal {H}$, then

$$\begin{aligned} {{\widetilde{J}}}(\xi ,\eta ) := (J\eta , J\xi ) \end{aligned}$$

defines a conjugation on $\mathcal {H}\oplus \mathcal {H}$, and ${{\widetilde{J}}}$ maps $\Gamma (\Delta ^{1/2})$ into itself if and only if the modularity relation $J\Delta J = \Delta ^{-1}$ holds.

Proof

If the modularity relation holds, then we also have $\Delta ^{-1/2} J= J \Delta ^{1/2}$, so that $J \mathcal {D}(\Delta ^{1/2}) = \mathcal {D}(\Delta ^{-1/2}) = \mathcal {R}(\Delta ^{1/2})$, and therefore

$$\begin{aligned} {{\widetilde{J}}}(\xi , \Delta ^{1/2}\xi ) = (J \Delta ^{1/2}\xi , J \xi ) = (\Delta ^{-1/2}J \xi , J \xi ) \in \Gamma (\Delta ^{1/2}) \quad \text{ for } \quad \xi \in \mathcal {D}(\Delta ^{1/2}).\end{aligned}$$

If, conversely, ${{\widetilde{J}}}$ preserves $\Gamma (\Delta ^{1/2})$, then

$$\begin{aligned} J \Delta ^{1/2} \xi \in \mathcal {D}(\Delta ^{1/2}) \quad \text{ and } \quad \Delta ^{1/2} J \Delta ^{1/2} \xi = J \xi \quad \text{ for } \quad \xi \in \mathcal {D}(\Delta ^{1/2}).\end{aligned}$$

This means that $J\xi \in \mathcal {D}(\Delta ^{-1/2})$ with $J \Delta ^{1/2} \xi = \Delta ^{-1/2} J \xi$. As J is an involution, $J\mathcal {D}(\Delta ^{1/2}) = \mathcal {D}(\Delta ^{-1/2})$ and $J \Delta ^{1/2} J = \Delta ^{-1/2}$. This implies $J\Delta J = \Delta ^{-1}$. $\quad \square$

If $J\Delta J = \Delta ^{-1}$, the preceding lemma shows that the closed subspace $\Gamma (\Delta ^{1/2})$ of $\mathcal {H}\oplus \mathcal {H}$ is invariant under ${{\widetilde{J}}}$. We also observe that the antilinear operator

$$\begin{aligned} T := J \Delta ^{1/2} = \Delta ^{-1/2} J\, :\,\mathcal {D}(\Delta ^{1/2}) \rightarrow \mathcal {D}(\Delta ^{1/2}) \end{aligned}$$

satisfies

$$\begin{aligned} {{\widetilde{J}}} \Psi (\xi ) = (J\Delta ^{1/2}\xi , J \xi ) = (T\xi , J \xi ) = (T \xi , \Delta ^{1/2}T \xi ) = \Psi (T\xi ) \quad \text{ for } \quad \xi \in \mathcal {D}(\Delta ^{1/2}). \end{aligned}$$

For the standard subspace $\mathtt{V}$ with $J_\mathtt{V}= J$ and $\Delta _\mathtt{V}= \Delta$, $T_\mathtt{V}:= T$ is the corresponding Tomita operator (Sect. A.1), and the relation

$$\begin{aligned} {{\widetilde{J}}} \circ \Psi = \Psi \circ T_\mathtt{V}\end{aligned}$$

implies that $\Psi (\mathtt{V}) = \Gamma (\Delta ^{1/2})^{{{\widetilde{J}}}}$. In particular, $T_\mathtt{V}$ is a conjugation for the complex Hilbert space structure on $\mathcal {D}(\Delta ^{1/2})\cong \Gamma (\Delta ^{1/2})$, whose fixed point space is $\mathtt{V}$.

Next we observe that, as J commutes with the unitary operators $\Delta ^{it}$, $t \in {{\mathbb {R}}}$,

$$\begin{aligned} ({\widehat{J}} f)(z) = J f(\pi i + \overline{z}) \end{aligned}$$

defines an isometric involution on the Hardy space $H^2(\mathcal {S}_\pi , \mathcal {H})^\Delta$. For $f \in H^2(\mathcal {S}_\pi ,\mathcal {H})^{\Delta }$ and $\xi := f(0) \in \mathcal {D}(\Delta ^{1/2})$, we have

$$\begin{aligned} {{\widetilde{J}}} \Phi (f) = (J f(\pi i), J f(0)) = \Phi ({\widehat{J}} f), \end{aligned}$$

so that $\Phi$ intertwines the conjugations ${{\widetilde{J}}}$ and ${\widehat{J}}$. We conclude in particular that

$$\begin{aligned} \Phi ^{-1} \Psi (\mathtt {V}) = \{ f \in H^2(\mathcal {S}_\pi ,\mathcal {H})^{\Delta } \,:\, {\widehat{J}}(f) = f \}. \end{aligned}$$

(A.6)

Lemma A.12

For $f \in H^2(\mathcal {S}_\pi ,\mathcal {H})^{\Delta }$, the following conditions are equivalent:

• (a) ${\hat{J}}(f) = f$, i.e., $f(\pi i + \overline{z}) = J f(z)$ for $z \in \mathcal {S}_\pi$.

• (b) $f(z) \in \mathcal {H}^J$ for Im $z = \frac{\pi }{2}$.

• (c) $f(0) \in \mathtt{V}$.

• (d) $f(\pi i) \in \mathtt{V}'$.

• (e) $f(\pi i) = J f(0)$.

Proof

The equivalence of (a) and (b) follows by uniqueness of analytic continuation from the line $\frac{\pi i}{2} + {{\mathbb {R}}}\subseteq ~\mathcal {S}_\pi .$ The equivalence of (a) and (c) follows from $\Psi ^{-1}\Phi (f) = f(0)$ and (A.6). As $f(\pi i) = \Delta ^{1/2} f(0)$ is contained in $\mathtt{V}' = J\mathtt{V}$ if and only if $J \Delta ^{1/2} f(0) \in \mathtt{V}$, which in turn is equivalent to $f(0) \in \mathtt{V}$, conditions (c) and (d) are also equivalent. The equivalence of (c) and (e) follows from Proposition 2.1.

The map $\mathop {\mathrm{ev}}\nolimits _0 = \Psi ^{-1}\Phi : \mathop {\mathrm{Fix}}\nolimits ({\hat{J}}) \rightarrow \mathtt{V}$ is an isometry of real Hilbert spaces because ${\hat{J}}(f) = f$ implies $\Vert f(0)\Vert = \Vert f(\pi i)\Vert$. In this sense every standard subspace can be realized in a natural way as a “real form” of a Hardy space on the strip $\mathcal {S}_\pi$.

B Wedges in euclidean Jordan algebras

We expect that the reader is familiar with the basic theory of simple euclidean Jordan algebras. We use [FK94] as a standard reference. For the basic definitions we refer to Sect. 6. From now on E is always a simple euclidean Jordan algebra with

$$\begin{aligned} \mathop {\mathrm{dim}}\nolimits (E) = n \quad \text{ and } \quad \mathop {\mathrm{rank}}\nolimits (E) = r, \end{aligned}$$

(B.1)

and $c_1,\ldots ,c_r$ is a Jordan frame. We then obtain the Pierce decomposition

$$\begin{aligned} E = \bigoplus _{j = 1}^r {{\mathbb {R}}}c_j \oplus \bigoplus _{i < j} E_{ij} \quad \text { with } \quad E_{ij} = \big \{ v \in E \,:\, c_i v = {\textstyle {\frac{1}{2}}}v, c_j v = {\textstyle {\frac{1}{2}}}v\big \} \end{aligned}$$

(B.2)

([FK94, §  IV.1]). The set $E^\times$ of invertible elements of E has $r+1$ connected components that can be described as follows. We fix a spectral decomposition $x = \sum _{j = 1}^r x_j {{\widetilde{c}}}_j$ of an element $x \in E$, where $({{\widetilde{c}}}_1,\ldots , {{\widetilde{c}}}_r)$ is a Jordan frame ( [FK94, Thm. III.1.1]). This means that, under the automorphism group $\mathop {\mathrm{Aut}}\nolimits (E)$, the element x is conjugate to $\sum _{j = 1}^r x_j c_j$. For $E = \mathop {\mathrm{Herm}}\nolimits _r({{\mathbb {K}}})$, this corresponds to the conjugation of a hermitian matrix x by $\mathrm{U{}}_r({{\mathbb {K}}})$ to a diagonal matrix, and for Minkowski space, it corresponds to conjugation of an element $x \in {{\mathbb {R}}}^{1,d-1}$ under $\mathrm{O{}}_{d-1}({{\mathbb {R}}})$ to one with $x_2 = \cdots = x_{d-1} = 0$.

We define (cf. [FK94, p. 29]):

• the index of x by $\mathop {\mathrm{ind}}\nolimits (x) := \sum _{j = 1}^r \mathop {\mathrm{sgn}}\nolimits (x_j) \in \{r,r-2,\ldots , -r\}.$

• the determinant of x by $\Delta (x) = \prod _{j=1}^r x_j$, and

• the trace of x by $\mathop {\mathrm{tr}}\nolimits (x) = \sum _{j=1}^r x_j$.

Then the connected components of $E^\times$ are the subsets

$$\begin{aligned} E^\times _j := \{ x \in E^\times \, :\,\mathop {\mathrm {ind}}\nolimits (x) = j\}, \qquad j = r,r-2, \ldots , -r. \end{aligned}$$

For a multiplication operator

$$\begin{aligned} h := \sum _{j = 1}^r a_j L(c_j), \end{aligned}$$

the Pierce decomposition (B.2) shows that the eigenvalues are $a_1, \ldots , a_r$ and $\frac{a_i + a_j}{2}$ for $i \not =j$. For $h \not =0$, it follows that the eigenvalues of h are contained in $\{-1,0,1\}$ if and only if $a_j \in \{\pm 1\}$. Reordering the Jordan frame, we see that, up to applying an automorphism of E, any such element is conjugate to one of the form

$$\begin{aligned} h := h_k := L(c_1 + \cdots + c_k) - L(c_{k+1} + \cdots + c_r) \quad \text{ for } \text{ some } \quad k \in \{0,\ldots , r\}.\nonumber \\ \end{aligned}$$

(B.3)

Then

$$\begin{aligned} E_1(h) = \bigoplus _{j = 1}^k {{\mathbb {R}}}c_j \oplus \bigoplus _{i< j \le k} E_{ij}, \quad E_0(h) = \bigoplus _{i \le k< j} E_{ij} \quad \text{ and } E_{-1}(h) = \bigoplus _{j = k+1}^r {{\mathbb {R}}}c_j \oplus \bigoplus _{k< i < j} E_{ij}, \end{aligned}$$

where $E_{\pm 1}(h)$ are Jordan subalgebras of E. Note that $E_1(-h) = E_{-1}(h)$.

We now observe that the quadruple $(E,C,h,\tau := e^{\pi i h})$ satisfies the assumptions (A1-3) in Sect. 3. Here (A1) and (A2) are obvious. To verify (A3), note that $e^{{{\mathbb {R}}}h} C = C$ follows from $e^{L(x)}C = C$ for every $x \in E$ ([FK94, p. 48]). Moreover, $\tau = e^{\pi i h} \in \mathop {\mathrm{Str}}\nolimits (E)$ ([FK94, Prop. VIII.2.8]) satisfies $\tau (e) = - e$, so that $\tau (C) = - C$. This proves (A3).

In this context, the constructions of Sect. 3 have a Jordan theoretic interpretation. The cones $C_+:= C \cap E_1(h)$ is the positive cone in the Jordan algebras $E_1(h)$ and $C_- = - C \cap E_{-1}(h)$ is the negative cone in $E_{-1}(h)$. The corresponding wedge is

$$\begin{aligned} W := W(h) := C_+^0 \oplus C_-^0 \oplus E_0(h). \end{aligned}$$

(B.4)

Note that $x \in W^c = E\setminus W$ if and only if $x_1 \not \in C_+^0$ or $x_{-1} \not \in C_-^0$. For the extremal situations $k = 0,r$, we obtain $W(h_r) = C^0$ and $W(h_0) = - C^0$.

Lemma B.1

$\mathop {\mathrm{tr}}\nolimits (h_k) = (2k-r)\frac{n}{r}$ for $n = \mathop {\mathrm{dim}}\nolimits (E)$ and $r = \mathop {\mathrm{rank}}\nolimits (E)$.

Proof

From the Pierce decomposition (B.2) it follows that

$$\begin{aligned} n = \mathop {\mathrm{dim}}\nolimits E= r + \frac{r(r-1)}{2}d = r\Big (1 + (r-1)\frac{d}{2}\Big ) \end{aligned}$$

(B.5)

and

$$\begin{aligned} \mathop {\mathrm{tr}}\nolimits (h_k)&= k - (r-k) + d\Big (\frac{k(k-1)}{2} - \frac{(r-k)(r-k-1)}{2}\Big ) \\&= 2k-r + \frac{d}{2}(k^2 - k + (r-k) - (r^2 - 2rk + k^2)) \\&= 2k-r + \frac{d}{2}(r- 2k + 2rk - r^2) \\&= 2k-r + \frac{d}{2}(2k-r)(r-1) = (2k-r)\Big (1 + (r-1)\frac{d}{2}\Big ) = (2k-r)\frac{n}{r}. \end{aligned}$$

$\square$

Remark B.2

We are interested in the parity of the numbers $\mathop {\mathrm{tr}}\nolimits (h_k)$. First, we observe that $\frac{n}{r} \in {\textstyle {\frac{1}{2}}}{{\mathbb {Z}}}$ by (B.5) and that this is an integer if and only if $(r-1)d$ is even. This is equivalent to d even or r odd. In this case $\mathop {\mathrm{tr}}\nolimits (h_k)$ is even if and only if n is even. In the other case the parity of $\mathop {\mathrm{tr}}\nolimits (h_k)$ depends on k if $\frac{2n}{r}$ is odd.

Lemma B.3

Let $p_1 \,:\, E \rightarrow E_1(h), x \mapsto x_1$, denote the projection map. Then $p_1(C) \subseteq C$ and $\mathop {\mathrm{rank}}\nolimits p_1(x) \le \mathop {\mathrm{rank}}\nolimits x$ for $x \in C$.

Proof

Let $m := \mathop {\mathrm{rank}}\nolimits x$ and $x \in C$. The subset $C_{\le m} := \{ w \in C \, :\,\mathop {\mathrm {rank}}\nolimits w \le m\}$ is a closed (non-convex) cone invariant under $e^{{{\mathbb {R}}}h}$. Therefore $x_1 = \lim _{t \rightarrow \infty } e^{-t} e^{th} x \in C_{\le m}.$

For an element $x \in E$, we write $x = x_+ - x_-$ with $x_\pm \in C$ and $x_+ x_- = 0$ for the canonical decomposition of x into positive and negative part which can be obtained from the spectral decomposition ([FK94, Thm. III.1.1]).

The following proposition and its corollary constitute the main result of this appendix. They are the key tool for the finer analysis of the support properties of the Fourier transforms ${{\hat{\mu }}}_s$ of the Riesz measures $\mu _s$.

Proposition B.4

Let $v = v_+ - v_-$ be the canonical decomposition of $v\in E$ into positive and negative part. Then $\mathop {\mathrm{Aut}}\nolimits (E)v \subseteq W(h_k)^c$ if and only if

$$\begin{aligned} \mathop {\mathrm{rank}}\nolimits v_+< k \quad \text{ or } \quad \mathop {\mathrm{rank}}\nolimits v_- < r-k.\end{aligned}$$

Proof

For $v \in W(h_k)$ we have $p_1(v) \in C_+^0$, so that $p_1(v_\pm ) \in C$ yields

$$\begin{aligned} p_1(v_+) = p_1(v) + p_1(v_-) \in C_+^0 + C_+ \subseteq C_+^0 \end{aligned}$$

is invertible in $E_1(h)$. Lemma B.3 thus implies that $\mathop {\mathrm{rank}}\nolimits v_+ \ge \mathop {\mathrm{rank}}\nolimits p_1(v_+) = k$. Therefore $\mathop {\mathrm{rank}}\nolimits v_+ < k$ entails $v \in W(h_k)^c$. For $g \in \mathop {\mathrm{Aut}}\nolimits (E)$, we have $\mathop {\mathrm{rank}}\nolimits (gv_+) = \mathop {\mathrm{rank}}\nolimits (v_+)$, so that $\mathop {\mathrm{rank}}\nolimits (v_+) < k$ implies $\mathop {\mathrm{Aut}}\nolimits (E)v \subseteq W(h_k)^c$. Likewise $\mathop {\mathrm{rank}}\nolimits v_- < r-k$ implies that $\mathop {\mathrm{Aut}}\nolimits (E)v \subseteq W(h_k)^c$.

Suppose, conversely, that $\mathop {\mathrm{Aut}}\nolimits (E)v \subseteq W(h_k)^c$. Then there exists a $g \in \mathop {\mathrm{Aut}}\nolimits (E)$ with

$$\begin{aligned} gv = \sum _{j = 1}^r \nu _j c_j \in \sum _{j = 1}^r {{\mathbb {R}}}c_j \subseteq E_1(h) \oplus E_{-1}(h) \quad \text{ and } \quad \nu _1 \ge \cdots \ge \nu _r.\end{aligned}$$

As $gv \not \in W(h_k)$, we have $(gv)_1 = \sum _{j \le k} \nu _j c_j \not \in C_+^0$ or $(gv)_{-1} = \sum _{j > k} \nu _j c_j \not \in C_-^0$. In the first case $\nu _k \le 0$, so that $\mathop {\mathrm{rank}}\nolimits v_+ < k$, and in the second case $\nu _{k+1} \ge 0$, so that $\mathop {\mathrm{rank}}\nolimits v_- < r-k$.

By negation we immediately obtain:

Corollary B.5

For $v = v_+ - v_-$ as in Proposition  B.4, the following are equivalent:

• (i) $\mathop {\mathrm{Aut}}\nolimits (E)v \cap W(h_k) \not =\emptyset$ for $W(h_k)$ as in (B.4).

• (ii) $\mathop {\mathrm{rank}}\nolimits (v_+) = k$ and $\mathop {\mathrm{rank}}\nolimits (v_-) = r-k$.

• (iii) v is invertible of index $\mathop {\mathrm{ind}}\nolimits (v) = 2k-r$.

In particular, $W(h_k) \subseteq E^\times _{2k-r}$.

Proof

The equivalence of (i) and (ii) follows from Proposition B.4. For the equivalence of (ii) and (iii), we note that $v = v_+ - v_-$ is invertible if and only if $\mathop {\mathrm {rank}}\nolimits (v_+) {+}\mathop {\mathrm {rank}}\nolimits (v_-) {=} r$. Then $\mathop {\mathrm{ind}}\nolimits (v) = \mathop {\mathrm{rank}}\nolimits (v_+) - \mathop {\mathrm{rank}}\nolimits (v_-) = 2k-r$ is equivalent to $\mathop {\mathrm{rank}}\nolimits (v_+) = k$ and $\mathop {\mathrm{rank}}\nolimits (v_-) = r-k$. $\quad \square$

Example B.6

(a) For $k = r$ we have $E = E_1(h)$ and $W(h_r) = C^0$. Therefore $\mathop {\mathrm{Aut}}\nolimits (E)v \subseteq W(h_r)^c$ is equivalent to $\mathop {\mathrm{rank}}\nolimits v_+ < r$, which is equivalent to $v \not \in C^0$.

(b) For $r = 2$ and $k = 1$ (Lorentz boost on Minkowski space), we obtain by $W(h_1)$ a wedge domain in the Minkowski space E. Then $\mathop {\mathrm{Aut}}\nolimits (E)W(h_1) = E^\times _0$ is the open subset of spacelike vectors whose complement is the closed double cone $\bigcap _{g \in \mathop {\mathrm{Aut}}\nolimits (E)} g W(h_1)^c = C \cup - C$ (cf. Example 5.5).