Cebysev subspaces of JBW*-triples

We describe the one-dimensional \v{C}eby\v{s}\"{e}v subspaces of a JBW$^*$-triple $M,$ by showing that for a non-zero element $x$ in $M$, $\mathbb{C}x$ is a \v{C}eby\v{s}\"{e}v subspace of $M$ if, and only if, $x$ is a Brown-Pedersen quasi-invertible element in ${M}$. We study the \v{C}eby\v{s}\"{e}v JBW$^*$-subtriples of a JBW$^*$-triple $M$. We prove that, for each non-zero \v{C}eby\v{s}\"{e}v JBW$^*$-subtriple $N$ of $M$, then exactly one of the following statements holds: $(a)$ $N$ is a rank one JBW$^*$-triple with dim$(N)\geq 2$ (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 $N=M$. We also provide new examples of \v{C}eby\v{s}\"{e}v subspaces of classic Banach spaces in connection with ternary rings of operators.


Introduction
Let V be a subspace of a Banach space X. The subspace V is called ǎ Cebyšëv (Chebyshev) subspace of X if and only if for each x ∈ X there exists a unique point x • ∈ V such that dist(x, V ) = x − x • .
Let K be a compact Hausdorff space. A classical theorem due to A. Haar establishes that an n-dimensional subspace V of the space C(K), of all continuous complex-valued functions on K, is aČebyšëv subspace of C(K) if, and only if, any non-zero f ∈ V admits at most n − 1 zeros (cf. [17] and the monograph [32, p. 215]). Having in mind the Riesz representation theorem, and the characterization of the extreme points of the closed unit ball in the dual space of C(K), we can easily see that, in the above conditions, V is an n-dimensionalČebyšëv subspace of C(K) if, and only if, there exists no set {δ t 1 , . . . , δ tn } of n-mutually orthogonal, pure states such that V ⊂ n i=1 ker(δ t i ). This result implies that any non-zero f in C(K) spans aČebyšëv subspace of the latter space if, and only if, f is invertible in the algebra C(K).
Later on, J.G. Stampfli proved in [33,Theorem 2], that the scalar multiples of the unit element in a von Neumann algebra M is aČebyšëv subspace of M . In [24], D.A. Legg, B.E. Scranton, and J.D. Ward characterize the semi-Čebyšëv and finite dimensionalČebyšëv subspaces of K(H), the algebra of compact operators on an infinite-dimensional Hilbert space H. They conclude that, for a separable Hilbert space H, there existČebyšëv subspaces of every finite dimension in K(H) [24,Theorem 3], when H is not separable K(H) has no finite-dimensionalČebyšëv subspaces [24,Corollary 2].
A.G. Robertson continued with the study onČebyšëv subspaces of von Neumann algebras in [28], where he established the following results: A.G. Robertson and D. Yost prove in [29,Corollary 1.4] that in an infinite dimensional C * -algebra A admits a finite dimensional * -subalgebra B which is also aČebyšëv in A if and only if A is unital and B = C1.
The results proved by Robertson and Yost were complement by G.K. Pedersen, who shows that if A is a C * -algebra without unit and B is ǎ Cebyšëv C * -subalgebra of A, then A = B (compare [27,Theorem 4]).
The previous results of Robertson [28] and Pedersen [27,Theorem 2] also prove the following equivalent reformulation of Theorem 1: for each nonzero element x in a von Neumann algebra M , the following statements are equivalent: A renewed interest onČebyšëv subspaces of C * -algebras has led M. Namboodiri, S. Pramod, and A. Vijayarajan to revisit and generalize the previous contributions of Robertson, Yost and Pedersen in [26].
On the other hand, C * -algebras can be regarded as elements in a strictly wider class of complex Banach spaces called JB * -triples (see §2 for the detailed definitions). Many geometric properties studied in the setting of C *algebras have been also explored in the bigger class of JB * -triples. Howeveř Cebyšëv subspaces and the theory of best approximations remains unexplored in the class of JB * -triples. In this note we present the first results aboutČebyšëv subspaces andČebyšëv subtriples in Jordan structures.
In Section 2 we prove that for a non-zero element x in a JBW * -triple M , Cx is aČebyšëv subspace of M if, and only if, x is a Brown-Pedersen quasi-invertible element in M (see Theorem 6). This result generalizes the result established by Robertson in Theorem 1 (cf. [28]), but it also add a new perspective from an independent argument.
In Section 3 we establish a precise description of the JBW * -subtriples of a JBW * -triple M which areČebyšëv subspaces in M . We should remark that in the setting of von Neumann algebras and C * -algebras, the scarcity of non-trivialČebyšëv * -subalgebras is endorsed with the following results: If an infinite dimensional von Neumann algebra, M , contains a finite dimensional von Neumann subalgebra N which is aČebyšëv subspace in M , then N must be one dimensional (compare Theorem 2 or [28,Theorem 6]). Furthermore, an infinite dimensional C * -algebra A admits a finite dimensional * -subalgebra B which is also aČebyšëv in A if and only if A is unital and B = C1 (cf. [29,Corollary 1.4]). If A is a C * -algebra without unit and B is aČebyšëv C * -subalgebra of A, then A = B (compare [27,Theorem 4]). The first main difference in the setting of JB * -triples is the existence ofČebyšëv JB * -subtriples with arbitrary dimensions; complex Hilbert spaces and spin factors give a complete list of examples (compare Remark 7 and comments before it).
In our main result aboutČebyšëv JBW * -subtriples (cf. Theorem 13), we establish the following criterium: Let N be a non-zeroČebyšëv JBW *subtriple of a JBW * -triple M . Then exactly one of the following statements holds: (a) N and M are rank one JBW * -triples. Moreover, M is a complex Hilbert space and N is a closed subspace of arbitrary dimension; (b) N = Ce, 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 3 and N = M .
It should be remarked at this point that the techniques applied by Robertson, Yost [28,29] and Pedersen [27] in the setting of von Neumann algebras do not make any sense in the wider setting of JBW * -triples. The techniques developed in this paper are completely independent and provide new arguments to understand theČebyšëv JBW * -subtriples of a von Neumann algebra (Corollary 14).

One-dimensionalČebyšëv subspaces and subtriples of JBW * -triples
A complex Jordan triple system is a complex linear space E equipped with a triple product which is bilinear and symmetric in the external variables and conjugate linear in the middle one and satisfies the Jordan identity: for all x, y, a, b, c ∈ E, where L(x, y) : E → E is the linear mapping given by L(x, y)z = {x, y, z}.
A JB * -triple is a complex Jordan triple system E which is a Banach space satisfying the additional "geometric" axioms: (a) For each x ∈ E, the operator L(x, x) is hermitian with non-negative spectrum; Every C * -algebra is a JB * -triple with respect to the triple product given by Every JB * -algebra (i.e. a complex Jordan Banach * -algebra satisfying 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 (2.2). In particular, every complex Hilbert space is a JB * -triple. Other examples of JB * -triples are given by the so-called Cartan factors. A Cartan factor of type 1 is a JB * -triple which coincides with the Banach space B(H, K) of bounded linear operators between two complex Hilbert spaces, H and K, where the triple product is defined by (2.2). Cartan factors of types 2 and 3 are JB * -triples which can be identified the subtriples of B(H) defined by II C = {x ∈ B(H) : x = −jx * j} and III C = {x ∈ B(H) : x = jx * j}, respectively, where j is a conjugation on H. A Cartan factor of type 4 or IV is a spin factor, that is, a complex Hilbert space provided with a conjugation x → x, where the triple product and the norm are defined by {x, y, z} = x/y z + z/y x − x/z ȳ, respectively. The Cartan factors of types 5 and 6 consist of finite dimensional spaces of matrices over the eight dimensional complex Cayley division algebra O; the type V I is the space of all hermitian 3x3 matrices over O, while the type V is the subtriple of 1x2 matrices with entries in O (compare [25], [16], and [12, §2.5]).
A JB * -triple W is called a JBW * -triple if it has a predual W * . It is known that a JBW * -triple admits a unique isometric predual and its triple product is separately σ(W, W * )-continuous (see [3]). The second dual E * * of a JB *triple E is a JBW * -triple with respect to a triple product which extends the triple product of E (cf. [13]).
For more detail of the properties of JB * -triples and JBW * -triples the reader is referred to the monographs [12] and [11].
Given an element a in a JB * -triple E, the symbol Q(a) will denote the conjugate linear operator on E defined by Q(a)(x) = {a, x, a}.
An element e ∈ E is called a tripotent when {e, e, e} = e. Each tripotent e ∈ E induces a decomposition of E, called the Peirce decomposition, in is the i 2 eigenspace of the operator L(e, e), i = 0, 1, 2. This decomposition satisfies the following Peirce rules: It is known that the Peirce-2 subspace E 2 (e) is a JB * -algebra with unit e, Jordan product x• e y := {x, e, y} and involution x * e := {e, x, e}, respectively. Since surjective linear isometries and triple isomorphisms on a JB * -triple coincide (cf. [22,Proposition 5.5]), the triple product in E 2 (e) is uniquely given by We shall make use of the following property: given a tripotent e ∈ E and an element λ in the unit sphere of C, the mapping: is a surjective linear isometry on E and a triple isomorphism (compare [15, A tripotent e ∈ E is said to be unitary if the operator L(e, e) coincides with the identity map I E on E; that is, E 2 (e) = E. We shall say that e is complete or maximal when E 0 (e) = E. When E 2 (e) = P 2 (e)(E) = Ce = {0}, we say that e is minimal.
The complete tripotents of a JB * -triple E coincide with the real and complex extreme points of its closed unit ball E 1 (cf. [ When a is an element in a JBW * -triple M , the sequence (a 1 2n−1 ) converges in the weak * -topology of M to a tripotent, denoted by r(a), called the range tripotent of a. The tripotent r(a) is the smallest tripotent e ∈ M satisfying that a is positive in the JBW * -algebra M 2 (e) (see [14, page 322]).
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 [22, 1.15]). Moreover, there exists a triple isomorphism Ψ : E a → C 0 (L) such that Ψ(a)(t) = t. Clearly, the range tripotent r(a) can be identified with the characteristic function We recall that an element x in a Jordan algebra J with unit e is called invertible if there exists an element y such that x • y = e and x 2 • y = x. The element y is called the inverse of x, and is denoted by x −1 . Inverse of any element x in a Jordan algebra J is unique whenever it exists. The set of all invertible elements in J is denoted by J −1 .
An element element a in a JB * -triple E is called von Neumann regular if and only if there exists b ∈ E such that When a is von Neumann regular, the (unique) element b ∈ E satisfying the above conditions is called the generalized inverse of a, and is denoted by a † . It is known that an element a ∈ E is von Neumann regular if, and only if, Q(a) has norm-closed image if, and only if, the range tripotent r(a) of a lies in E and a is positive and invertible element of the JB * -algebra E 2 (r(a)) (compare [10]). Furthermore, when a is von Neumann regular, Q(a)Q(a † ) = Q(a † )Q(a) = P 2 (r(a)) and L(a, a † ) = L(a † , a) = L(r(a), r(a)) [10, page 192].
Given a pair of elements a, b in a JB * -triple E, the Bergmann operator associated to a and b is the mapping [12, page 22]).
An element a in a JB * -triple E is said to be Brown-Pedersen quasiinvertible (BP-quasi-invertible for short) when it is von Neumann regular with generalized inverse b such that the Bergman operator B(a, b) vanishes; in such a case, b is called the BP-quasi inverse of a. The set of BP-quasi invertible elements in E is denoted by E −1 q [34]. It is established in [34] that an element a ∈ E is BP-quasi-invertible if, and only if, one of the following equivalent statements holds: (i) a is von Neumann regular, and its range tripotent r(a) is an extreme point of the closed unit ball E 1 of E (i.e. r(a) is a complete tripotent of E); (ii) There exists a complete tripotent e ∈ E such that a is positive and invertible in the JB * -algebras E 2 (e).
We recall that two elements a, b in a JB * -triple, E, are said to be orthogonal (written a ⊥ b) if L(a, b) = 0. Lemma 1 in [8] shows that a ⊥ b if and only if one of the following nine statements holds: Let e be a tripotent in a JB * -triple E. Lemma 1.3(a) in [15] shows that for every x 2 ∈ E 2 (e) and every x 0 ∈ E 0 (e). Combining this result with the equivalences in (2.5) we see that  Proof. Arguing by contradiction, we suppose that V ∩ M −1 q = ∅. Let us take x ∈ V with x = 1. By assumptions, x / ∈ M −1 q . Under these conditions, the range complete tripotent of x, r(x) is not complete in M or x is not invertible in the JBW * -algebra M 2 (r(x)). By [19,Lemma 3.12], there exists a complete tripotent e in M such that r(x) ≤ e.
We shall identify the JB * -subtriple, M x , of M generated by x with some C 0 (L) where 1 = x ∈ L ⊆ [0, 1 ] with L ∪ {0} compact (cf. [22, 1.15]). We further know that there exists a triple isomorphism Ψ : M x → C 0 (L) such that Ψ(x)(t) = t, and the range tripotent r(x) identifies with the characteristic function χ (0, x ]∩L ∈ C 0 (L) * * (see page 2). It is clear that, under this identification, for every |λ| ≤ 1 in C. When e = r(x), the element x is not invertible in the JBW * -algebra M 2 (r(x)), and hence e − x = r(x) − x = 1. When e r(x), we have e − r(x) = 1. Thus, applying e − r(x) ⊥ r(x) and (2.6), we further known that We observe that, since e is a complete tripotent, e ∈ M −1 q , and hence e / ∈ V . Since V is aČebyšëv subspace, there exists a unique best approximation, If dist(e, V ) = e − c V (e) ≥ 1, we would have 1 = e ≥ dist(e, V ) = 1, and 1 = e − c V (e) = dist(e, V ) = e − λx , for every |λ| ≤ 1, contradicting the uniqueness of the best approximation of e in V . We can therefore assume that dist(e, V ) < 1. Consequently, there exits y ∈ V with e−y < 1. Corollary 2.4. in [20] implies that y ∈ M −1 q ∩V , which is impossible.
Let e be a tripotent in a JB * -triple E. Let us recall that e is a tripotent in the JBW * -triple E * * , and that Peirce projections associated with e on E * * are weak * -continuous. Goldstine's theorem assures that E is weak * -dense in E * * , and hence, E * * k (e) coincides with the weak * -closure of E k (e) in E * * , for every k = 0, 1, 2. In particular, e is complete in E * * whenever e is a complete tripotent in E. Moreover, since the orthogonal complement of a tripotent e in a JB * -triple F coincides with F 0 (e), we have: The following technical result is part of the folklore in the theory of best approximation (see [28,Lemma 3] or [32, Theorem 2.1]).

Lemma 5. ([28, Lemma 3]). Let x be an element in complex a
Banach space X such that Cx is not aČebyšëv subspace of X. Then there exists an extreme point φ of the closed unit ball of X * , a vector y ∈ X and a scalar We can characterize now the one dimensionalČebyšëv subspaces of a JBW * -triple. Suppose that Cx is not aČebyšëv subspace of M . By Lemma 5 there exists an extreme point φ of the closed unit ball of M * , λ ∈ C\{0}, and y ∈ M such that φ(x) = 0 and φ(y) = y = y − λx .
The above Theorem 6 generalizes the previously commented results obtained by Robertson [28] (compare Theorem 1). In order to find a triple version of the reformulation established by Pedersen in [27, Theorem 2], stated as statement (c) in page 2, we recall some notation.
For each functional ϕ in the predual of a JBW * -triple W , and for each z in W with ϕ(z) = ϕ , and z = 1, the mapping x → x ϕ := (ϕ{x, x, z}) 1/2 defines a pre-Hilbertian semi-norm on W . Moreover, ϕ{x, x, w} = ϕ{x, x, z} whenever w ∈ W with ϕ(w) = ϕ and w = 1 (cf. [1, Proposition 1.2]). It is known that for every x ∈ W (see [2, page 258]). The inequality in (2.7) together with Lemma 5 imply the following property: Let x be a non-zero element in a JBW * -triple M such that Cx is ǎ Cebyšëv subspace of M . Then for each extreme point ϕ of the closed unit ball of M * we have x ϕ 0. It would be interesting to know under what additional hypothesis, the condition x ϕ 0, for every extreme point ϕ of the closed unit ball of M * , implies that x is BP-quasi invertible.

3.Čebyšëv subtriples of JBW * -triples
In this section, we shall determine the JBW * -subtriples of a JBW * -triple M which areČebyšëv subspaces in M . Let us recall that in the case of an infinite dimensional von Neumann algebra M , if a finite dimensional von Neumann subalgebra N of M is aČebyšëv subspace in M then N must be one dimensional (compare Theorem 2 or [28, Theorem 6]). Furthermore, an infinite dimensional C * -algebra A admits a finite dimensional * -subalgebra B which is also aČebyšëv in A if and only if A is unital and B = C1 (cf. [29,Corollary 1.4]). The scarcity of non-trivialČebyšëv C * -subalgebras in general C * -algebras can be better understood with the following result due to G.K. Pedersen: If A is a C * -algebra without unit and B is aČebyšëv C * -subalgebra of A, then A = B (compare [27,Theorem 4]).
The first main difference in the setting of JB * -triples is the existence of Cebyšëv JB * -subtriples with arbitrary dimensions. For example, let E = H be a complex Hilbert space regarded as a type 1 Cartan factor with the Hilbert norm and the product where ., . denotes the inner product of H. The Orthogonal Projection theorem tells that any closed subspace of H is aČebyšëv subspace of H and clearly a JB * -subtriple.
The following remark provides an additional example.
Remark 7. Let E be a spin factor with triple product and norm given by {x, y, z} = x/y z + z/y x − x/z ȳ, x is a conjugation on E, and ./. denotes the inner product of E. Let K be a closed subspace of E with K = K. Clearly, K is a JB * -subtriple of E. Since K is a closed subspace of the complex Hilbert space E, there exists an orthogonal projection P of E onto K. Since E = K H, where H = (I − P )(E) with K/H = 0. Since K = K, we also have H = H. Given η ∈ K and ξ ∈ H, it is easy to check that Moreover, η + ξ = η if and only if ξ = 0. This shows that P : E → E is a bi-contractive for the norm . , and for each x ∈ E, P (x) is the unique best approximation of x in K. Therefore, K is aČebyšëv JB * -subtriple of E. We observe that the dimensions of E and K can be arbitrarily big.
We can present now our conclusions onČebyšëv JB * -subtriples.
The next property ofČebyšëv subspaces is probably part of the folklore in the theory of best approximation in normed spaces, but we couldn't find an exact reference.
for every x ∈ X.
Proof. Let x be an element in X. The condition P ≤ 1 implies that The element P c V (P (x)) ∈ P (V ) ⊆ V . Thus, the uniqueness of the best approximation in V proves that P c V (P (x)) = c V (P (x)).
Proposition 9. Let F be aČebyšëv JB * -subtriple of a JB * -triple E. Suppose e is a non-zero tripotent in F . Then E 0 (e) = F 0 (e). Consequently, every complete tripotent in F is complete in E.
Proof. Since e is a tripotent in F and the latter is a JB * -subtriple of E, e is a tripotent in E and F 0 (e) ⊆ E 0 (e). Arguing by contradiction, let us assume Having in mind that e ∈ E 2 (e) ⊥ E 0 (e) ∋ b − c F (b), we deduce, via (2.6), that for every |λ| ≤ dist(b, F ). This contradicts the uniqueness of the best approximation, Proposition 10. Let F be aČebyšëv JB * -subtriple of a JB * -triple E. Suppose e is a tripotent in F with F 0 (e) = {e} ⊥ F = 0. Then E 2 (e) = F 2 (e). Proof. Clearly F 2 (e) ⊆ E 2 (e). We have to show that E 2 (e) ⊆ F 2 (e). Suppose, on the contrary, that E 2 (e)\F 2 (e) = E 2 (e)\F = ∅. Pick b ∈ E 2 (e)\F .
By Lemma 8 applied to P = P 2 (e), X = E and V = F , we deduce that P 2 (e)(c F (b)) = c F (b).
By hypothesis, F 0 (e) = {e} ⊥ F = 0. So, there exists a norm-one element z ∈ F 0 (e). The conditions b, ∈ E 2 (e), c F (b) ∈ F 2 (e) and z ∈ F 0 (e) combined with 2.6 give for every |λ| ≤ dist(b, F ), which contradicts the uniqueness of the best approximation of b in F because c F (b) − λz ∈ F , for every λ in the above conditions.
Let e and v be tripotents in a JB * -triple E. We shall say that v ≤ e, when e − v is a tripotent in E with e − v ⊥ v (compare the notation in [15]).
Let E be a JB * -triple. A subset S ⊆ E is said to be orthogonal if 0 / ∈ S and x ⊥ y for every x = y in S. The minimal cardinal number r satisfying card(S) ≤ r for every orthogonal subset S ⊆ E is called the rank of E (and will be denoted by r(E)). Given a tripotent e ∈ E, the rank of the Peirce-2 subspace E 2 (e) will be called the rank of e.
Theorem 3.1 in [4] combined with Proposition 4.5.(iii) in [6] assure that a JB * -triple is reflexive if and only if it is isomorphic to a Hilbert space if, and only if, it has finite rank.
Suppose E is a rank-one JB * -triple. The above comments show that E is reflexive and hence a JBW * -triple. Let e be a complete tripotent in E. Since the rank of e is smaller than the rank of E, we deduce that e is a minimal tripotent in E. Proposition 3.7 in [9] and its proof show that E = {e} ⊥⊥ = {0} ⊥ is a rank-one Cartan factor of the form L(H, C), where H is a complex Hilbert space or a type 2 Cartan factor II 3 (it is known that II 3 is JB * -triple isomorphic to a 3-dimensional complex Hilbert space). This shows the following: Proof. We can always find a complete tripotent e in N (see the comments in page 6). Proposition 9 implies that e is complete in M (i.e. M 0 (e) = {0}). We have three possibilities: (i) e has rank one in N ; (ii) e has rank 2 in N ; (iii) e has rank greater or equal than 3 in N .
(i) Suppose first that e has rank one in N . In this case, e is a minimal and complete tripotent in N . Therefore, N is a complex Hilbert space regarded as a type 1 Cartan factor (cf. Lemma 12 or Proposition 3.7 in [9]).
We claim that e has rank one in M or N 1 (e) = {0}. Suppose, on the contrary to our claim, that e is not rank one in M and N = Ce We define a 0 = 0 0 1 0 0 1 0 0 ∈ C ≡ B ⊆ M.
Considering the identification of B and C given above, it is a good exercise to see that dist(a 0 , N ) = dist(a 0 , Ce ⊕ Ch)) = dist( a 0 , C e ⊕ C h) which contradicts that a 0 admits a unique best approximation in N . This proves the claim. If e has rank one in M , then M is a rank one JB * -triple and hence a complex Hilbert space regarded as a type 1 Cartan factor and we are in statement (a). If N = Ce (i.e. N 1 (e) = {0}) we are in case (b).
(ii) We assume now that e has rank 2 in N . Then there exist two nonzero minimal, mutually orthogonal tripotents e 1 , e 2 ∈ N with e = e 1 + e 2 . Propositions 9 and 10 show that M 2 (e j ) = N 2 (e j ), and M 0 (e j ) = N 0 (e j ), for every j in {1, 2}. Since e is complete in M , we also have M = M 2 (e) ⊕ M 1 (e). We shall prove the desired statement by showing that e 1 and e 2 are minimal in M . The statement concerning the dimension of N follows from the example in Remark 7.
We have therefore shown that x = P 1 (e 2 )(x) + P 0 (e 2 )(x) ∈ N , which implies that M 1 (e 1 ) ⊆ N and consequently M = N . This concludes the proof.
Let us recall that a C * -algebra is reflexive if and only if it if finite dimensional (cf. [30,Proposition 2]). Consequently, a C * -algebra has finite rank if and only if it is finite dimensional. In particular, the result established by Robertson in [28, Theorem 6] (see Theorem 2) is a direct consequence of our last corollary: