Inner ideals, compact tripotents and Čebyšëv subtriples of JB$^{*}$-triples and C$^*$-algebras

Abstract

The aim of this note is to study Čebyšëv JB$^*$-subtriples of general JB$^*$-triples. It is established that if F is a non-zero Čebyšëv JB$^*$-subtriple of a JB$^*$-triple E, then exactly one of the following statements holds:

• (a) F is a rank one JBW$^*$-triple with dim $(F)\ge 2$ (i.e. a complex Hilbert space regarded as a type 1 Cartan factor). Moreover, F may be a closed subspace of arbitrary dimension and E may have arbitrary rank;

• (b) $F= \mathbb {C} e$, where e is a complete tripotent in E;

• (c) E and F are rank two JBW$^*$-triples, but F may have arbitrary dimension;

• (d) F has rank greater or equal than three and $E=F$.

1 Introduction

It is known that certain problems on operator algebras are more feasible when the algebra under study is a von Neumann algebra (i.e. a C$^*$-algebra which is also a dual Banach space). For example, A.G. Robertson gave in [35] a complete description of one-dimensional Čebyšëv subspaces, and of finite dimensional Čebyšëv hermitian subalgebras with dimension bigger than 1 of a general von Neumann algebra. Concretely, for a non-zero element x in a von Neumann algebra M, the subspace $\mathbb {C} x$ is a Čebyšëv subspace of M if and only if there is a projection p in the center of M such that p x is left invertible in p M and $(1-p)x$ is right invertible in $(1-p){M}$ (cf. [35, Theorem 1]). A finite dimensional $^*$-subalgebra N of an infinite dimensional von Neumann algebra M with dim $(N)>1$ never is a Čebyšëv subspace of M (see [35, Theorem 6]).

Two years later A.G. Robertson and D. Yost proved in [36, Corollary 1.4] that an infinite dimensional C$^*$-algebra A admits a finite dimensional $^*$-subalgebra B which is also a Čebyšëv subspace in A if and only if A is unital and $B=\mathbb {C} 1$. The results proved by Robertson and Yost were complemented by G.K. Pedersen, who shows that if A is a C$^*$-algebra without unit and B is a Čebyšëv C$^*$-subalgebra of A, then $A=B$ (compare [34, Theorem 4]).

We recall that a subspace V of a Banach space X is called a Čebyšëv (Chebyshev) subspace of X if for each $x\in X$ there exists a unique point $\pi _{_V} (x)\in V$ such that $\hbox {dist}(x,V)=\left\| x-\pi _{_V} (x)\right\|$. Throughout this note the symbol $\pi _{_V} (x)$ will denote the best approximation of an element x in X in a Čebyšëv subspace V of X. For more information on Čebyšëv and best approximation theory we refer to the monograph [37].

Similar benefits to those obtained working with von Neumann algebras re-appear when studying Čebyšëv subspaces of Ternary Rings of Operators (TRO’s) of a given von Neumann algebra, or when exploring Čebyšëv JBW$^*$-subtriples of a given JBW$^*$-triple (see Sect. 2 for definitions). In a previous paper, we establish the following description of Čebyšëv JBW$^*$-subtriples of a JBW$^*$-triple.

Theorem 1

[26, Theorem 13] Let N be a non-zero Čebyšëv JBW$^*$-subtriple of a JBW$^*$-triple M. Then exactly one of the following statements holds:

• (a) N is a rank one JBW$^*$-triple with dim $(N)\ge 2$ (i.e. a complex Hilbert space regarded as a type 1 Cartan factor). Moreover, N may be a closed subspace of arbitrary dimension and M may have arbitrary rank;

• (b) $N= \mathbb {C} e$, where e is a complete tripotent in M;

• (c) N and M have rank two, but N may have arbitrary dimension;

• (d) N has rank greater or equal than three and $N=M$. $\Box$

We refer to [21, Preliminaries] and [11, Example 2.5.31] for the definition of Cartan factors.

The question whether in the above theorem JBW$^*$-triples and subtriples can be replaced with JB$^*$-triples and subtriples remained as an open problem. The techniques employed in [26] rely heavily on the rich geometric properties of JBW$^*$-triples. In this note we study this problem in the more general setting of JB$^*$-triples. We combine here new arguments involving inner ideals and compact tripotents in the bidual of a JB$^*$-triple. The main result of this note shows that the conclusion of the above Theorem also holds when N is a JB$^*$-subtriple of a general JB$^*$-triple M (see Theorem ).

Among the new results proved in this note we also establish that a Čebyšëv C$^*$-subalgebra B (respectively, a Čebyšëv JB$^*$-subtriple) of a C$^*$-algebra A with rank$(B)\ge 3$ coincides with the whole A (see Corollary 10).

2 Preliminaries

The multiple attempts to understand a Riemann mapping theorem type for complex Banach spaces of dimension bigger or equal than 2, led some mathematicians to the study of bounded symmetric domains (compare [10, 23, 24, 33] and [29]). The definite answer was given by W. Kaup, who showed the existence of a set of algebraic-geometric-analytic axioms which determine a class of complex Banach spaces, the class of JB$^*$-triples, whose open unit balls are bounded symmetric domains, and every bounded symmetric domain in a complex Banach space is biholomorphically equivalent to the open unit ball of a JB$^*$-triple; in this way, the category of all bounded symmetric domains with base point is equivalent to the category of JB$^*$-triples.

A JB $^{*}$-triple is a complex a Banach space E with a continuous triple product $(a,b,c) \mapsto \{a,b,c\}$, which is bilinear and symmetric in the external variables and conjugate linear in the middle one and satisfies:

• (a) (Jordan identity)

  $$\begin{aligned} L(x,y)\{a,b,c\}=\{L(x,y)a,b,c\}-\{a,L(y,x)b,c\}+\{a,b,L(x,y)c\}, \end{aligned}$$

  for all $x,y,a,b,c\in E$, where $L(x,y):E\rightarrow E$ is the linear mapping given by $L(x,y)z=\{x,y,z\}$;

• (b) For each $x\in E$, the operator L(x, x) is hermitian with non-negative spectrum;

• (c) $\left\| \{x,x,x\}\right\| =\left\| x\right\| ^{3}$ for all $x\in E$.

Given an element a in a JB$^*$-triple E, the symbol Q(a) will denote the conjugate linear map on E defined by $Q(a) (x) := \{a,x,a\}$.

The class of JB$^*$-triples includes all C$^*$-algebras when the latters are equipped with the triple product given by

$$\begin{aligned} \{a,b,c\} = \frac{1}{2} (a b^* c+ c b^* a). \end{aligned}$$

(2.1)

The space B(H, K) of all bounded linear operators between complex Hilbert spaces, although rarely is a C$^*$-algebra, is a JB$^*$-triple with the product defined in (2.1). In particular, every complex Hilbert space is a JB$^*$-triple. Thus, the class of JB$^*$-triples is strictly wider than the class of C$^*$-algebras.

A JBW$^*$-triple is a JB$^*$-triple which is also a dual Banach space (with a unique isometric predual [1]). The triple product of every JBW$^*$-triple is separately weak$^*$ continuous (cf. [1]). The second dual, $E^{**},$ of a JB$^*$-triple, E,  is a JBW$^*$-triple with a certain triple product extending the product of E (cf. [12]).

Let E be a JB$^*$-triple. An element $e\in E$ is called a tripotent if $\left\{ e,e,e \right\} =e$. For each tripotent $e\in E$, the eigenspaces of the operator L(e, e) induce a decomposition (called Peirce decomposition) of E in the form

$$\begin{aligned} E= E_{2} (e) \oplus E_{1} (e) \oplus E_0 (e), \end{aligned}$$

where for $i=0,1,2,$ $E_i (e)=\{ x\in E : L(e,e) (x) = \frac{i}{2} x\}$ (compare [33, Theorem 25]). The natural projections of E onto $E_i(e)$ will be denoted by $P_i(e)$. It is known that this decomposition satisfies the following multiplication rules:

$$\begin{aligned} \left\{ E_{i}(e),E_{j} (e),E_{k} (e) \right\} \subseteq E_{i-j+k} (e), \end{aligned}$$

if $i-j+k \in \{0,1,2\}$ and is zero otherwise. In addition,

$$\begin{aligned} \left\{ E_{2} (e),E_{0}(e),E \right\} = \left\{ E_{0} (e),E_{2}(e),E \right\} =0. \end{aligned}$$

A tripotent e in E is called complete (respectively, minimal) if the equality $E_0(e)=0$ (respectively, $E_2(e)=\mathbb {C} e \ne \{0\}$) holds.

The connections between JB$^*$-triples and JB$^*$-algebras are very deep. Every JB$^*$-algebra is a JB$^*$-triple under the triple product defined by

$$\begin{aligned} \left\{ x,y,z \right\} = (x\circ y^*) \circ z + (z\circ y^*)\circ x - (x\circ z)\circ y^*. \end{aligned}$$

(2.2)

The Peirce 2-subspace $E_2 (e)$ is a JB$^*$-algebra with product $x\circ _e y := \left\{ x,e,y \right\}$ and involution $x^{\sharp _e} := \left\{ e,x,e \right\}$.

Let a be an element in a JB$^*$-triple E. It is known that the JB$^*$-subtriple, $E_{a}$, generated by a, identifies with some $C_0(L_a)$ where $\Vert a\Vert \in L_a\subseteq [0,\Vert a\Vert ]$ with $L_a\cup \{0\}$ compact, and the element a is associated with a positive generating element in $C_0(L_a)$ (cf. [29, 1.15]). The above identification lifts to that of the bidual of the JB$^*$-triple generated by a with a commutative von Neumann algebra the identity element of which, r(a), is called the range tripotent of a in $E^{**}$, and we note that a is positive in $E^{**}_{2} (r(a))$ (see [13, §3]).

A (closed) subtriple I of a JB$^*$-triple E is said to be a triple ideal or simply an ideal of E if $\left\{ E,E,I \right\} + \left\{ E,I,E \right\} \subseteq I$. If we only have $\left\{ I,E,I \right\} \subseteq I$ we say that I is an inner ideal of E. Following standard notation, given an element a in E, we denote by E(a) the norm-closure of $Q(a) (E) = \{a,E,a\}$ in E. It is known that E(a) is precisely the norm-closed inner ideal of E generated by a (cf. [6]). The identity $2 Q(a,b) = Q(a+b) -Q(a) -Q(b)$ ($a,b\in E$) implies that a JB$^*$-subtriple I of E is an inner ideal if and only if I contains E(a) for all a in I. It is also known that

$$\begin{aligned} E(a)^{**} = \overline{E(a)}^{\sigma (E^{**},E^*)} = E^{**}_{2} (r(a)) \end{aligned}$$

(2.3)

(see [6, Proposition 2.1 and comments prior to it]).

If e and a are contained in a JB$^*$-subtriple F of E we have

• (*) $F_k (e) = E_k (e)$ if and only if $E_k (e)$ is contained in F;

• (**) $F(a) = E(a)$ if and only if E(a) is contained in F.

The first of these equivalences is immediate upon application of the projection $P_k(e)$. As for $(**)$, F(a) is contained in E(a), by definition, and if F contains E(a) then $F^{**}$ contains $E(a)^{**} = E^{**}_2(r(a))$ so that $F(a)^{**} = E(a)^{**}$, by (a), implying the result. A similar argument shows that F(a) is the intersection of F with E(a).

We recall that two elements a, b in a JB$^*$-triple E are orthogonal (written as $a\perp b$) if $L(a, b) = 0$ (see [7, Lemma 1] for several equivalent reformulations). Given a subset $M\subseteq E$, we write $M_{E}^{\perp }$ (or simply $M^{\perp }$) for the (orthogonal) annihilator of M defined by $M_{E}^{\perp }=\{y\in E:y\perp x,\forall x\in M\}$. If $e\in E$ is a tripotent, then $\{e\}_{E}^{\perp }=$ $E_{0 }(e)$, and $\{a\}_{E}^{\perp }=$ $(E^{**})_{0 }(r(a))\cap E$, for every $a\in E$ (cf. [8, Lemma 3.2]).

It is known that

$$\begin{aligned} \Vert a + b\Vert = \max \{ \Vert a\Vert , \Vert b\Vert \}, \end{aligned}$$

(2.4)

whenever a and b are orthogonal elements in a JB$^*$-triple (cf. [20, Lemma 1.3(a)]). A subset $S \subseteq E$ is said to be orthogonal if $0 \notin S$ and $x \perp y$ for every $x \ne y$ in S. The minimal cardinal number r satisfying $card(S) \le r$ for every orthogonal subset $S \subseteq E$ is called the rank of E (and will be denoted by r(E)). Given a tripotent $e\in E$, the rank of the Peirce 2-subspace $E_2(e)$ will be called the rank of e.

We shall also make use of a natural partial order defined on the set of tripotents (see Corollary 1.7 and comments preceding it in [20]). Given two tripotents e, u in a JB$^*$-triple E, we say that $e\le u$ if $u-e$ is a tripotent in E with $u-e\perp e$.

We finally, recall that an element x in a JB$^*$-triple E is called Brown-Pedersen quasi-invertible (BP quasi-invertible for short) if there exists $y\in E$ such that $B(x,y)=0$ (cf. [27]), where B(x, y) denotes the Bergmann operator $B(x,y)=I_E-2L(x,y)+Q(x)Q(y)$. Theorems 6 and 11 in [27] prove that an element x in E is Brown-Pedersen quasi-invertible if, and only if, x is von Neumann regular in the sense of [9, 15, 30] and its range tripotent is an extreme point of the closed unit ball of E, equivalently, there exists a complete tripotent $v\in E$ such that x is positive and invertible in $E_2(v)$. In particular, every extreme point of the closed unit ball of E is BP quasi-invertible. The symbol $E_q^{-1}$ will denote the set of BP quasi-invertible elements in E.

3 Čebyšëv subtriples of JB$^*$-triples

The following auxiliary results were established in [26, §3]

Proposition 2

[26, Propositions 9 and 10] Let F be a Čebyšëv JB$^*$-subtriple of a JB$^*$-triple E. Suppose e is a non-zero tripotent in F. Then the following statements hold:

• (a) $E_0 (e) = F_0 (e)$, and consequently, every complete tripotent in F is complete in E.

• (b) If $\{e\}^{\perp }_{F}\ne 0$, then $E_2 (e) = F_2 (e)$.$\Box$

We continue, in this paper, our study on Čebyšëv subtriples of general JB$^*$-triples. The first part of the next proposition owes much to the arguments developed by Pedersen in [34, Lemma 4].

Proposition 3

Let a belong to a Čebyšëv JB$^*$-subtriple F of a JB$^*$-triple E. The following statements hold:

• (a) If a is not BP quasi-invertible we have $F(a) = E(a)$;

• (b) If F contains no BP quasi-invertible elements we have that F is an inner ideal of E.

Proof

(a) Since a is in $F\backslash F_q^{-1}$ we have two possibilities either a is not von Neumann regular or a is von Neumann regular and its range tripotent is not an extreme point of the closed unit ball of F. We deal with each case separately. We can assume that $\Vert a\Vert =1$. Suppose first that a is not von Neumann regular. Then 0 is a non-isolated point in the triple spectrum $L_{a}$ of a. We know that in this case, $0,1\in L_a\subseteq [0,1]$, with $L_a$ compact. Regarding $F_a$ as a commutative C$^*$-algebra by its identification with $C_{0}(L_a)$ in the standard way, given a positive $\varepsilon$, as in [34, Lemma 4] we can choose positive norm one elements x, y and z in $F_a$ such that $\{y,x,y\} = x$ such that y (and hence, x) is orthogonal to z and $\Vert a - x\Vert <\varepsilon$. In addition, we have that r(x) is orthogonal to $y - r(x)$ (in $F^{**}$). In particular, $Q(y)^2$ must restrict to the identity map on E(x). We claim that

$$\begin{aligned} F \hbox { contains } E(x). \end{aligned}$$

To see which, let w belong to E(x). With $c = \pi _{F} (w)$ we have c lies in F and $\left\| w - Q(y)^2 (c)\right\| = \left\| Q(y)^2 ( w - c)\right\| \le \left\| w - c \right\| ,$ so that $c = Q(y)^2 (c)$ by uniqueness of best approximation, which further implies that c is orthogonal to z. If c is not equal to w choose any positive real $\lambda \le \Vert w-c\Vert$ to give

$$\begin{aligned} \Vert w -(\lambda z + c)\Vert = \Vert (w-c)-\lambda z\Vert = \max \{ \Vert w- c\Vert , \lambda \} = \Vert w-c\Vert , \end{aligned}$$

contradicting unique approximation. Thus $w = c$ and so belongs to F, which proves the claim. Since the above argument is true for all positive $\varepsilon$ we deduce that E(a) is contained in F, as desired.

If a is von Neumann regular, so that its annihilator is nonzero and r(a) lies in F then, since a belongs to $E_2 (r(a))$, F must contain E(a) by Proposition 2(b). The statement (a) follows from $(*)$.

(b) Is immediate from (a) and the comments in page 4. $\square$

Lemma 4

Let F be a JB$^*$-subtriple of a JB$^*$-triple E. Suppose F contains no BP quasi-invertible elements (equivalently, $\partial _e (F_1)=\emptyset$). Then for each $e\in \partial _e(E_1)$ we have $\hbox \mathrm{dist}(e,F)= 1$. If F is a Čebyšëv subspace of E, we have $\pi _{_F} (e) =0$, for every e as above.

Proof

Suppose we can find $x\in F$ satisfying $\Vert e-x\Vert < 1$. Then

$$\begin{aligned} \left\| e-P_2(e) (x)\right\| = \left\| P_2 (e) (e-x)\right\| \le \Vert e-x\Vert <1. \end{aligned}$$

Since e is the unit element of the JB$^*$-algebra $E_2(e)$, we deduce that $P_2 (e) (x)$ is an invertible element in $E_2(e)$. Lemma 2.2 in [25] implies that x is BP quasi-invertible in E, and hence BP quasi-invertible in F, which is impossible. The second statement is clear because dist$(e,F)=1=\Vert e\Vert$.

We establish now an strengthened version of Proposition 2. The result is inspired by an argument in [34, Theorem 4].

Proposition 5

Let F be a Čebyšëv JB$^*$-subtriple of a JB$^*$-triple E. Suppose that a is a non-zero element in F. Then $\{a\}_{E}^{\perp } \subseteq F$.

Proof

Arguing by contradiction, we suppose the existence of an element $x\in \{a\}_{E}^{\perp } \backslash F.$ Fix a real number t and consider the automorphism of F (and E) given by $S_{t}=\exp (it L(a,a))$. It is clear that $S_t (\pi _{_F} (x)) = \pi _{_F} ( S_t (x)),$ for every $t\in \mathbb {R}.$ Having in mind that $a\perp x$ it follows that $L(a,a)^n (x) =0,$ for every natural n, which shows that $S_t (x) = x$ for every real t. Therefore

$$\begin{aligned} \pi _{_F} (x ) =\pi _{_F} (x) + it L(a,a) (\pi _{_F} (x)) + \sum _{n=2}^{\infty } \frac{i^n t^n}{n!} L(a,a) (\pi _{_F} (x)). \end{aligned}$$

Differentiating at $t=0$ we conclude that $L(a,a) (\pi _{_F} (x))=0$, or equivalently $a\perp \pi _{_F} (x)$ (cf. [7, Lemma 1]).

We have proved that $a\perp x, \pi _{_F} (x).$ Therefore $\pi _{_F} (x)+\mu a\in F$, for every $\mu \in \mathbb {C}$ and, by orthogonality,

$$\begin{aligned} 0<\hbox {dist} (x,F)= \Vert x-\pi _{_F} (x)\Vert = \max \{ \Vert x-\pi _{_F} (x)\Vert ,\Vert \mu a\Vert \} = \Vert x-\pi _{_F} (x) -\mu a\Vert , \end{aligned}$$

for every $\mu \in \mathbb {C}$ with $\Vert \mu a\Vert \le \Vert x-\pi _{_F} (x)\Vert ,$ contradicting the uniqueness of the best approximation of x in F. $\square$

We recall that a tripotent u in the bidual of a JB$^*$-triple E is said to be open when $E^{**}_2 (u)\cap E$ is weak$^*$ dense in $E^{**}_2 (u)$ (see [14]). A tripotent e in $E^{**}$ is said to be compact-$G_{\delta }$ (relative to E) if there exists a norm one element a in E such that e coincides with s(a), the support tripotent of a (see [14]). A tripotent e in $E^{**}$ is said to be compact (relative to E) if there exists a decreasing net $(e_{\lambda })$ of tripotents in $E^{**}$ which are compact-$G_{\delta }$ with infimum e, or if e is zero.

Closed and bounded tripotents in $E^{**}$ were introduced and studied in [16] and [17]. A tripotent e in $E^{**}$ such that $E^{**}_0 (e)\cap E$ is weak$^*$ dense in $E^{**}_0(e)$ is called closed relative to E. When there exists a norm one element a in E such that $a= e + P_0 (e) (a)$, the tripotent e is called bounded (relative to E) (cf. [16]). Theorem 2.6 in [16] (see also [19, Theorem 3.2]) asserts that a tripotent e in $E^{**}$ is compact if, and only if, e is closed and bounded.

Corollary 6

Let F be a Čebyšëv JB$^*$-subtriple of a JB$^*$-triple E. Let e be a tripotent in $F^{**}$ satisfying that $F^{**}_2 (e) \cap F\ne \{0\}$. Then $\{e\}_{E}^{\perp } = \{x\in E : x\perp e\} = E\cap E^{**}_0 (e) \subseteq F$. Furthermore, if e is closed in $E^{**}$ relative to E we also have $E^{**}_0 (e) = F^{**}_0(e)$.

Proof

By hypothesis, the set $F\cap F^{**}_2 (e)$ is non-zero, thus, there exists a non-zero element $a\in F\cap F^{**}_2 (e)$. It is easy to check that $\{e\}_{E}^{\perp } \subseteq \{a\}_{E}^{\perp }$, and the latter is contained in F by Proposition 5.

We have already proved that $\{e\}_{E}^{\perp } = E\cap E^{**}_0 (e) \subseteq F$, which implies that $E\cap E^{**}_0 (e) = F\cap F^{**}_0 (e)$. Since e is closed in $E^{**}$, we can assure that

$$\begin{aligned} E^{**}_0 (e) = \overline{E\cap E^{**}_0 (e)}^{\sigma (E^{**},E^*)}= \overline{F\cap F^{**}_0 (e)}^{\sigma (E^{**},E^*)} \subseteq F^{**}_0 (e) \subseteq E^{**}_0 (e). \end{aligned}$$

$\square$

Corollary 7

Let F be a Čebyšëv JB$^*$-subtriple of a JB$^*$-triple E. Let a be a non-zero element in F and let r(a) denote the range tripotent of a in $F^{**}$. Suppose that $\{a\}^{\perp }_{F}\ne \{0\}$. Then $E^{**}_0 (r(a)) = F^{**}_0(r(a))$.

Proof

Proposition 3(a) implies that $E(a) = F(a).$ Therefore $E(a)^{**}=F(a)^{**}$ is an open JB$^*$-subtriple of $E^{**}$ relative to E in the sense employed in [16, 18, 19]. Proposition 3.3 in [19] (or [16, Corollary 2.9]) implies that every compact tripotent in $F(a)^{**}$ is compact in $E^{**}.$ Let us take a compact tripotent $e\in F(a)^{**}$ satisfying that $e\le r(a)$ and $F^{**}_2 (e) \cap F\ne \{0\}$. Since e is compact, and hence closed in $E^{**}$ (cf. [16, Theorem 2.6]), Corollary 6 proves that $E^{**}_0 (e) = F^{**}_0(e).$ Finally, it is easy to see that, since $r(a)\ge e$, $E^{**}_0 (r(a))\subseteq E^{**}_0 (e) = F^{**}_0(e)\subseteq F^{**},$ and hence $E^{**}_0 (r(a)) = F^{**}_0(r(a))$.

We turn now our focus to the Peirce 1-subspace associated with a range tripotent. For this purpose we state the following technical lemma.

Lemma 8

Let e and f be orthogonal tripotents in a JB$^*$-subtriple F of a JB$^*$-triple E such that $E_0 (e)$, $E_0(f)$ and $E_2(e + f)$ are contained in F. Then $E = F$.

Proof

It is sufficient to show that the Peirce 1-subspace of $e + f$ is contained in F (since the other two Peirce subspaces of $e + f$ are automatically in F). It is known (elementary calculation) that $E_1 (e + f)$ is always contained in $E_0(e) + E_0(f)$, which is contained in F by hypothesis, giving the result. $\square$

We can establish now our first main result on Čebyšëv JB$^*$-subtriples of a general JB$^*$-triple.

Theorem 9

Let F be a Čebyšëv JB$^*$-subtriple of a JB$^*$-triple E. Suppose F has rank greater or equal than three. Then $E=F$.

Proof

Since F has rank greater or equal than three, we can find mutually orthogonal norm-one elements a, b, c in F.

Proposition 3(a) yields $E(a+b) = F(a+b)$. From (2.3) we conclude that

$$\begin{aligned} E_2^{**} (r(a+b)) = \overline{E(a+b)}^{\sigma (E^{**},E^*)} = \overline{F(a+b)}^{\sigma (F^{**},F^*)} \subseteq F^{**}. \end{aligned}$$

Corollary 7 now implies that $E_0^{**} (r(a))= F_0^{**} (r(a))$ and $E_0^{**} (r(b))= F_0^{**} (r(b))$. We deduce from Lemma 8 that $E^{**} =F^{**}.$ Finally, as a consequence of the Hahn-Banach Theorem, it is easy to check that $E=F,$ as desired. $\square$

Corollary 10

Let B be a Čebyšëv JB$^*$-subtriple of a C$^*$-algebra A. Suppose B has rank greater or equal than three. Then $A=B$.

It remains to study Čebyšëv JB$^*$-subtriples of rank smaller or equal than two. In this case, the conclusion will follow from the main result in [26] and the studies about finite rank JB$^*$-triples developed in [5] and [2].

Theorem 11

Let F be a non-zero Čebyšëv JB$^*$-subtriple of a JB$^*$-triple E. Then exactly one of the following statements holds:

• (a) F is a rank one JBW$^*$-triple with dim $(F)\ge 2$ (i.e. a complex Hilbert space regarded as a type 1 Cartan factor). Moreover, F may be a closed subspace of arbitrary dimension and E may have arbitrary rank;

• (b) $F= \mathbb {C} e$, where e is a complete tripotent in E;

• (c) E and F are rank two JBW$^*$-triples, but F may have arbitrary dimension;

• (d) F has rank greater or equal than three and $E=F$.

Proof

If F has rank $\ge 3$, Theorem 9 implies that $E=F$. We may therefore assume that F has rank $\le 2$. It follows from [5, Proposition 4.5 and comments at the beggining of §4] (see also [2, §3]) that F is reflexive. So, F is a reflexive JBW$^*$-triple of rank $\le 2$.

We shall adapt next the arguments in the proof of [26, Theorem 13], providing a simplified argument. Every JBW$^*$-triple admits an abundant collection of complete tripotents or extreme points of its closed unit ball (cf. [3, Lemma 4.1] and [32, Proposition 3.5] or [11, Theorem 3.2.3]). Thus, we can find a complete tripotent e in F. There are only two possibilities: either e is minimal in F or e has rank two in F.

When e is rank two in F, we can write $e= e_1+e_2$ with $e_1,e_2$ mutually orthogonal minimal tripotents in F. Proposition 2 proves that $E_2 (e_j) = F_2 (e_j) = \mathbb {C} e_j$, $E_0 (e_j) = F_0 (e_j)$, and $E_0 (e_1+e_2) = F_0 (e_1+e_2)=\{0\},$ which proves that $e_1$ and $e_2$ are minimal tripotents in E, e is complete in E, and E is a rank-2 JBW$^*$-triple.

We finally assume that e is minimal and complete in F. If dim $(F)=1$, then $F= \mathbb {C} e$, and we are in case (b), otherwise we are in case (a). $\square$

It should be remarked here that Remark 7 in [26] provides an example of an infinite dimensional rank-one Čebyšëv JB$^*$-subtriple of a JB$^*$-triple, while [26, Remark13] gives an example of a rank-one Čebyšëv JB$^*$-subtriple of a rank-n JBW$^*$-triple, where n is an arbitrary natural number.