- Research
- Open Access
Čebyšëv subspaces of JBW^{∗}-triples
- Fatmah B Jamjoom^{1},
- Antonio M Peralta^{2, 3}Email author,
- Akhlaq A Siddiqui^{1} and
- Haifa M Tahlawi^{1}
https://doi.org/10.1186/s13660-015-0813-2
© Jamjoom et al. 2015
- Received: 8 June 2015
- Accepted: 3 September 2015
- Published: 17 September 2015
Abstract
- (a)
N is a rank-one JBW^{∗}-triple with \(\dim(N)\geq2\) (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 ≥2;
- (d)
N has rank greater than or equal to three, and \(N=M\).
We also provide new examples of Čebyšëv subspaces of classic Banach spaces in connection with ternary rings of operators.
Keywords
- Čebyšëv/Chebyshev subspace
- JBW^{∗}-triples
- Čebyšëv/Chebyshev subtriple
- von Neumann algebra
- Brown-Pedersen quasi-invertibility
- spin factor
- minimum covering sphere
MSC
- 41A50
- 41A52
- 41A65
- 46L10
- 17C65
- 46L05
1 Introduction
Let V be a subspace of a Banach space X. The subspace V is called a Čebyšëv (Chebyshev) subspace of X if and only if for each \(x\in X\) there exists a unique point \(x_{0 }\in V\) such that \(\operatorname {dist}(x,V)=\Vert x-x_{0}\Vert \). The uniqueness of \(x_{0}\) plays a key role in this paper (see, for example, Lemma 3 and Proposition 9).
Let K be a compact Hausdorff space. A classical theorem due to Haar establishes that an n-dimensional subspace V of the space \(C(K)\), of all continuous complex-valued functions on K, is a Čebyšëv subspace of \(C(K)\) if and only if any non-zero \(f\in V\) admits at most \(n-1\) zeros (cf. [1] and the monograph [2], 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-dimensional Čebyšëv subspace of \(C(K)\) if and only if for every set \(\{\delta_{t_{1}},\ldots, \delta _{t_{n}}\}\) of n-mutually orthogonal pure states, we have \(V\cap\bigcap_{i=1}^{n} \ker(\delta_{t_{i}}) = \{0\}\). This result implies that any non-zero f in \(C(K)\) spans a Čebyšëv subspace of the latter space if and only if f is invertible in the algebra \(C(K)\).
Later on, Stampfli proved in [3], Theorem 2, that the scalar multiples of the unit element in a von Neumann algebra M is a Čebyšëv subspace of M. In [4], Legg et al. characterize the semi-Čebyšëv and finite dimensional Čebyšëv 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 exist Čebyšëv subspaces of every finite dimension in \(K(H)\) [4], Theorem 3, when H is not separable, \(K(H)\) has no finite-dimensional Čebyšëv subspaces [4], Corollary 2.
Robertson continued with the study on Čebyšëv subspaces of von Neumann algebras in [5], where he established the following results.
Theorem 1
([5], Theorem 6)
Let x be a non-zero element in a von Neumann algebra M. Then the one-dimensional 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 px is left invertible in pM and \((1-p)x\) is right invertible in \((1-p){M}\).
Theorem 2
([5], Theorem 6)
Let N be a finite dimensional ^{∗}-subalgebra of an infinite dimensional von Neumann algebra M. Suppose that N has dimension >1. Then N is not a Čebyšëv subspace of M.
Robertson and Yost prove in [6], Corollary 1.4, that an infinite dimensional C^{∗}-algebra A admits a finite dimensional ^{∗}-subalgebra B which is also a Čebyšëv in A if and only if A is unital and \(B=\mathbb{C} 1\).
The results proved by Robertson and Yost were complemented by 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 [7], Theorem 4).
- (a)
\(\mathbb{C} x\) is a Čebyšëv subspace of M;
- (b)
x is Brown-Pedersen quasi-invertible in M (see page 6 for the precise definition of this notion);
- (c)
For each pure state (i.e., for each extreme point of the positive part of the closed unit ball of \(M^{*}\)) \(\varphi\in M^{*}\), and for each unitary \(u\in M\), we have \(\varphi(x^{*} x) +\varphi(u x x^{*} u ) >0\).
A renewed interest in Čebyšëv subspaces of C^{∗}-algebras has led Namboodiri, Pramod and Vijayarajan to revisit and generalize the previous contributions of Robertson, Yost and Pedersen in [9].
On the other hand, C^{∗}-algebras can be regarded as elements in a strictly wider class of complex Banach spaces called JB^{∗}-triples (see Section 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. However, Čebyšëv subspaces and the theory of best approximations remains unexplored in the class of JB^{∗}-triples. In this note we present the first results about Čebyšëv subspaces and Čebyšëv subtriples in Jordan structures.
In Section 2 we prove that for a non-zero element x in a JBW^{∗}-triple M, \(\mathbb{C}x\) is a Čebyšëv subspace of M if and only if x is a Brown-Pedersen quasi-invertible element in M (see Theorem 6). This theorem generalizes the result established by Robertson in Theorem 1 (cf. [5]), but it also adds 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 are Čebyšëv subspaces in M. We should remark that in the setting of von Neumann algebras and C^{∗}-algebras, the scarcity of non-trivial Čebyšëv ^{∗}-subalgebras is endorsed by Theorems 1 and 2 and [6, 7]. The first main difference in the setting of JB^{∗}-triples is the existence of Čebyšëv 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 we give a complete description of all Čebyšëv JBW^{∗}-subtriples of an arbitrary JBW^{∗}-triple (see Theorem 14). We provide examples of infinite dimensional proper Čebyšëv JBW^{∗}-subtriples of JBW^{∗}-triples (see Remark 7). We apply the solution of the minimum covering sphere problem in the Euclidean space \(\ell _{2}^{m}\) to present new examples of Čebyšëv subspaces of classical Banach spaces (cf. Remark 12) and to construct an example of a rank-one Hilbert space which is a Čebyšëv JBW^{∗}-subtriple of a rank-n JBW^{∗}-triple, where n is an arbitrary natural number (cf. Remark 13).
It should be remarked at this point that the techniques applied by Robertson, Yost [5, 6] and Pedersen [7] 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 the Čebyšëv von Neumann subalgebras of a von Neumann algebra (Corollary 15).
2 One-dimensional Čebyšëv subspaces of JBW^{∗}-triples
- (a)
For each \(x\in E\), the operator \(L(x,x)\) is hermitian with non-negative spectrum;
- (b)
\(\Vert \{x,x,x\}\Vert =\Vert x\Vert ^{3}\) for all \(x\in E\).
A JB^{∗}-triple W is called a JBW ^{∗} -triple if it has a predual \(W_{\ast}\). It is known that a JBW^{∗}-triple admits a unique isometric predual, and its triple product is separately \(\sigma (W,W_{\ast})\)-continuous (see [14]). The second dual \(E^{\ast \ast}\) of a JB^{∗}-triple E is a JBW^{∗}-triple with respect to a triple product which extends the triple product of E (cf. [15]).
For more details of the properties of JB^{∗}-triples and JBW^{∗}-triples, the reader is referred to the monographs [13] and [16].
Given an element a in a JB^{∗}-triple E, the symbol \(Q(a)\) will denote the conjugate linear operator on E defined by \(Q(a)(x)=\{ a,x,a\}\).
The separate weak^{∗}-continuity of the triple product of a JBW^{∗}-triple M implies that Peirce projections associated with a tripotent e in M are weak^{∗}-continuous.
A tripotent \(e\in E\) 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 \(E_{2} (e) = P_{2}(e) (E) =\mathbb {C}e\neq\{0\}\), 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. [19], Lemma 4.1 and [20], Proposition 3.5 or [13], Theorem 3.2.3). Consequently, the Krein-Milman theorem assures that every JBW^{∗}-triple admits an abundant set of complete tripotents [13], Corollary 3.2.4.
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 \(C_{0}(L)\), where \(\Vert a\Vert \in L\subseteq[0,\Vert a\Vert ]\) with \(L\cup\{0\}\) compact (cf. [18], Corollary 1.15). Moreover, there exists a triple isomorphism \(\Psi: E_{a} \to C_{0}(L)\) such that \(\Psi(a) (t) =t\).
When a is an element in a JBW^{∗}-triple M, the sequence \((a^{\frac{1}{2n-1}})\) 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 \(e\in M\) satisfying that a is positive in the JBW^{∗}-algebra \(M_{2}(e)\) (see [21], p.322). Clearly, the range tripotent \(r(a)\) can be identified with the characteristic function \(\chi_{{(0,\Vert a\Vert ]\cap L}}\in C_{0}(L)^{**}\) (see [22], beginning of Section 2).
We recall that an element x in a Jordan algebra \(\mathcal{J}\) with unit e is called invertible if there exists an element y such that \(x\circ y=e\) and \(x^{2}\circ y=x\). The element y is called the inverse of x and is denoted by \(x^{-1}\). The inverse of any element x in a Jordan algebra \(\mathcal{J}\) is unique whenever it exists. The set of all invertible elements in \(\mathcal{J}\) is denoted by \(\mathcal {J}^{-1}\).
Given a pair of elements a, b in a JB^{∗}-triple E, the Bergmann operator associated to a and b is the mapping \(B(a,b):E \rightarrow L(E)\) defined by \(B(a,b)=\mathit{Id}_{E}-2L(a,b)+Q(a)Q(b)\) (cf. [13], p.22).
- (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 \(e\in E\) such that a is positive and invertible in the JB^{∗}-algebra \(E_{2}(e)\).
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 \(\{e\}^{\perp}=\) \(E_{0 }(e)\) and \(\{a\}^{\perp}=\) \((E^{\ast\ast })_{0 }(r(a))\cap E\) for every \(a\in E\) (cf. [26], Lemma 3.2).
Lemma 3
Let V be a non-zero Čebyšëv subspace of a JBW ^{∗}-triple M. Then \(V\cap M_{q}^{-1}\neq\emptyset\), where \(M_{q}^{-1}\) denotes the set of BP-quasi-invertible elements of M.
Proof
Arguing by contradiction, we suppose that \(V\cap M_{q}^{-1}= \emptyset\).
Let us take \(x\in V\) with \(\Vert x\Vert=1\). By assumptions, \(x\notin M_{q}^{-1}\). By [27], Lemma 3.12, there exists a complete tripotent e in M such that \(r(x)\leq e\), where \(r(x)\) denotes the range tripotent of x.
We observe that, since e is a complete tripotent, \(e\in M_{q}^{-1}\), and hence \(e\notin V\). Since V is a Čebyšëv subspace, there exists a unique best approximation \(c_{V} (e)\in V\) of e in V satisfying \(\operatorname {dist}(e,V)=\Vert e- c_{V} (e)\Vert >0\).
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.
Lemma 4
Let e be a complete tripotent in a JB ^{∗}-triple E. Then \(\{e\} ^{\perp}_{{E^{**}}}=\{0\}\), that is, e is not orthogonal to any non-zero element in \(E^{**}\).
The following technical result is part of the folklore in the theory of best approximation (see [5], Lemma 3 or [2], Theorem 2.1).
Lemma 5
([5], Lemma 3)
- (a)
\(\phi(x)=0\);
- (b)
\(\phi(y)=\) \(\Vert y\Vert =\Vert y-\lambda x\Vert \).
We can characterize now the one-dimensional Čebyšëv subspaces of a JBW^{∗}-triple.
Theorem 6
- (a)
\(\mathbb{C}x\) is a Čebyšëv subspace of M;
- (b)
x is a Brown-Pedersen quasi-invertible element in M.
Proof
The implication (a) ⇒ (b) follows from Lemma 3.
(b) ⇒ (a) Suppose that x is BP-quasi-invertible in M. We note that the support tripotent \(r(x)\) of x is complete in M, and hence a complete tripotent in \(M^{\ast\ast}\) (cf. Lemma 4 and comments before it).
Suppose that \(\mathbb{C}x\) is not a Čebyšëv subspace of M. By Lemma 5 there exists an extreme point ϕ of the closed unit ball of \(M^{\ast}\), \(\lambda\in \mathbb{C}\backslash\{0\}\), and \(y\in M\) such that \(\phi(x)=0\) and \(\phi(y)=\Vert y\Vert=\Vert y-\lambda x\Vert\).
The support tripotent \(\upsilon=s(\phi)\) of ϕ in \(M^{\ast\ast }\) is a (non-zero) minimal tripotent in \(M^{**}\) satisfying \(\phi =P_{2}(\upsilon)^{*} \phi=\phi P_{2}(\upsilon)\) and \(\phi(z) \upsilon =P_{2}(\upsilon) (z)\), \(\forall z\in M^{\ast\ast}\) (cf. [17], Proposition 4). Therefore, \(P_{2}(\upsilon)(x)=\phi (x)\upsilon=0\).
We may suppose that \(\Vert y\Vert=1\). Since \(P_{2}(\upsilon)(y)=\phi (y) \upsilon=\upsilon\), Lemma 1.6 in [17] implies that \(P_{1}(\upsilon)(y)=0\), which shows that \(y=\upsilon+P_{0}(\upsilon )y\). We similarly get \(P_{1}(\upsilon)(y-\lambda x)=0\) (we simply observe that \(\phi(y-\lambda x) = \Vert y\Vert = \Vert y-\lambda x\Vert =1\)). Therefore, \(P_{1}(\upsilon)(x)=0\), and \(x=P_{0}(\upsilon)x\in (M^{\ast\ast})_{0}(\upsilon )=((M^{\ast\ast})_{2}(\upsilon))^{\perp}\), implying that \(x\perp \upsilon\). The equivalent statements in (2.5) prove that \(r(x)\perp\upsilon\), which contradicts Lemma 4. □
The above Theorem 6 generalizes the previously commented results obtained by Robertson in [5] (compare Theorem 1). We have been unable to find a triple version of the reformulation established by Pedersen in [7], Theorem 2, stated as statement (c) on page 2. However, we do have a partial result in that direction.
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 \(\mathbb{C}x\) is a Čebyšëv subspace of M. Then, for each extreme point φ of the closed unit ball of \(M^{*}\), we have \(\Vert x \Vert _{\varphi }\gneqq0\). It would be interesting to know under what additional hypothesis the condition \(\Vert x \Vert _{\varphi}\gneqq0\) 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 are Čebyšëv subspaces in M. The scarcity of non-trivial Čebyšëv C^{∗}-subalgebras in general C^{∗}-algebras can be better understood with the following result due to Pedersen: If A is a C^{∗}-algebra without unit and B is a Čebyšëv C^{∗}-subalgebra of A, then \(A=B\) (compare [7], Theorem 4).
The following remark provides an additional example.
Remark 7
We can present now our conclusions on Čebyšëv JB^{∗}-subtriples.
The next property of Čebyšëv subspaces is probably part of the folklore in the theory of best approximation in normed spaces, but we could not find an exact reference.
Lemma 8
Proof
Proposition 9
Let F be a Čebyšëv JB ^{∗}-subtriple of a JB ^{∗}-triple E. Suppose that 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 \(F_{0} (e) \subseteq E_{0}(e)\). Arguing by contradiction, let us assume that there exists \(b\in E_{0} (e) \backslash F_{0} (e) = E_{0} (e) \backslash F \neq \emptyset\). Since \(\operatorname {dist}(b,F) >0\) and F is a Čebyšëv subspace, there exists a unique \(c_{F} (b) \in F\) such that \(\Vert b-c_{F}(b)\Vert = \operatorname {dist}(b,F)\).
Proposition 10
Let F be a Čebyšëv JB ^{∗}-subtriple of a JB ^{∗}-triple E. Suppose that e is a tripotent in F with \(F_{0}(e) = \{e\}^{\perp}_{F}\neq0\). Then \(E_{2} (e) = F_{2} (e)\).
Proof
Clearly \(F_{j} (e) \subseteq E_{j} (e)\) for \(j=0,1,2\). We have to show that \(E_{2} (e) \subseteq F_{2} (e)\). Suppose, on the contrary, that \(E_{2} (e) \backslash F_{2} (e) = E_{2} (e) \backslash F\neq\emptyset\). Pick \(b\in E_{2} (e) \backslash F\). Since F is a Čebyšëv subspace of E, there exists a unique \(c_{{F}} (b)\in F\) satisfying \(\Vert b- c_{F}(b) \Vert = \operatorname {dist}(b,F)>0\).
By Lemma 8 applied to \(P= P_{2} (e)\), \(X= E\) and \(V=F\), we deduce that \(P_{2} (e) ( c_{F}(b)) = c_{F}(b)\).
Let e and v be tripotents in a JB^{∗}-triple E. We shall say that \(v\leq e\), when \(e-v\) is a tripotent in E with \(e-v\perp v\) (compare the notation in [17]).
Let E be a JB^{∗}-triple. A subset \(S \subseteq E\) is said to be orthogonal if \(0 \notin S\) and \(x \perp y\) for every \(x \neq y\) in S. The minimal cardinal number r satisfying \(\operatorname {card}(S) \leq r\) 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.
Theorem 3.1 in [31] combined with Proposition 4.5(iii) in [32] assures 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 that 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 [26] and its proof show that \(E= \{e\}^{\perp\perp} = \{ 0\} ^{\perp}\) is a rank-one Cartan factor of the form \(L(H,\mathbb{C})\), 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 three-dimensional complex Hilbert space). We have proved the following.
Lemma 11
Every JB ^{∗}-triple of rank one is JB ^{∗}-isomorphic (and hence isometric) to a complex Hilbert space regarded as a type 1 Cartan factor. □
The above result is also stated in [33], Corollary in p.308.
We have shown several examples of Hilbert spaces (regarded as a type 1 Cartan factor) which are Čebyšëv JB^{∗}-subtriples of JB^{∗}-triples of rank one and two. We present next more examples of Hilbert spaces which are Čebyšëv JB^{∗}-subtriples of JB^{∗}-triples having a bigger rank. The first example is a construction with classical Banach spaces and the second one is an isometric translation to the setting of JB^{∗}-triples.
Remark 12
It is well known that a solution to the minimum covering sphere problem always exists, the center \((\lambda,\mu)\) and the radius ρ are unique (cf. [34, 35]). This shows that every element \(x=(\lambda^{1}_{1} \xi_{1} + \lambda^{1}_{2} \xi _{2},\ldots, \lambda^{n}_{1} \xi_{1} + \lambda^{n}_{2} \xi_{2})\) in X admits a unique best approximation in V, which proves the claim.
Remark 13
Let e and u be two colinear complete tripotents in a JB^{∗}-triple E. Let us assume that we can find two sets \(\{e_{1},\ldots,e_{n}\}\) and \(\{ u_{1},\ldots,u_{n}\}\) of mutually orthogonal tripotents in \(E_{2}(e)\) and \(E_{2}(u)\), respectively, such that \(e_{i}\mathop{\top} u_{i}\) for all i and \(u_{i}\perp e_{j}\) for every \(i\neq j\). Take, for example, \(E=M_{n\times (2n)} (\mathbb{C})\), \(e=\sum_{i=1}^{n} w_{i,i}\), \(u=\sum_{i=1}^{n} w_{i,i+n}\), \(e_{i}= w_{i,i}\) and \(u_{i} = e=w_{i,i+n}\), where \(w_{i,j}\) is the matrix with entry 1 at the position i, j and zero elsewhere.
The theorem describing the Čebyšëv JBW^{∗}-subtriples of a JBW^{∗}-triple can be stated now. We shall show that the examples given in Remarks 7 and 13 are essentially the unique examples of non-trivial Čebyšëv JBW^{∗}-subtriples.
Theorem 14
- (a)
N is a rank-one JBW ^{∗}-triple with \(\dim(N)\geq2\) (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 ≥2;
- (d)
N has rank greater than or equal to three, and \(N=M\).
Proof
- (i)
e has rank one in N;
- (ii)
e has rank two in N;
- (iii)
e has rank greater than or equal to three in N.
(i) Suppose first that e has rank one in N. In this case, e is a minimal and complete tripotent in N and a complete tripotent in M. Therefore, N is a complex Hilbert space regarded as a type 1 Cartan factor (cf. Lemma 11 or Proposition 3.7 in [26]). If \(\dim N= 1\), then (b) holds. If \(\dim N \geq2\), (a) holds.
In the latter case, the examples given before Remark 7 and in Remark 13 show that N may have arbitrary dimension and M may have rank as big as desired.
(ii) We assume now that e has rank two in N. Then there exist two non-zero minimal, mutually orthogonal tripotents \(e_{1},e_{2}\in N\) with \(e= e_{1}+e_{2}\). Propositions 9 and 10 show that \(M_{2} (e_{j})= N_{2} (e_{j})\), and \(M_{0} (e_{j})= N_{0} (e_{j})\neq\{0\}\) for every j in \(\{1,2\}\). Since \(M_{2} (e_{j})= N_{2} (e_{j}) = \mathbb{C} e_{j}\), we deduce that \(e_{1}\) and \(e_{2}\) are minimal tripotents in M. We also know that \(e= e_{1} +e_{2}\) is a complete tripotent in M (i.e., \(M= M_{2} (e) \oplus M_{1}(e)\)), which proves that M has rank two. The statement concerning the dimension of N follows from the example in Remark 7. Thus (c) holds.
Let us recall that a C^{∗}-algebra is reflexive if and only if it is finite dimensional (cf. [36], Proposition 2). Consequently, a C^{∗}-algebra has finite rank if and only if it is finite dimensional. It is further known that a C^{∗}-algebra A has rank one if and only if \(A= \mathbb{C} 1\). In particular, the result established by Robertson in [5], Theorem 6 (see Theorem 2) is a direct consequence of our last theorem.
Corollary 15
Let M be an infinite dimensional von Neumann algebra. Let N be a Čebyšëv von Neumann subalgebra of M. Then \(N = \mathbb{C} 1\) or \(M=N\). □
We have already seen that, for each natural n, we can find a complex Hilbert space (of dimension two) which is a Čebyšëv JB^{∗}-subtriple of a JB^{∗}-triple having rank n. It is natural to ask whether we can find a precise description of those complex Hilbert spaces which are Čebyšëv JBW^{∗}-subtriples of a JBW^{∗}-triple. Another general question that remains open in this paper is the following:
Problem 16
Determine the Čebyšëv JB^{∗}-subtriples of a general JB^{∗}-triple.
Declarations
Acknowledgements
We would like to thank the anonymous referees for their useful suggestions and comments. Their careful and thorough reviews improved the final version of this paper and solved a gap in the original arguments in the proof of Lemma 3.
The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No. RG-1435-020. The second author also is partially supported by the Spanish Ministry of Economy and Competitiveness project No. MTM2014-58984-P.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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