# On the Geometry of the Unit Ball of aJB*-Triple

Authors: Haifa M. Tahlawi; Akhlaq A. Siddiqui; Fatmah B. Jamjoom
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We explore aJB*-triple analogue of the notion of quasi invertible elements, originally studied by Brown and Pedersen in the setting of C*-algebras. This class of BP-quasi invertible elements properly includes all invertible elements and all extreme points of the unit ball and is properly included in von Neumann regular elements in a JB*-triple; this indicates their structural richness. We initiate a study of the unit ball of a JB*-triple investigating some structural properties of the BP-quasi invertible elements; here and in sequent papers, we show that various results on unitary convex decompositions and regular approximations can be extended to the setting of BP-quasi invertible elements. Some C*-algebra and JB*-algebra results, due to Kadison and Pedersen, Rørdam, Brown, Wright and Youngson, and Siddiqui, including the Russo-Dye theorem, are extended to JB*-triples.


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## Body

## 1. Introduction

Brown and Pedersen [1] introduced a notion of quasi invertible elements in a C*-algebra. As is well explained in [1, 2], the Brown-Pedersen quasi (in short, BP-quasi) invertible elements bear many interesting properties similar to those of invertible elements. They have successfully demonstrated the significant role of BP-quasi invertible elements in studying geometry of the unit ball of a C*-algebra; in particular, they obtained several results verifying that the relationships between the extreme convex decomposition theory and the distance αq(x) from an element x to the set of BP-quasi invertible elements are analogous with the relationships in the earlier C*-algebra theory of unitary convex decompositions and regular approximations.It is widely believed that the underlying structure making several interesting results onC*-algebras hold is not the presence of an associative product xy but the presence of the Jordan triple product {xyz}=(1/2)(xyz+zyx). This provided one of the stimuli for the development of Jordan algebra or Jordan triple product generalizations of C*-algebras; these include JB*-algebras and JB*-triples together with their subclasses which are Banach dual spaces (cf. [3]). In [4, 5], we introduced an exact analogue of the BP-quasi invertible elements for JB*-triples. By using certain identities concerning the Bergman operators, we established that in case of nondegenrate Jordan triples B(a,b)=0⇒B(b,a)=0 and {aba}=a and that a is BP-quasi invertible with BP-quasi inverse b⇔B(a,b)=0 [5]. Our aim in this paper is to study geometric consequences of these facts in the setting of complex JB*-triples. After discussing some basics, we begin by recording a known result: {aba}=a⇒a is von Neumann regular that admits a tripotent ea, called its range tripotent. We prove that B(a,b)=0⇔ea is an extreme point of the unit ball and that the set of BP-quasi invertible elements of a JB*-triple is open in the norm topology. Then we investigate the natural analogue of the Russo-Dye theorem [6, 7] on the representability of the elements with means of extreme points of the unit ball. Later on, we look at convex combinations of extreme points of the unit ball. In the course of our analysis, we obtain JB*-triple analogues of some C*-algebra results; this approach provides alternative proofs for some of the corresponding known results for C*-algebras.A Jordan triple system is a vector space𝒥 over a field of characteristic not 2, endowed with a triple product {xyz} which is linear and symmetric in the outer variables x,z, and linear or antilinear in the inner variable y satisfying the Jordan triple identity: {xu{yvz}}+{{xvy}uz}-{yv{xuz}}={x{uyv}z} [3]. A JB*-triple is a complex Banach space 𝒥 together with a continuous, sesquilinear, operator-valued map (x,y)∈𝒥×𝒥↦L(x,y) that defines a triple product L(x,y)z:={xy*z} in 𝒥 making it a Jordan triple system such that each L(x,x) is a positive hermitian operator on 𝒥 and ∥{xx*x}∥=∥x∥3 for all x∈𝒥. Any bounded symmetric domain in a complex Banach space is biholomorphically equivalent to the open unit ball of a JB*-triple [8, 9]. An important example, from the viewpoint of the classical theory of operators and matrices, is the triple system ℬ(ℋ,𝒦) of all bounded linear operators between complex Hilbert spaces ℋ and 𝒦 under the product {xyz}=(1/2)(xy*z+zy*x) and the operator norm, where x* denotes the Hilbert adjoint of x (cf. [10]). In the finite-dimensional case, this can be viewed as adding an algebraic product to a linear matrix space ℳmn(ℂ) of all m×n matrices with complex number entries, namely, the triple product {xyz}=(1/2)(xytz+zytx) where yt denotes the conjugate transpose of the matrix y and the norm defined by ∥z∥=zzt (see [11, Example  4.7]). Any JB*-algebra (cf. [3]) is a JB*-triple under the triple product {xy*z}:=(x∘y*)∘z+(z∘y*)∘x-(x∘z)∘y*, where “∘” denotes the underlying Jordan binary product.A basic operatorP(x,y) on the JB*-triple 𝒥 is defined by P(x,y)z={xz*y} for all x,y,z∈𝒥; we write P(x,x) in short as P(x). The Bergman operator B(x,y) is defined on 𝒥 by B(x,y)=I-2L(x,y)+P(x)P(y), where I is the identity operator [12, 2.11]. The operators L(x,y) and P(x,y) are the JB*-triple analogues of the usual Jordan algebra operators Vx,yz:={xyz} and Ux,yz:={xzy}, respectively; in fact, L(x,y)=Vx,y* and P(x,y)z=Ux,yz* for all z∈𝒥. The Bergman operator takes the form B(x,y)=I-2Vx,y*+UxUy*, which translates to B(x,y)z = (1-xy*)z(1-y*x) for the C*-algebra case.


## 2. BP-Quasi Invertible Elements

We begin with the following characterization of BP-quasi invertible elements of aJB*-triple (cf. [5, Theorem  6]).Definition 1.
In anyJB*-triple, an element x is BP-quasi invertible with BP-quasi inverse y if (and only if) the Bergman operator B(x,y)=0.In anyJB*-triple 𝒥, B(x,y)=0⇔B(y,x)=0 by [5, Theorem  2], and so the relation of being BP-quasi inverse of some element is symmetric in 𝒥. Generally, BP-quasi inverse in not unique: if x∈𝒥q-1 with BP-quasi inverse y, then P(y)x is also a BP-quasi inverse of x since B(x,P(y)x) = I-2L(x,P(y)x) + P(x)P(P(y)x) = I-2L(P(x)y,y) + P(x)P(y)P(x)P(y) = I-2L(x,y) + P(x)P(y) (cf. [12])  =B(x,y)=0. We denote the set of BP-quasi invertible elements in 𝒥 by 𝒥q-1. 𝒥q-1 includes the set of all invertible elements 𝒥-1 and the set of all extreme points ℰ(𝒥)1 of the closed unit ball (𝒥)1; and in case of a JB*-algebra 𝒥, the BP-quasi invertibility is invariant under the involution “*” (cf. [4]).In theJB*-triple ℳ23(ℂ), all matrices of the form [α000β0], where α,β are any nonzero complex numbers, are BP-quasi invertible with BP-quasi inverse of the form [1/α0001/β0].An elementx in a JB*-triple 𝒥 is called von Neumann regular if there exists y∈𝒥 with x=P(x)y. Such an element y is called generalized inverse of x [13]. If 𝒥 is a JB*-algebra, then x=P(x)y={xy*x}, and so y is a generalized inverse of x in 𝒥 considered as a JB*-triple if and only if y* is a generalized inverse of x in the JB*-algebra 𝒥. From [5, Theorem  3], we know that x and y are von Neumann regular with generalized inverses of each other if B(x,y)=0. Hence, any BP-quasi invertible element is necessarily von Neumann regular. For a von Neumann regular element in the C*-algebra ℳ2(ℂ) of 2×2 matrices that is not BP-quasi invertible, see [5, Example  9]. Thus, the class of BP-quasi invertible elements is a proper subclass of the von Neumann regular elements.For any fixed elementa in a JB*-triple 𝒥, the underlying linear space 𝒥 becomes a complex Jordan algebra 𝒥[a] with respect to the Jordan product x·ay:={xay}, called a-homotope of 𝒥 (cf. [3]). An element u in a JB*-triple 𝒥 is called unitary if L(u,u)z=z [3]; the set of all unitary elements is denoted by 𝒰(𝒥). For any u∈𝒰(𝒥), the u-homotope of 𝒥 is a JB*-algebra with respect to the original norm and the involution “*u” given by x*u=P(u)x (cf. [3]); such a homotope is denoted by 𝒥[u], called a unitary isotope of 𝒥. The symbols Pu(x,y), Lu(x,y), Bu(x,y) will denote the respective analogues of the operators P(x,y), L(x,y), and B(x,y) on the isotope 𝒥[u]. Thus, Pu(x,y)z={xz*uy}u and Lu(x,y)z={xy*uz}u for all x,y,z∈𝒥; in particular, Pu(x)z=Pu(x,x)z={xz*ux}u. Here, {⋯}u denotes the induced Jordan triple product in 𝒥[u].Theorem 2.
For any fixed unitary elementu of a JB*-triple 𝒥 and for all x,y,z∈𝒥, one has the following:(i)
the Jordan triple product{xy*uz}u in 𝒥[u] coincides with {xy*z};(ii)
Pu(x)=P(x);(iii)
Lu(x,y)=L(x,y);(iv)
Bu(x,y)=B(x,y);(v)
(𝒥[u])q-1=(𝒥)q-1.Proof.
(i) By using the basic Jordan triple identity, we get that(1){xy*uz}u={x{u(y*u)*u}*z}+{y*uu*{xu*z}}-{{xu*y*u}u*z}-{y*uu*{xu*z}}+{zu*{xu*y*u}}={x{u(y*u)*u}*z}={x((y*u)*u)*z}={xy*z}.
(ii) By the part (i),Pu(x)z={xz*ux}u={xz*x}=P(x)z for all x∈𝒥.
(iii) For anya,b∈𝒥, P(a,b)=P(a+b)-P(a)-P(b) (cf. [12]). This together with the part (ii) gives Lu(x,y)z=Pu(x,z)y=Pu(x+z)y-Pu(x)y-Pu(z)y=P(x+z)y-P(x)y-P(z)y=P(x,z)y = L(x,y)z for all x,y,z∈𝒥.
(iv) Follows from parts (ii) and (iii) sinceB(x,y)=I-2L(x,y)+P(x)P(y).
(v)x∈𝒥q-1⇔B(x,y)=0⇔Bu(x,y)=0 (by (iv)) ⇔x∈(𝒥[u])q-1.We close this section with the following observation on images of BP-quasi invertible elements under triple homomorphisms.Theorem 3.
For any closed idealV in JB*-triple 𝒥, the triple homomorphism ϕ:𝒥→𝒥/V given by ϕ(x)=x+V preserves the BP-quasi invertibility.Proof.
V being closed ideal of 𝒥 is a JB*-triple, and so is the quotient 𝒥/V. The quotient JB*-triple 𝒥/V admits the canonical surjective triple homomorphism ϕ:𝒥→𝒥/V defined by ϕ(x)=x+V such that ϕ({xyz})={ϕ(x)ϕ(y*)ϕ(z)} for all x,y,z∈𝒥 [8, Proposition 5.5].
Now, for any fixedx∈𝒥q-1 with inverse y∈𝒥 and for all z∈𝒥, B(ϕ(x),ϕ(y))ϕ(z) = ϕ(z)-2{ϕ(x)(ϕ(y))*ϕ(z)} + {ϕ(x){ϕ(y)(ϕ(z))*ϕ(y)}*ϕ(x)} = ϕ(z)-2ϕ({xy*z}) + ϕ({x{yz*y}*x}) = ϕ(B(x,y)(z)) = ϕ(0)=0. Hence, ϕ(x)∈(𝒥/V)q-1.


## 3. Positivity of BP-Quasi Invertibles

In this section, we prove that any BP-quasi invertible element in aJB*-triple 𝒥 is positive invertible in the Peirce 1-space 𝒥1(v):={x∈𝒥∣L(v,v)x=x} of the operator L(v,v) for certain extreme point v of the closed unit ball. Besides other results, we obtain another characterization of the BP-quasi invertible elements. The following result is known in pieces; we include it with a unified proof.Theorem 4.
For each von Neumann regular elementx of a JB*-triple 𝒥, there exists a unique tripotent ex∈𝒥 such that x is positive invertible in 𝒥1(ex).Proof.
Sincex is von Neumann regular, there exists a unique tripotent ex∈𝒥, called the range tripotent of x (cf. [10]) such that x is invertible in the Peirce space 𝒥1(ex) (cf. [10, Lemma  3.2], [14, page 540], and [15, Theorem  1]). Let 𝒥(x) denotes the norm closure of P(x)(𝒥)={x𝒥x}, which is the smallest norm closed inner ideal of 𝒥 containing x (cf. [16, pages 19-20]). From [10, Lemma  3.2], we see that P(x)(𝒥) is norm closed in 𝒥. Hence, 𝒥(x)=P(x)(𝒥)=𝒥1(ex) since x is von Neumann regular in 𝒥 (cf. [10, page 572]). Thus, x is positive in 𝒥1(ex) by [16, Proposition  2.1].Remark 5.
The range tripotentex of x is defined as the least tripotent for which x is a positive element in the JB*-algebra 𝒥1(ex). If u and v are two tripotents in 𝒥 such that x is positive invertible in both the Peirce spaces 𝒥1(u) and 𝒥1(v), then u=v; for details, see [17, Lemma  3.3] and [16, Proposition  2.1].Theorem 6.
Let𝒥 be a JB*-triple and x∈𝒥q-1. Then the range tripotent ex is a unique extreme point of (𝒥)1 such that x is positive invertible in 𝒥1(ex). Moreover, there exists a unique y∈𝒥q-1 satisfying P(x)y=x, P(y)x=y, P(x)P(y)=P(y)P(x), and L(x,y)=L(ex,ex)=L(y,x).Proof.
Sincex being BP-quasi invertible is a von Neumann regular element, Theorem 4 gives the existence of a unique tripotent ex in 𝒥 such that x is positive invertible in 𝒥1(ex). By [10, Lemma  3.2], there exists a unique generalized inverse y∈𝒥 satisfying all the conditions of the theorem; in fact, this generalized inverse is called the Moore-Penrose inverse, usually denoted by x†. Since {xx†x}=x, {zx†x} = {zx†{xx†x}} = 2{{zx†x}x†x}-{x{x†zx†}x} for all z∈𝒥. So that P(x)P(x†) = 2L(x,x†)2-L(x,x†). Hence, B(x,x†) = 1-3L(x,x†)+2L(x,x†)2 = (2L(x,x†)-1)(L(x,x†)-1). In view of [18, Lemma  2.1] and [10, page 573], we have L(x,x†)=L(ex,ex)=L(x†,x) since ex is the unique von Neumann regular element in (𝒥)1(ex). Therefore, B(x,x†)=B(ex,ex)=Π0, where Πμ stands for the Peirce projection associated with the tripotent ex onto the subtriple 𝒥μ(ex)={z:L(ex,ex)z=μz} (cf. [12]). In [19, page 192], authors show that ex is unitary in P(x)𝒥, hence P(x)𝒥=𝒥1(ex). Therefore, ex∈ℰ(𝒥)1 by [20, Lemma  4]. Thus, B(x,x†)=B(ex,ex)=0. We conclude that y∈𝒥q-1.Remark 7.
The unique BP-quasi inverse of any BP-quasi invertible matrixx in the JB*-triple ℳ23(ℂ), satisfying the conditions of Theorem 6, is precisely the Moore-Penrose inverse x† (cf. [10, Lemma  3.2]), which can be calculated according to the rank of the matrix (cf. [21]). If rank(x)=2 then xxt is invertible in ℳ2(ℂ), and the Moore-Penrose inverse of x is given by x†=xt(xxt)-1. For example, let x=[1-10011]. Then x is BP-quasi invertible and its unique BP-quasi inverse, via Theorem 6, is given by x†=(1/3)[2-11112]. Since x(x†)t=(x†)tx=e (the unit in ℳ2(ℂ)), x(x†)ty=(x†)txy=y for all y∈M23 and so L(x,x†)=I. Thus, B(x,x†)=0.Next result establishes an important topological property of the BP-quasi invertible elements in aJB*-triple.Theorem 8.
For anyJB*-triple 𝒥, the set 𝒥q-1 is open in the norm topology.Proof.
AssumeB(a,c)=0 where c=a†+z for some nonzero z∈𝒥, and a† denotes the Moore-Penrose type inverse of a (cf. [4, 10]). Consequently, P(a)c={ac*a}=a, and so a = {a(a†+z)*a} = {a(a†)*a}+{az*a} = a+{az*a}. Thus, P(a)z={az*a}=0, which means P(a†)P(a)z=0. Recall that Π1=P2(v)=P(a†)P(a) (see [10] and [19, page 192]). Also, since {a(Π1h)*a}=0 implies that Π1h=0, the nonzero element z∉range(Π1). Moreover, as a projection, Π0 is the identity operator on 𝒥0(v), so Π0B(a,c)Π0=IΠ0=Π0=0 since B(a,c)=0. Thus, range(Π0)=0 and z∈range(Π1/2)=𝒥1/2(v). Conversely, if a is BP-quasi invertible with the unique BP-quasi inverse a† and z∈range(Π1/2)=𝒥1/2(v), then B(a†,a+z) = 1-3L(a†,a+z)+2L(a†,a+z)2 = 1-3L(a†,a) + 2L(a†,a)2-3L(a†,z) + 2(L(a†,a)L(a†,z) + L(a†,z)L(a†,a) + L(a†,z)2) = B(a†,a) + 2(L(a†,a)L(a†,z) + L(a†,z)L(a†,a) + L(a†,z)2). Since B(a†,a)=0 and {𝒥1𝒥0𝒥}=0 [12, Theorem  5.4 (9)], L(a†,z)=0, and we get the value 0 for the right hand side. Hence a+z is BP-quasi invertible. The elements x∈𝒥1(v):=range(Π1) with spectrum Sp(L(x,x))|𝒥1>0 form an open subset of 𝒥1. It follows that {x+z:Sp(L(x,x))|𝒥1(v)>0,z∈𝒥1/2(v)} is an open subset of 𝒥. Thus, 𝒥q-1 is an open set.Now, we give the following improvement of [22, Theorem  4.12].Theorem 9.
Letx be an invertible element in a unital JB*-algebra 𝒥. Then there exists unique u∈𝒰(𝒥) such that x is positive and invertible in 𝒥[u].Proof.
Sincex∈𝒥-1, there is a unique element y∈𝒥 with x∘y=e and x2∘y=x where “∘” denotes the Jordan binary product and e is the unit in 𝒥. Since 𝒥-1⊆𝒥q-1 and that any JB*-algebra is a JB*-triple under the triple product {ab*c}:=a∘(b*∘c)-b*∘(a∘c)+c∘(a∘b*), so, by Theorem 6, there exists a unique u∈ℰ(𝒥)1 (viz, the range tripotent ex) such that x is positive and invertible in 𝒥1(u) with the inverse of the same y in 𝒥1(u) (cf. [15, Theorem  6]). Moreover, x†=x-1, and so Vu,u*=L(u,u)=L(x,x†)=I. Hence, the Peirce decomposition of 𝒥 reduces to 𝒥=𝒥1(u). Of course, the product and the involution defined on both the JB*-algebra 𝒥1(u) and the unitary isotope 𝒥[u] of 𝒥 are the same. Further, e = e∘uu = u∘ue = {uu*e} = u∘u* and u=u∘uu = {uu*u} = 2u∘(u∘u*)-u2∘u*, so that u2∘u*=u. Hence, u∈𝒥-1 with inverse u*; that is, u∈𝒰(𝒥). We conclude that 𝒥1(u) and 𝒥[u] coincide as JB*-algebras.The following result shows that positive invertibility of an elementx in the Peirce 1-space for some extreme point of the unit ball is sufficient for the BP-quasi invertibility of x.Theorem 10.
Let𝒥 be a JB*-triple and x∈𝒥∖𝒥q-1. Then x is not positive invertible in the Peirce 1-space 𝒥1(v), for all v∈ℰ(𝒥)1.Proof.
Supposex is positive invertible in 𝒥1(v) for some v∈ℰ(𝒥)1. Then there exists a unique inverse y∈𝒥1(v) satisfying x·vy=y·vx=v and x2·vy=x. Since (𝒥(v))-1⊆(𝒥(v))q-1, x is von Neumann regular with generalized inverse y satisfying Pv(x)y=x and Pv(y)x=y, where Pv(a)b={ab*va}v. Being positive, x is self-adjoint in 𝒥1(v); that is, x=x*v. However, the invertibility is invariant under the involution (see above). Therefore, x∈(𝒥1(v))-1 with inverse y such that y=x-1v = (x*v)-1v = (x-1v)*v = y*v=Pvy by Theorem 2. This together with the von Neumann regularity of x in 𝒥1(v) gives x=Pv(x)y = P(x)P(v)y = P(x)y. Thus, the element x is von Neumann regular in 𝒥.
Now, by Theorem4, there exists a (unique) tripotent ex∈𝒥 such that x is positive invertible in 𝒥1(ex). Hence, by Remark 5 and our supposition on x, we get the coincidence 𝒥1(ex)=𝒥1(v).
Next, by Theorem6, y is the unique generalized inverse of x satisfying P(x)y=x, P(y)x=y, and L(x,y)=L(v,v)=L(y,x). Hence, B(x,y)=B(v,v)=0. This means x∈𝒥q-1, a contradiction to the hypothesis.The above result together with Theorem6 gives another characterization of the BP-quasi invertibility, as follows.Theorem 11.
An elementx in a JB*-triple 𝒥 is BP-quasi invertible if and only if x is positive and invertible in the Peirce 1-space 𝒥1(v) for some extreme point v∈ℰ(𝒥)1.


## 4. An Analogue of the Russo-Dye Theorem

We investigate the natural analogue of the Russo-Dye theorem on the representability of the elements with arithmetic means of extreme points of the unit ball in aJB*-triple. We extend [7, Theorem  2.2] for JB*-triples as follows.Theorem 12.
Let𝒥 be a JB*-triple with v∈ℰ(𝒥)1 and s∈(𝒥1(v))1∘ (the open unit ball of 𝒥1(v)). Then for any positive integer n, v+(n-1)s=∑i=1nvi with vi∈ℰ(𝒥)1 for each i=1,…,n.Proof.
Sincev∈ℰ(𝒥)1, 𝒥1(v) is a JB*-algebra with unit v. Since ∥-s∥<1 in 𝒥1(v), v+s and so (1/2)(v+s) are invertible elements in 𝒥1(v) (cf. [22, Lemma  2.1(iii)]). Clearly, ∥(1/2)(v+s)∥<1. Hence, by [7, Lemma  2.1], there exist unitaries v1,v2∈𝒰(𝒥1(v)) with v+s=v1+v2. Moreover, v1,v2∈ℰ(𝒥)1 by [20, Lemma  4]. The required result now follows by induction on n.Definition 13.
For any elementx in a JB*-triple 𝒥, the number em(x) is defined by em(x):=min{n:x=(1/n)∑j=1n‍vj,vj∈ℰ(𝒥)1}. If x has no such decomposition, then em(x):=∞.Next result describes a basic connection betweenem(x) and the distance to the set ℰ(𝒥)1.Theorem 14.
Let𝒥 be a JB*-triple, v∈ℰ(𝒥)1 and x∈𝒥 with nx-v∈𝒥1(v) satisfying ∥nx-v∥<n-1 for some n≥2. Then em(x)≤n. On the other hand, if em(x)≤n, then dist(nx,ℰ(𝒥)1)≤n-1.Proof.
Clearly,(n-1)-1(nx-v)∈(𝒥1(v))1∘. Replacing s by (n-1)-1(nx-v) in Theorem 12, we get x=(1/n)(v+(n-1)s)=(1/n)∑i=1nvi with vi∈ℰ(𝒥)1 for i=1,…,n. Thus, em(x)≤n. On the other hand, suppose em(x)≤n. Then x=r-1∑i=1rvi for some 1≤r≤n with vis in ℰ(𝒥)1. Then ∥x∥≤r-1∑i=1r∥vi∥=1. Further, ∥rx-v1∥ = ∥∑i=2rvi∥≤r-1. Hence, ∥nx-v1∥ = ∥(n-r)x+rx-v1∥≤∥(n-r)x∥ + ∥rx-v1∥≤n-r + r-1 = n-1 since ∥x∥≤1. Thus, dist(nx,ℰ(𝒥)1) = infw∈ℰ(𝒥)1∥nx-w∥≤∥nx-v1∥≤n-1.For anyv∈ℰ(𝒥)1, the set ℰ(𝒥)1 includes the set of all unitaries in 𝒥1(v), as seen above from [20, Lemma  4]; the following example shows that the inclusion generally is proper.Example 15.
Let𝒥 be the JB*-triple ℳ12(ℂ), and let v denotes the matrix [10]. Then ∥v∥=1 and vvtv=v. So, v is a norm one tripotent matrix. Moreover, for any matrix z=[z1z2], we have Lv,vz=[z1(1/2)z2]. Also note that Pvz=[z1¯0], where zi¯ denotes the complex conjugate of zi. So that, PvPvz=[z10]. Hence, B(v,v)z=[z1z2]-2[z1(1/2)z2]+[z10]=[00]. Thus, v∈ℰ(𝒥)1.
Next, consider the Peirce spaces of𝒥, relative to the extreme point matrix v=[10], given by 𝒥1(v)={z:z=[z10],z1∈ℂ}, 𝒥1/2(v)={z:z=[0z2],z2∈ℂ} and 𝒥0(v)={[00]}. Clearly, both the balls (𝒥1(v))1 and (𝒥1/2(v))1 are nontrivial. It is easily seen that the matrix [01]∈ℰ(𝒥)1∖ℳ1(v).From [6, 7], we know different proofs of strict analogue of the Russo-Dye theorem for general (unital) JB*-algebras. Unfortunately, there exists no exact analogue of the Russo-Dye theorem for nonunital JB*-triples. The following result is an appropriate analogue for an arbitrary JB*-triple, where coℰ(𝒥)1 and co¯ℰ(𝒥)1 denote the convex hull of ℰ(𝒥)1 and its norm closure, respectively.Theorem 16 (A Russo-Dye Theorem type forJB*-triples).
Let𝒥 be a JB*-triple, v∈ℰ(𝒥)1. (i)
For anyx∈𝒥1(v) with ∥x∥<1-2n-1 for some n≥3, there exists vi∈ℰ(𝒥)1 for all i=1,2,3,…,n such that x=(1/n)∑i=1nvi.(ii)
(𝒥1(v))1∘⊆coℰ(𝒥)1.(iii)
(𝒥1(v))1⊆co¯ℰ(𝒥)1.Proof.
(i) Since∥x∥<1-2n-1, we have ∥nx∥<n-2, so that ∥(n-1)-1(nx-v)∥<1. Also, note that (n-1)-1(nx-v)∈𝒥1(v); recall that x is in the JB*-algebra 𝒥1(v). Hence, by taking v the same as in Theorem 12 while s=(n-1)-1(nx-v), we get nx=∑i=1nvi for some extremes vis in 𝒥.
(ii) Supposex∈(𝒥1(v))1∘. Then ∥x∥<1-2n-1 for some positive integer n≥3. Therefore, x∈coℰ(𝒥)1 by the part (i).
(iii) Using the part (ii), we get(𝒥1(v))1=(𝒥1(v)¯)1∘⊆co¯ℰ(𝒥)1.Remark 17.
If aJB*-triple 𝒥 has a BP-quasi invertible element then ℰ(𝒥)1 is a nonempty set by Theorem 6. Hence, the above theorem holds in this case.Corollary 18.
For any extreme pointv of the closed unit ball in a JB*-triple 𝒥, (𝒥1(v))1⊂co¯ℰ(𝒥)1. If v is a unitary element in 𝒥, then (𝒥)1=co¯ℰ(𝒥)1=co¯𝒰(𝒥). Moreover, ⋃w∈ℰ(𝒥)1‍(𝒥1(w))1 = co¯𝒰(𝒥) = co¯ℰ(𝒥)1 = (𝒥)1.Proof.
We know from [20, Lemma  4] that 𝒰(𝒥1(v))⊆ℰ(𝒥)1. So, by the Russo-Dye Theorem for JB*-algebras [7, Theorem  2.3], (𝒥1(v))1=(𝒥1(v))1∘¯=co¯𝒰(𝒥1(v))⊆co¯ℰ(𝒥)1. By Example 15, there exists a JB*-triple 𝒥 with u,v∈ℰ(𝒥)1 such that u∉𝒥1(v). Hence, we have (𝒥1(v))1⊂co¯ℰ(𝒥)1.
Ifv∈𝒰(𝒥), then Lv,v=I (the identity operator on 𝒥)  (cf. [3]). So, 𝒥=𝒥1(v) which is a unital JB*-algebra, and hence 𝒥 is a JB*-algebra. Then, (𝒥)1=(𝒥1(v))1⊆co¯ℰ(𝒥)1⊆(𝒥)1=co¯𝒰(𝒥) by [7, Theorem  2.3]. Hence, (𝒥)1=co¯ℰ(𝒥)1=co¯𝒰(𝒥).
As mentioned above, every unitary elementv in a JB*-triple 𝒥 satisfies Lv,v=I, and so P(v)P(v)=I (cf. [10, page  582]). Hence, v is an extreme point of (𝒥)1 by [23, Lemma  3.2 and Proposition  3.5]. Thus, for any v∈𝒰(𝒥), (𝒥)1 = (𝒥1(v))1⊆⋃w∈ℰ(𝒥)1‍(𝒥1(w))1⊆(𝒥)1 = co¯𝒰(𝒥). Therefore, ⋃w∈ℰ(𝒥)1‍(𝒥1(w))1 = co¯𝒰(𝒥) = co¯ℰ(𝒥)1 = (𝒥)1.Corollary 19.
Each element of the Peirce 1-space𝒥1(v) in a JB*-triple 𝒥 with v∈ℰ(𝒥)1 is a positive multiple of the sum of three extreme points of (𝒥)1.Proof.
Letx∈𝒥1(v) and ϵ>0. Let y=(3∥x∥+ϵ)-1x. Then ∥y∥<1/3. Hence, by Theorem 16(i), there exist three extreme points u1,u2,u3 in 𝒥 such that y=(1/3)(u1+u2+u3). Thus, x=(∥x∥+ϵ/3)(u1+u2+u3).


## 5. Convex Combinations of Extreme Points

We continue investigating convex combinations, not necessarily means, of extreme points of the closed unit ball. The following result gives an analogue of [7, Lemma  2.1] for BP-quasi invertible elements in a JB*-triple.Theorem 20.
In anyJB*-triple 𝒥, 𝒥q-1∩(𝒥)1⊆(1/2)(ℰ(𝒥)1+ℰ(𝒥)1).Proof.
Letx∈𝒥q-1∩(𝒥)1. Then, there is a unique v∈ℰ(𝒥)1 such that x is positive and invertible in 𝒥1(v) by Theorem 6. In particular, x is self-adjoint in the JB*-algebra 𝒥1(v). This together with ∥x∥≤1 gives the existence of unitaries v1,v2∈𝒥1(v) satisfying x=(1/2)(v1+v2) by [24, Theorem  2.11]. The result now follows from [20, Lemma  4].The unit ball of aJB*-triple 𝒥 often has extreme points in abundance, as in the case when 𝒥 is a Hilbert space H; however, the set ℰ(𝒥)1 may be empty as in the JB*-triple 𝒥=C∘(H) of compact operators on H [25, page 151].Theorem 21.
Let𝒥 be a JB*-triple and v∈ℰ(𝒥)1. Let x∈(𝒥1(v))1 with dist(x,𝒰(𝒥1(v))<2α and α<1/2. Then x∈αℰ(𝒥)1+(1-α)ℰ(𝒥)1.Proof.
Since𝒥1(v) is a JB*-algebra with unit v, x∈α𝒰(𝒥1(v))+(1-α)𝒰(𝒥1(v)) by [26, Corollary  3.4]. Hence, x∈αℰ(𝒥)1+(1-α)ℰ(𝒥)1 by [20, Lemma  4].Let𝒥 be a JB*-triple and x∈(𝒥)1. We define V(x):={β≥1:x∈coβℰ(𝒥)1}, where coβℰ(𝒥)1:={β-1∑i=1n-1vi+β-1(1+β-n)vn:vj∈ℰ(𝒥)1,j=1,…,n} in which the positive integer n satisfies the condition n-1<β≤n. We observe the following immediate connection of these constructs with the number em(x).Lemma 22.
Let𝒥 be a JB*-triple. Let x∈𝒥 and n∈ℕ (the set of positive integers). Then(i)
x∈conℰ(𝒥)1⇔em(x)≤n.(ii)
n∈V(x)⇔em(x)≤n.(iii)
em(x)=min(V(x)∩ℕ).An extension of [27, Proposition  3.1] for unital JB*-algebras appeared in [28, Theorem  2.2]. We use Theorem 6, to deduce an analogue of the same result for JB*-triples, as follows.Theorem 23.
Let𝒥 be a JB*-triple, v∈ℰ(𝒥)1 and x∈(𝒥)1∩𝒥1(v).(a)
Let∥γx-v0∥≤γ-1 for some γ≥1 and some v0∈𝒰(𝒥1(v)). Let (δ2,…,δm)∈ℝm-1 such that 0≤δj<γ-1 for all j=2,…,m-1 and γ-1+∑j=2m-1δj=1. Then there exist v1,…,vm∈ℰ(𝒥)1 satisfying x=γ-1v1+∑j=2mδjvj. Moreover, ]γ,∞[⊆V(x).(b)
On the other hand, if]γ,∞[⊆V(x) holds, then for each r>γ, there is v1∈ℰ(𝒥)1 such that ∥rx-v1∥≤r-1.Proof.
Since𝒥1(v) is a unital JB*-algebra, the part (a) follows from [20, Theorem  2.2] and [20, Lemma  4].
(b) Suppose]γ,∞[⊆V(x). If r>γ then x∈corℰ(𝒥)1 with x=r-1(v1+⋯+vn-1+(1+r-n)vn) for some v1,…,vn∈ℰ(𝒥)1 and positive integer n satisfying n-1<r≤n. Hence, ∥rx-v1∥=∥∑j=2n-1‍vj+(1+r-n)vn∥≤(n-2)+(1+r-n)=r-1.Corollary 24.
For anyJB*-triple 𝒥, coγℰ(𝒥)1⊆coδℰ(𝒥)1 if 1≤γ≤δ. In particular, for each x∈(𝒥)1∩𝒥1(v), V(x) is either empty or equal to [γ,∞) or (γ,∞) for some γ≥1.


## 6. Distance to the Extreme Points

In this section, we prove some results on distances from an element of aJB*-triple 𝒥 to the set ℰ(𝒥)1 and to the set 𝒥q-1. We define the function αq:𝒥→ℝ by αq(x):=inf{∥x-u∥:u∈𝒥q-1}.Lemma 25.
Let𝒥 be a JB*-triple with nonempty ℰ(𝒥)1 and x∈𝒥. Then(i)
αq(rx)=|r|αq(x) for all r∈ℂ;(ii)
αq(x)≤∥x∥;(iii)
|αq(x)-αq(y)|≤∥x-y∥ for all y∈𝒥;(iv)
if𝒥 is a JB*-algebra, then αq(x)=αq(x*).Proof.
(i) For any complex numberr≠0, x∈𝒥q-1 if and only if rx∈𝒥q-1. So, the part (i) is clear for all nonzero complex numbers. If r=0, then αq(rx)=αq(0)=0, one may consider the BP-quasi invertible element (1/n)v with v∈ℰ(𝒥)1 and positive integer n→0.
(ii) Since for anyv∈ℰ(𝒥)1 and any positive integer n, (1/n)v∈𝒥q-1; hence, αq(x) = dist(x,𝒥q-1)≤∥x-(1/n)v∥≤∥x∥+1/n. It follows that αq(x)≤∥x∥.
(iii) Since the setℰ(𝒥)1 is nonempty, and since any extreme point is BP-quasi invertible, α(x)<∞. Now, by definition of αq(x), for each n∈ℕ, there exists an element xn∈𝒥q-1 such that ∥x-xn∥≤αq(x)+1/n. Hence, αq(y)≤∥y-xn∥≤∥y-x∥+∥x-xn∥≤∥x-y∥+αq(x)+1/n for all n∈ℕ, so that αq(y)≤∥x-y∥+αq(x). Interchanging x and y, we get αq(x)≤∥x-y∥+αq(y). Thus, |αq(x)-αq(y)|≤∥x-y∥ for all y∈𝒥.
(iv) Now, let the𝒥 be a JB*-algebra. If x∈𝒥q-1, then x*∈𝒥q-1 by [4, Theorem  3.2], and hence αq(x)=0=αq(x*). Next, if x is not BP-quasi invertible, then αq(x) = dist(x,𝒥q-1) = inf{∥x-u∥:u∈𝒥q-1} = inf{∥x*-u∥:u∈𝒥q-1} = αq(x*) since the involution “*” is an isometry.α
q
(
x
) provides information about dist(x,ℰ(𝒥)1) as follows.Theorem 26.
Let𝒥 be a JB*-triple, v∈ℰ(𝒥)1 and x∈𝒥1(v)∖𝒥q-1. Then dist(x,ℰ(𝒥)1) ≥ max{αq(x)+1,∥x∥-1}.Proof.
Sincev∈ℰ(𝒥)1, ∥x-v∥≥∥x∥-1. We assume ∥x-v∥<1, then the fact that v is the unit of the JB*-algebra 𝒥1(v) gives the invertibility of x in 𝒥1(v), and so x is positive and invertible in the unitary isotope of 𝒥1(v) induced by certain unitary element u in 𝒥1(v); hence, u∈ℰ(𝒥)1 by [20, Lemma  4]. For any z∈𝒥1(v)[u], we have L(v,v)z=z and Lv(u,u)=I on 𝒥1(v) since u is unitary in 𝒥1(v), so that L(u,u)z=Lv(u,u)z=z by Theorem 2. Hence, 𝒥1(v)[u]⊆𝒥1(u). For the reverse inclusion, recall that u∈ℰ(𝒥)1, and so 𝒥1(u)=P(u)(𝒥) (see the above proof of Theorem 4); similarly, 𝒥1(v)=P(v)(𝒥), and so u=P(v)w for some w∈𝒥. Hence, for any fixed z∈𝒥1(u), there exists s∈𝒥 with z=P(u)s=P(P(v)w)s=P(v)P(w)P(v)s∈P(v)𝒥=𝒥1(v). Thus, 𝒥1(u)⊆𝒥1(v)[u]. Therefore, 𝒥1(v)[u]=𝒥1(u) as sets. Moreover, we observe that a∘vub={au*vb}v={au*b}=a∘ub and a*vu={ua*vu}v={ua*u}=a*u by Theorem 2. We conclude that (𝒥1(v))[u] and 𝒥1(u) coincide as JB*-triples and x is positive invertible in the Peirce 1-space 𝒥1(u). So, x∈𝒥q-1 by Theorem 10; a contradiction to the hypothesis. Thus, ∥x-v∥≥1.
By settingy=∥x-v∥-1(x-v)+v, we get y∈𝒥1(v) since x,v∈𝒥1(v). Then ∥y-v∥=∥x-v∥-1∥x-v∥=1. However, the open unit ball 𝔅(v;1) in 𝒥1(v) about v is included in (𝒥1(v))-1 (cf. [22, Lemma  2.1]). Therefore, y∈(𝒥1(v))-1¯. From our above discussion, we also have (𝒥1(v))-1⊆𝒥q-1. Thus, αq(x) = infz∈𝒥q-1∥x-z∥≤infw∈(𝒥1(v))-1∥x-w∥≤∥x-y∥ = ∥x-∥x-v∥-1(x-v)-v∥ = ∥x-v∥(1-∥x-v∥-1) = ∥x-v∥-1 as ∥x-v∥≥1; which means αq(x)+1≤∥x-v∥. However, ∥x∥-1≤∥x-v∥, as seen above. We conclude that dist(x,ℰ(𝒥)1)≥max{αq(x)+1,∥x∥-1}.At present, we do not know whether inequality in the above theorem can be replaced with equality. To give some partial affirmative answer to this question, we need the following result.Theorem 27.
If𝒥 is a JB*-triple, then dist(x,ℰ(𝒥)1)≤max{1,∥x∥-1} for all x∈𝒥q-1¯.Proof.
Letx∈𝒥q-1. Then by Theorem 6, x is positive invertible in the JB*-algebra 𝒥1(v) for some v∈ℰ(𝒥)1, which is a JB*-algebra with unit v. Hence, x being positive satisfies ∥x-v∥=dist(x,ℰ(𝒥)1) by [26, Proposition  3.2]. By the functional calculus for positive elements in JB*-algebras,
(2)∥x-v∥≤{1if∥x∥≤2;∥x∥-1if∥x∥≥2.
Hence,dist(x,ℰ(𝒥)1)≤max{1,∥x∥-1} for all x∈𝒥q-1. Next, suppose (xn) is a sequence in 𝒥q-1 such that ∥xn-x∥<(1/n) for all n∈ℕ. By Theorem 6, there exists a unique vn∈ℰ(𝒥)1 corresponding to each xn. Hence, dist(xn,ℰ(𝒥)1)≤∥xn-vn∥ for all n. By (2), ∥xn-vn∥≤max{1,∥xn∥-1} for all n. Hence, for large n, we get dist(x,ℰ(𝒥)1)≤∥x-vn∥=∥xn-vn∥+∥x-xn∥≤max{1,∥xn∥-1}+(1/n) by using (2). It follows that dist(x,ℰ(𝒥)1)≤max{1,∥x∥-1}.The following result gives some cases in which the inequality appearing in Theorem26 becomes equality.Theorem 28.
Let𝒥 be a JB*-triple, v∈ℰ(𝒥)1, and x∈𝒥1(v)∖𝒥q-1.(i)
Ifαq(x)=∥x∥ then dist(x,ℰ(𝒥)1)=αq(x)+1. In particular, αq(x)=∥x∥=1 implies dist(x,ℰ(𝒥)1)=2.(ii)
Ifαq(x)<∥x∥=1, then dist(x,ℰ(𝒥)1)=αq(x)+1.Proof.
(i) Since0∈𝒥q-1¯,dist(x,ℰ(𝒥)1) ≤ ∥x-0∥ + dist(0,ℰ(𝒥)1) ≤ αq(x)+max{1,∥0∥-1} = αq(x)+1 by Theorem 27. On the other hand, dist(x,ℰ(𝒥)1) ≥ αq(x)+1 by Theorem 26. Hence, the part (i) is proved.
(ii) Letϵ>0 with αq(x)+ϵ<1. Then there exists y∈𝒥q-1 such that ∥y-x∥<αq(x)+ϵ. So, ∥y∥≤∥y-x∥+∥x∥<αq(x)+ϵ+1<2. Since y∈𝒥q-1, y is positive and invertible in 𝒥1(v) for some v∈ℰ(𝒥)1 by Theorem 6; 𝒥1(v) is a JB*-algebra with unit v. By the continuous functional calculus, we get ∥y-v∥≤∥y∥-1<2-1=1 since ∥y∥<2. Therefore, ∥x-v∥≤∥x-y∥+∥y-v∥<αq(x)+ϵ+1. Hence, dist(x,ℰ(𝒥)1)≤αq(x)+1. Moreover, dist(x,ℰ(𝒥)1)≥αq(x)+1 by Theorem 26.The next result gives some conditions sufficient for the existence of extreme point approximants.Corollary 29.
Let𝒥 be a JB*-triple, v∈ℰ(𝒥)1 and x∈𝒥 with ∥x∥≥2.(i)
Ifx is positive in 𝒥1(v), then ∥x-v∥=∥x∥-1.(ii)
Ifx∈𝒥q-1, then there is u∈ℰ(𝒥)1 such that ∥x-u∥=∥x∥-1.Proof.
(i) Sincev being an extreme point of the closed unit ball, v is a tripotent in 𝒥, v=Pvv={vv*v}. Hence, ∥v∥=∥{vv*v}∥=∥v∥3, so that ∥v∥=1. Now, since x is positive in 𝒥1(v) and since ∥x∥≥2, we get by the functional calculus for positive elements that ∥x-v∥=∥x∥-1.
(ii) Supposex∈𝒥q-1. By Theorem 6, x is positive in 𝒥1(v) for some v∈ℰ(𝒥)1. Hence, ∥x-u∥=∥x∥-1 by Part (i).



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