On the Geometry of the Unit Ball of a 𝐽𝐵^∗-Triple

Abstract

We explore a 𝐽𝐵^∗-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.

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 𝛼_𝑞(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 on 𝐶^∗-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 𝑒_𝑎, called its range tripotent. We prove that 𝐵(𝑎,𝑏)=0⇔𝑒_𝑎 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 space 𝒥 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 ℳ_𝑚𝑛(ℂ) of all 𝑚×𝑛 matrices with complex number entries, namely, the triple product {𝑥𝑦𝑧}=(1/2)(𝑥𝑦^𝑡𝑧+𝑧𝑦^𝑡𝑥) where 𝑦^𝑡 denotes the conjugate transpose of the matrix 𝑦 and the norm defined by ‖𝑧‖=√𝑧𝑧^𝑡 (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 operator 𝑃(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 𝑉_𝑥,𝑦𝑧∶={𝑥𝑦𝑧} and 𝑈_𝑥,𝑦𝑧∶={𝑥𝑧𝑦}, respectively; in fact, 𝐿(𝑥,𝑦)=𝑉_𝑥,𝑦∗ and 𝑃(𝑥,𝑦)𝑧=𝑈_𝑥,𝑦𝑧^∗ for all 𝑧∈𝒥. The Bergman operator takes the form 𝐵(𝑥,𝑦)=𝐼−2𝑉_𝑥,𝑦∗+𝑈_𝑥𝑈_𝑦∗, 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 a 𝐽𝐵^∗-triple (cf. [5, Theorem  6]).

Definition 1. In any 𝐽𝐵^∗-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 any 𝐽𝐵^∗-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∈𝒥_𝑞⁻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 𝒥_𝑞⁻¹. 𝒥_𝑞⁻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 the 𝐽𝐵^∗-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 element 𝑥 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 element 𝑎 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 𝑃_𝑢(𝑥,𝑦), 𝐿_𝑢(𝑥,𝑦), 𝐵_𝑢(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, 𝑃_𝑢(𝑥,𝑦)𝑧={𝑥𝑧^∗𝑢𝑦}_𝑢 and 𝐿_𝑢(𝑥,𝑦)𝑧={𝑥𝑦^∗𝑢𝑧}_𝑢 for all 𝑥,𝑦,𝑧∈𝒥; in particular, 𝑃_𝑢(𝑥)𝑧=𝑃_𝑢(𝑥,𝑥)𝑧={𝑥𝑧^∗𝑢𝑥}_𝑢. Here, {⋯}_𝑢 denotes the induced Jordan triple product in 𝒥^[𝑢].

Theorem 2. For any fixed unitary element 𝑢 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)𝑃_𝑢(𝑥)=𝑃(𝑥);(iii)𝐿_𝑢(𝑥,𝑦)=𝐿(𝑥,𝑦);(iv)𝐵_𝑢(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 𝑥𝑦^∗𝑢𝑧_𝑢=𝑥𝑢𝑦^∗𝑢∗𝑢^∗𝑧+𝑦^∗𝑢𝑢^∗𝑥𝑢^∗𝑧−𝑥𝑢^∗𝑦^∗𝑢𝑢^∗𝑧−𝑦^∗𝑢𝑢^∗𝑥𝑢^∗𝑧+𝑧𝑢^∗𝑥𝑢^∗𝑦^∗𝑢=𝑥𝑢𝑦^∗𝑢∗𝑢^∗𝑧=𝑥𝑦^{∗𝑢∗𝑢∗}𝑧=𝑥𝑦^∗𝑧. (1)
(ii) By the part (i), 𝑃_𝑢(𝑥)𝑧={𝑥𝑧^∗𝑢𝑥}_𝑢={𝑥𝑧^∗𝑥}=𝑃(𝑥)𝑧 for all 𝑥∈𝒥.
(iii) For any 𝑎,𝑏∈𝒥, 𝑃(𝑎,𝑏)=𝑃(𝑎+𝑏)−𝑃(𝑎)−𝑃(𝑏) (cf. [12]). This together with the part (ii) gives 𝐿_𝑢(𝑥,𝑦)𝑧=𝑃_𝑢(𝑥,𝑧)𝑦=𝑃_𝑢(𝑥+𝑧)𝑦−𝑃_𝑢(𝑥)𝑦−𝑃_𝑢(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) since 𝐵(𝑥,𝑦)=𝐼−2𝐿(𝑥,𝑦)+𝑃(𝑥)𝑃(𝑦).
(v) 𝑥∈𝒥_𝑞⁻¹⇔𝐵(𝑥,𝑦)=0⇔𝐵_𝑢(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 ideal 𝑉 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 fixed 𝑥∈𝒥_𝑞⁻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 a 𝐽𝐵^∗-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 element 𝑥 of a 𝐽𝐵^∗-triple 𝒥, there exists a unique tripotent 𝑒_𝑥∈𝒥 such that 𝑥 is positive invertible in 𝒥₁(𝑒_𝑥).

Proof. Since 𝑥 is von Neumann regular, there exists a unique tripotent 𝑒_𝑥∈𝒥, called the range tripotent of 𝑥 (cf. [10]) such that 𝑥 is invertible in the Peirce space 𝒥₁(𝑒_𝑥) (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(𝑒_𝑥) since 𝑥 is von Neumann regular in 𝒥 (cf. [10, page 572]). Thus, 𝑥 is positive in 𝒥₁(𝑒_𝑥) by [16, Proposition  2.1].

Remark 5. The range tripotent 𝑒_𝑥 of 𝑥 is defined as the least tripotent for which 𝑥 is a positive element in the 𝐽𝐵^∗-algebra 𝒥₁(𝑒_𝑥). 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. Let 𝒥 be a 𝐽𝐵^∗-triple and 𝑥∈𝒥_𝑞⁻¹. Then the range tripotent 𝑒_𝑥 is a unique extreme point of (𝒥)₁ such that 𝑥 is positive invertible in 𝒥₁(𝑒_𝑥). Moreover, there exists a unique 𝑦∈𝒥_𝑞⁻¹ satisfying 𝑃(𝑥)𝑦=𝑥, 𝑃(𝑦)𝑥=𝑦, 𝑃(𝑥)𝑃(𝑦)=𝑃(𝑦)𝑃(𝑥), and 𝐿(𝑥,𝑦)=𝐿(𝑒_𝑥,𝑒_𝑥)=𝐿(𝑦,𝑥).

Proof. Since 𝑥 being BP-quasi invertible is a von Neumann regular element, Theorem 4 gives the existence of a unique tripotent 𝑒_𝑥 in 𝒥 such that 𝑥 is positive invertible in 𝒥₁(𝑒_𝑥). 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(𝑒_𝑥,𝑒_𝑥)=𝐿(𝑥^†,𝑥) since 𝑒_𝑥 is the unique von Neumann regular element in (𝒥)₁(𝑒_𝑥). Therefore, 𝐵(𝑥,𝑥^†)=𝐵(𝑒_𝑥,𝑒_𝑥)=Π₀, where Π_𝜇 stands for the Peirce projection associated with the tripotent 𝑒_𝑥 onto the subtriple 𝒥_𝜇(𝑒_𝑥)={𝑧∶𝐿(𝑒_𝑥,𝑒_𝑥)𝑧=𝜇𝑧} (cf. [12]). In [19, page 192], authors show that 𝑒_𝑥 is unitary in 𝑃(𝑥)𝒥, hence 𝑃(𝑥)𝒥=𝒥₁(𝑒_𝑥). Therefore, 𝑒_𝑥∈ℰ(𝒥)₁ by [20, Lemma  4]. Thus, 𝐵(𝑥,𝑥^†)=𝐵(𝑒_𝑥,𝑒_𝑥)=0. We conclude that 𝑦∈𝒥_𝑞⁻¹.

Remark 7. The unique BP-quasi inverse of any BP-quasi invertible matrix 𝑥 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 𝑥𝑥^𝑡 is invertible in ℳ₂(ℂ), and the Moore-Penrose inverse of 𝑥 is given by 𝑥^†=𝑥^𝑡(𝑥𝑥^𝑡)⁻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 a 𝐽𝐵^∗-triple.

Theorem 8. For any 𝐽𝐵^∗-triple 𝒥, the set 𝒥_𝑞⁻¹ is open in the norm topology.

Proof. Assume 𝐵(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(Π_1/2)=𝒥_1/2(𝑣). Conversely, if 𝑎 is BP-quasi invertible with the unique BP-quasi inverse 𝑎^† and 𝑧∈range(Π_1/2)=𝒥_1/2(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∈𝒥_1/2(𝑣)} is an open subset of 𝒥. Thus, 𝒥_𝑞⁻¹ is an open set.

Now, we give the following improvement of [22, Theorem  4.12].

Theorem 9. Let 𝑥 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. Since 𝑥∈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⊆𝒥_𝑞⁻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 𝑒_𝑥) 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 𝑉_𝑢,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 element 𝑥 in the Peirce 1-space for some extreme point of the unit ball is sufficient for the BP-quasi invertibility of 𝑥.

Theorem 10. Let 𝒥 be a 𝐽𝐵^∗-triple and 𝑥∈𝒥⧵𝒥_𝑞⁻¹. Then 𝑥 is not positive invertible in the Peirce 1-space 𝒥₁(𝑣), for all 𝑣∈ℰ(𝒥)₁.

Proof. Suppose 𝑥 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 𝑃_𝑣(𝑥)𝑦=𝑥 and 𝑃_𝑣(𝑦)𝑥=𝑦, where 𝑃_𝑣(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=𝑃_𝑣𝑦 by Theorem 2. This together with the von Neumann regularity of 𝑥 in 𝒥₁(𝑣) gives 𝑥=𝑃_𝑣(𝑥)𝑦 = 𝑃(𝑥)𝑃(𝑣)𝑦 = 𝑃(𝑥)𝑦. Thus, the element 𝑥 is von Neumann regular in 𝒥.
Now, by Theorem 4, there exists a (unique) tripotent 𝑒_𝑥∈𝒥 such that 𝑥 is positive invertible in 𝒥₁(𝑒_𝑥). Hence, by Remark 5 and our supposition on 𝑥, we get the coincidence 𝒥₁(𝑒_𝑥)=𝒥₁(𝑣).
Next, by Theorem 6, 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∈𝒥_𝑞⁻¹, a contradiction to the hypothesis.

The above result together with Theorem 6 gives another characterization of the BP-quasi invertibility, as follows.

Theorem 11. An element 𝑥 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 a 𝐽𝐵^∗-triple. We extend [7, Theorem  2.2] for 𝐽𝐵^∗-triples as follows.

Theorem 12. Let 𝒥 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=∑^𝑛_𝑖=1𝑣_𝑖 with 𝑣_𝑖∈ℰ(𝒥)₁ for each 𝑖=1,…,𝑛.

Proof. Since 𝑣∈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 element 𝑥 in a 𝐽𝐵^∗-triple 𝒥, the number 𝑒_𝑚(𝑥) is defined by 𝑒_𝑚(𝑥)∶=min{𝑛∶𝑥=(1/𝑛)∑^𝑛_𝑗=1𝑣_𝑗,𝑣_𝑗∈ℰ(𝒥)₁}. If 𝑥 has no such decomposition, then 𝑒_𝑚(𝑥)∶=∞.

Next result describes a basic connection between 𝑒_𝑚(𝑥) and the distance to the set ℰ(𝒥)₁.

Theorem 14. Let 𝒥 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 𝑒_𝑚(𝑥)≤𝑛. On the other hand, if 𝑒_𝑚(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)∑^𝑛_𝑖=1𝑣_𝑖 with 𝑣_𝑖∈ℰ(𝒥)₁ for 𝑖=1,…,𝑛. Thus, 𝑒_𝑚(𝑥)≤𝑛. On the other hand, suppose 𝑒_𝑚(𝑥)≤𝑛. Then 𝑥=𝑟⁻¹∑^𝑟_𝑖=1𝑣_𝑖 for some 1≤𝑟≤𝑛 with 𝑣_𝑖s in ℰ(𝒥)₁. Then ‖𝑥‖≤𝑟⁻¹∑^𝑟_𝑖=1‖𝑣_𝑖‖=1. Further, ‖𝑟𝑥−𝑣₁‖ = ‖∑^𝑟_𝑖=2𝑣_𝑖‖≤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) = inf_{𝑤∈ℰ(𝒥)1}‖𝑛𝑥−𝑤‖≤‖𝑛𝑥−𝑣₁‖≤𝑛−1.

For any 𝑣∈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. Let 𝒥 be the 𝐽𝐵^∗-triple ℳ₁₂(ℂ), and let 𝑣 denotes the matrix [10]. Then ‖𝑣‖=1 and 𝑣𝑣^𝑡𝑣=𝑣. So, 𝑣 is a norm one tripotent matrix. Moreover, for any matrix 𝑧=𝑧₁𝑧₂, we have 𝐿_𝑣,𝑣𝑧=𝑧₁(1/2)𝑧₂. Also note that 𝑃_𝑣𝑧=𝑧₁0, where 𝑧_𝑖 denotes the complex conjugate of 𝑧_𝑖. So that, 𝑃_𝑣𝑃_𝑣𝑧=𝑧₁0. Hence, 𝐵(𝑣,𝑣)𝑧=𝑧₁𝑧₂−2𝑧₁(1/2)𝑧₂+𝑧₁0=00. Thus, 𝑣∈ℰ(𝒥)₁.
Next, consider the Peirce spaces of 𝒥, relative to the extreme point matrix 𝑣=[10], given by 𝒥₁(𝑣)={𝑧∶𝑧=[𝑧₁0],𝑧₁∈ℂ}, 𝒥_1/2(𝑣)={𝑧∶𝑧=[0𝑧₂],𝑧₂∈ℂ} and 𝒥₀(𝑣)={[00]}. Clearly, both the balls (𝒥₁(𝑣))₁ and (𝒥_1/2(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 coℰ(𝒥)₁ denote the convex hull of ℰ(𝒥)₁ and its norm closure, respectively.

Theorem 16 (A Russo-Dye Theorem type for 𝐽𝐵^∗-triples). Let 𝒥 be a 𝐽𝐵^∗-triple, 𝑣∈ℰ(𝒥)₁. (i)For any 𝑥∈𝒥₁(𝑣) with ‖𝑥‖<1−2𝑛⁻¹ for some 𝑛≥3, there exists 𝑣_𝑖∈ℰ(𝒥)₁ for all 𝑖=1,2,3,…,𝑛 such that 𝑥=(1/𝑛)∑^𝑛_𝑖=1𝑣_𝑖. (ii)(𝒥₁(𝑣))^∘₁⊆coℰ(𝒥)₁.(iii)(𝒥₁(𝑣))₁⊆coℰ(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=∑^𝑛_𝑖=1𝑣_𝑖 for some extremes 𝑣_𝑖s in 𝒥.
(ii) Suppose 𝑥∈(𝒥₁(𝑣))^∘₁. Then ‖𝑥‖<1−2𝑛⁻¹ for some positive integer 𝑛≥3. Therefore, 𝑥∈coℰ(𝒥)₁ by the part (i).
(iii) Using the part (ii), we get (𝒥₁(𝑣))₁=(𝒥₁(𝑣))^∘₁⊆coℰ(𝒥)₁.

Remark 17. If a 𝐽𝐵^∗-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 point 𝑣 of the closed unit ball in a 𝐽𝐵^∗-triple 𝒥, (𝒥₁(𝑣))₁⊂coℰ(𝒥)₁. If 𝑣 is a unitary element in 𝒥, then (𝒥)₁=coℰ(𝒥)₁=co𝒰(𝒥). Moreover, ⋃_{𝑤∈ℰ(𝒥)1}(𝒥₁(𝑤))₁ = co𝒰(𝒥) = coℰ(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=co𝒰(𝒥₁(𝑣))⊆coℰ(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⊂coℰ(𝒥)₁.
If 𝑣∈𝒰(𝒥), then 𝐿_𝑣,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⊆coℰ(𝒥)₁⊆(𝒥)₁=co𝒰(𝒥) by [7, Theorem  2.3]. Hence, (𝒥)₁=coℰ(𝒥)₁=co𝒰(𝒥).
As mentioned above, every unitary element 𝑣 in a 𝐽𝐵^∗-triple 𝒥 satisfies 𝐿_𝑣,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 = co𝒰(𝒥). Therefore, ⋃_{𝑤∈ℰ(𝒥)1}(𝒥₁(𝑤))₁ = co𝒰(𝒥) = coℰ(𝒥)₁ = (𝒥)₁.

Corollary 19. Each element of the Peirce 1-space 𝒥₁(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. Let 𝑥∈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 any 𝐽𝐵^∗-triple 𝒥, 𝒥_𝑞⁻¹∩(𝒥)₁⊆(1/2)(ℰ(𝒥)₁+ℰ(𝒥)₁).

Proof. Let 𝑥∈𝒥_𝑞⁻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 a 𝐽𝐵^∗-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. Let 𝒥 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. Since 𝒥₁(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].

Let 𝒥 be a 𝐽𝐵^∗-triple and 𝑥∈(𝒥)₁. We define 𝑉(𝑥)∶={𝛽≥1∶𝑥∈co_𝛽ℰ(𝒥)₁}, where co_𝛽ℰ(𝒥)₁∶={𝛽⁻¹∑^𝑛−1_𝑖=1𝑣_𝑖+𝛽⁻¹(1+𝛽−𝑛)𝑣_𝑛∶𝑣_𝑗∈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 𝑒_𝑚(𝑥).

Lemma 22. Let 𝒥 be a 𝐽𝐵^∗-triple. Let 𝑥∈𝒥 and 𝑛∈ℕ (the set of positive integers). Then(i)𝑥∈co_𝑛ℰ(𝒥)₁⇔𝑒_𝑚(𝑥)≤𝑛.(ii)𝑛∈𝑉(𝑥)⇔𝑒_𝑚(𝑥)≤𝑛.(iii) 𝑒_𝑚(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. Let 𝒥 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,…,𝛿_𝑚)∈ℝ^𝑚−1 such that 0≤𝛿_𝑗<𝛾−1 for all 𝑗=2,…,𝑚−1 and 𝛾⁻¹+∑^𝑚−1_𝑗=2𝛿_𝑗=1. Then there exist 𝑣₁,…,𝑣_𝑚∈ℰ(𝒥)₁ satisfying 𝑥=𝛾⁻¹𝑣₁+∑^𝑚_𝑗=2𝛿_𝑗𝑣_𝑗. 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. Since 𝒥₁(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∈co_𝑟ℰ(𝒥)₁ with 𝑥=𝑟⁻¹(𝑣₁+⋯+𝑣_𝑛−1+(1+𝑟−𝑛)𝑣_𝑛) for some 𝑣₁,…,𝑣_𝑛∈ℰ(𝒥)₁ and positive integer 𝑛 satisfying 𝑛−1<𝑟≤𝑛. Hence, ‖𝑟𝑥−𝑣₁‖=‖∑^𝑛−1_𝑗=2𝑣_𝑗+(1+𝑟−𝑛)𝑣_𝑛‖≤(𝑛−2)+(1+𝑟−𝑛)=𝑟−1.

Corollary 24. For any 𝐽𝐵^∗-triple 𝒥, co_𝛾ℰ(𝒥)₁⊆co_𝛿ℰ(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 a 𝐽𝐵^∗-triple 𝒥 to the set ℰ(𝒥)₁ and to the set 𝒥_𝑞⁻¹. We define the function 𝛼_𝑞∶𝒥→ℝ by 𝛼_𝑞(𝑥)∶=inf{‖𝑥−𝑢‖∶𝑢∈𝒥_𝑞⁻¹}.

Lemma 25. Let 𝒥 be a 𝐽𝐵^∗-triple with nonempty ℰ(𝒥)₁ and 𝑥∈𝒥. Then(i)𝛼_𝑞(𝑟𝑥)=|𝑟|𝛼_𝑞(𝑥) for all 𝑟∈ℂ;(ii)𝛼_𝑞(𝑥)≤‖𝑥‖;(iii)|𝛼_𝑞(𝑥)−𝛼_𝑞(𝑦)|≤‖𝑥−𝑦‖ for all 𝑦∈𝒥;(iv) if 𝒥 is a 𝐽𝐵^∗-algebra, then 𝛼_𝑞(𝑥)=𝛼_𝑞(𝑥^∗).

Proof. (i) For any complex number 𝑟≠0, 𝑥∈𝒥_𝑞⁻¹ if and only if 𝑟𝑥∈𝒥_𝑞⁻¹. So, the part (i) is clear for all nonzero complex numbers. If 𝑟=0, then 𝛼_𝑞(𝑟𝑥)=𝛼_𝑞(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 any 𝑣∈ℰ(𝒥)₁ and any positive integer 𝑛, (1/𝑛)𝑣∈𝒥_𝑞⁻¹; hence, 𝛼_𝑞(𝑥) = dist(𝑥,𝒥_𝑞⁻¹)≤‖𝑥−(1/𝑛)𝑣‖≤‖𝑥‖+1/𝑛. It follows that 𝛼_𝑞(𝑥)≤‖𝑥‖.
(iii) Since the set ℰ(𝒥)₁ is nonempty, and since any extreme point is BP-quasi invertible, 𝛼(𝑥)<∞. Now, by definition of 𝛼_𝑞(𝑥), for each 𝑛∈ℕ, there exists an element 𝑥_𝑛∈𝒥_𝑞⁻¹ such that ‖𝑥−𝑥_𝑛‖≤𝛼_𝑞(𝑥)+1/𝑛. Hence, 𝛼_𝑞(𝑦)≤‖𝑦−𝑥_𝑛‖≤‖𝑦−𝑥‖+‖𝑥−𝑥_𝑛‖≤‖𝑥−𝑦‖+𝛼_𝑞(𝑥)+1/𝑛 for all 𝑛∈ℕ, so that 𝛼_𝑞(𝑦)≤‖𝑥−𝑦‖+𝛼_𝑞(𝑥). Interchanging 𝑥 and 𝑦, we get 𝛼_𝑞(𝑥)≤‖𝑥−𝑦‖+𝛼_𝑞(𝑦). Thus, |𝛼_𝑞(𝑥)−𝛼_𝑞(𝑦)|≤‖𝑥−𝑦‖ for all 𝑦∈𝒥.
(iv) Now, let the 𝒥 be a 𝐽𝐵^∗-algebra. If 𝑥∈𝒥_𝑞⁻¹, then 𝑥^∗∈𝒥_𝑞⁻¹ by [4, Theorem  3.2], and hence 𝛼_𝑞(𝑥)=0=𝛼_𝑞(𝑥^∗). Next, if 𝑥 is not BP-quasi invertible, then 𝛼_𝑞(𝑥) = dist(𝑥,𝒥_𝑞⁻¹) = inf{‖𝑥−𝑢‖∶𝑢∈𝒥_𝑞⁻¹} = inf{‖𝑥^∗−𝑢‖∶𝑢∈𝒥_𝑞⁻¹} = 𝛼_𝑞(𝑥^∗) since the involution “∗” is an isometry.

𝛼_𝑞(𝑥) provides information about dist(𝑥,ℰ(𝒥)₁) as follows.

Theorem 26. Let 𝒥 be a 𝐽𝐵^∗-triple, 𝑣∈ℰ(𝒥)₁ and 𝑥∈𝒥₁(𝑣)⧵𝒥_𝑞⁻¹. Then dist(𝑥,ℰ(𝒥)₁) ≥ max{𝛼_𝑞(𝑥)+1,‖𝑥‖−1}.

Proof. Since 𝑣∈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 𝐿_𝑣(𝑢,𝑢)=𝐼 on 𝒥₁(𝑣) since 𝑢 is unitary in 𝒥₁(𝑣), so that 𝐿(𝑢,𝑢)𝑧=𝐿_𝑣(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∈𝒥_𝑞⁻¹ by Theorem 10; a contradiction to the hypothesis. Thus, ‖𝑥−𝑣‖≥1.
By setting 𝑦=‖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⊆𝒥_𝑞⁻¹. Thus, 𝛼_𝑞(𝑥) = inf_{𝑧∈𝒥𝑞}⁻¹‖𝑥−𝑧‖≤inf_{𝑤∈(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 𝛼_𝑞(𝑥)+1≤‖𝑥−𝑣‖. However, ‖𝑥‖−1≤‖𝑥−𝑣‖, as seen above. We conclude that dist(𝑥,ℰ(𝒥)₁)≥max{𝛼_𝑞(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. If 𝒥 is a 𝐽𝐵^∗-triple, then dist(𝑥,ℰ(𝒥)₁)≤max{1,‖𝑥‖−1} for all 𝑥∈𝒥_𝑞⁻¹.

Proof. Let 𝑥∈𝒥_𝑞⁻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, ‖‖≤1𝑥−𝑣if‖𝑥‖‖𝑥‖≤2;−1if‖𝑥‖≥2. (2)
Hence, dist(𝑥,ℰ(𝒥)₁)≤max{1,‖𝑥‖−1} for all 𝑥∈𝒥_𝑞⁻¹. Next, suppose (𝑥_𝑛) is a sequence in 𝒥_𝑞⁻¹ such that ‖𝑥_𝑛−𝑥‖<(1/𝑛) for all 𝑛∈ℕ. By Theorem 6, there exists a unique 𝑣_𝑛∈ℰ(𝒥)₁ corresponding to each 𝑥_𝑛. Hence, dist(𝑥_𝑛,ℰ(𝒥)₁)≤‖𝑥_𝑛−𝑣_𝑛‖ for all 𝑛. By (2), ‖𝑥_𝑛−𝑣_𝑛‖≤max{1,‖𝑥_𝑛‖−1} for all 𝑛. Hence, for large 𝑛, we get dist(𝑥,ℰ(𝒥)₁)≤‖𝑥−𝑣_𝑛‖=‖𝑥_𝑛−𝑣_𝑛‖+‖𝑥−𝑥_𝑛‖≤max{1,‖𝑥_𝑛‖−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 Theorem 26 becomes equality.

Theorem 28. Let 𝒥 be a 𝐽𝐵^∗-triple, 𝑣∈ℰ(𝒥)₁, and 𝑥∈𝒥₁(𝑣)⧵𝒥_𝑞⁻¹.(i) If 𝛼_𝑞(𝑥)=‖𝑥‖ then dist(𝑥,ℰ(𝒥)₁)=𝛼_𝑞(𝑥)+1. In particular, 𝛼_𝑞(𝑥)=‖𝑥‖=1 implies dist(𝑥,ℰ(𝒥)₁)=2.(ii) If 𝛼_𝑞(𝑥)<‖𝑥‖=1, then dist(𝑥,ℰ(𝒥)₁)=𝛼_𝑞(𝑥)+1.

Proof. (i) Since 0∈𝒥_𝑞⁻¹,dist(𝑥,ℰ(𝒥)₁) ≤ ‖𝑥−0‖ + dist(0,ℰ(𝒥)₁) ≤ 𝛼_𝑞(𝑥)+max{1,‖0‖−1} = 𝛼_𝑞(𝑥)+1 by Theorem 27. On the other hand, dist(𝑥,ℰ(𝒥)₁) ≥ 𝛼_𝑞(𝑥)+1 by Theorem 26. Hence, the part (i) is proved.
(ii) Let 𝜖>0 with 𝛼_𝑞(𝑥)+𝜖<1. Then there exists 𝑦∈𝒥_𝑞⁻¹ such that ‖𝑦−𝑥‖<𝛼_𝑞(𝑥)+𝜖. So, ‖𝑦‖≤‖𝑦−𝑥‖+‖𝑥‖<𝛼_𝑞(𝑥)+𝜖+1<2. Since 𝑦∈𝒥_𝑞⁻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‖<𝛼_𝑞(𝑥)+𝜖+1. Hence, dist(𝑥,ℰ(𝒥)₁)≤𝛼_𝑞(𝑥)+1. Moreover, dist(𝑥,ℰ(𝒥)₁)≥𝛼_𝑞(𝑥)+1 by Theorem 26.

The next result gives some conditions sufficient for the existence of extreme point approximants.

Corollary 29. Let 𝒥 be a 𝐽𝐵^∗-triple, 𝑣∈ℰ(𝒥)₁ and 𝑥∈𝒥 with ‖𝑥‖≥2.(i) If 𝑥 is positive in 𝒥₁(𝑣), then ‖𝑥−𝑣‖=‖𝑥‖−1.(ii) If 𝑥∈𝒥_𝑞⁻¹, then there is 𝑢∈ℰ(𝒥)₁ such that ‖𝑥−𝑢‖=‖𝑥‖−1.

Proof. (i) Since 𝑣 being an extreme point of the closed unit ball, 𝑣 is a tripotent in 𝒥, 𝑣=𝑃_𝑣𝑣={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) Suppose 𝑥∈𝒥_𝑞⁻¹. By Theorem 6, 𝑥 is positive in 𝒥₁(𝑣) for some 𝑣∈ℰ(𝒥)₁. Hence, ‖𝑥−𝑢‖=‖𝑥‖−1 by Part (i).

Acknowledgments

This work was supported by King Saud University, Deanship of Scientific Research, College of Science, Research Center. The authors thank the anonymous referees for their helpful comments for improving the presentation of this work.