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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 891249, 8 pages
On the Geometry of the Unit Ball of a -Triple
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Received 14 February 2013; Revised 5 May 2013; Accepted 8 May 2013
Academic Editor: Ngai-Ching Wong
Copyright © 2013 Haifa M. Tahlawi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We explore a -triple analogue of the notion of quasi invertible elements, originally studied by Brown and Pedersen in the setting of -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 -triple; this indicates their structural richness. We initiate a study of the unit ball of a -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 -algebra and -algebra results, due to Kadison and Pedersen, Rørdam, Brown, Wright and Youngson, and Siddiqui, including the Russo-Dye theorem, are extended to -triples.
Brown and Pedersen  introduced a notion of quasi invertible elements in a -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 -algebra; in particular, they obtained several results verifying that the relationships between the extreme convex decomposition theory and the distance from an element to the set of BP-quasi invertible elements are analogous with the relationships in the earlier -algebra theory of unitary convex decompositions and regular approximations.
It is widely believed that the underlying structure making several interesting results on -algebras hold is not the presence of an associative product but the presence of the Jordan triple product . This provided one of the stimuli for the development of Jordan algebra or Jordan triple product generalizations of -algebras; these include -algebras and -triples together with their subclasses which are Banach dual spaces (cf. ). In [4, 5], we introduced an exact analogue of the BP-quasi invertible elements for -triples. By using certain identities concerning the Bergman operators, we established that in case of nondegenrate Jordan triples and and that is BP-quasi invertible with BP-quasi inverse . Our aim in this paper is to study geometric consequences of these facts in the setting of complex -triples. After discussing some basics, we begin by recording a known result: is von Neumann regular that admits a tripotent , called its range tripotent. We prove that is an extreme point of the unit ball and that the set of BP-quasi invertible elements of a -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 -triple analogues of some -algebra results; this approach provides alternative proofs for some of the corresponding known results for -algebras.
A Jordan triple system is a vector space over a field of characteristic not 2, endowed with a triple product which is linear and symmetric in the outer variables , and linear or antilinear in the inner variable satisfying the Jordan triple identity: . A -triple is a complex Banach space together with a continuous, sesquilinear, operator-valued map that defines a triple product in making it a Jordan triple system such that each is a positive hermitian operator on and for all . Any bounded symmetric domain in a complex Banach space is biholomorphically equivalent to the open unit ball of a -triple [8, 9]. An important example, from the viewpoint of the classical theory of operators and matrices, is the triple system of all bounded linear operators between complex Hilbert spaces and under the product and the operator norm, where denotes the Hilbert adjoint of (cf. ). In the finite-dimensional case, this can be viewed as adding an algebraic product to a linear matrix space of all matrices with complex number entries, namely, the triple product where denotes the conjugate transpose of the matrix and the norm defined by (see [11, Example 4.7]). Any -algebra (cf. ) is a -triple under the triple product , where “” denotes the underlying Jordan binary product.
A basic operator on the -triple is defined by for all ; we write in short as . The Bergman operator is defined on by , where is the identity operator [12, 2.11]. The operators and are the -triple analogues of the usual Jordan algebra operators and , respectively; in fact, and for all . The Bergman operator takes the form , which translates to = for the case.
2. BP-Quasi Invertible Elements
We begin with the following characterization of BP-quasi invertible elements of a -triple (cf. [5, Theorem 6]).
Definition 1. In any -triple, an element is BP-quasi invertible with BP-quasi inverse if (and only if) the Bergman operator .
In any -triple , by [5, Theorem 2], and so the relation of being BP-quasi inverse of some element is symmetric in . Generally, BP-quasi inverse in not unique: if with BP-quasi inverse , then is also a BP-quasi inverse of since = + = + = + (cf. ) . We denote the set of BP-quasi invertible elements in by . includes the set of all invertible elements and the set of all extreme points of the closed unit ball ; and in case of a -algebra , the BP-quasi invertibility is invariant under the involution “” (cf. ).
In the -triple , all matrices of the form , where are any nonzero complex numbers, are BP-quasi invertible with BP-quasi inverse of the form .
An element in a -triple is called von Neumann regular if there exists with . Such an element is called generalized inverse of . If is a -algebra, then , and so is a generalized inverse of in considered as a -triple if and only if is a generalized inverse of in the -algebra . From [5, Theorem 3], we know that and are von Neumann regular with generalized inverses of each other if . Hence, any BP-quasi invertible element is necessarily von Neumann regular. For a von Neumann regular element in the -algebra of 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 in a -triple , the underlying linear space becomes a complex Jordan algebra with respect to the Jordan product , called -homotope of (cf. ). An element in a -triple is called unitary if ; the set of all unitary elements is denoted by . For any , the -homotope of is a -algebra with respect to the original norm and the involution “” given by (cf. ); such a homotope is denoted by , called a unitary isotope of . The symbols , , will denote the respective analogues of the operators , , and on the isotope . Thus, and for all ; in particular, . Here, denotes the induced Jordan triple product in .
Theorem 2. For any fixed unitary element of a -triple and for all , one has the following:(i) the Jordan triple product in coincides with ;(ii);(iii);(iv);(v).
Proof. (i) By using the basic Jordan triple identity, we get that
(ii) By the part (i), for all .
(iii) For any , (cf. ). This together with the part (ii) gives = for all .
(iv) Follows from parts (ii) and (iii) since .
(v) (by (iv)) .
We close this section with the following observation on images of BP-quasi invertible elements under triple homomorphisms.
Theorem 3. For any closed ideal in -triple , the triple homomorphism given by preserves the BP-quasi invertibility.
Proof. being closed ideal of is a -triple, and so is the quotient . The quotient -triple admits the canonical surjective triple homomorphism defined by such that for all [8, Proposition 5.5].
Now, for any fixed with inverse and for all , = + = + = = . Hence, .
3. Positivity of BP-Quasi Invertibles
In this section, we prove that any BP-quasi invertible element in a -triple is positive invertible in the Peirce -space of the operator for certain extreme point 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 of a -triple , there exists a unique tripotent such that is positive invertible in .
Proof. Since is von Neumann regular, there exists a unique tripotent , called the range tripotent of (cf. ) such that is invertible in the Peirce space (cf. [10, Lemma 3.2], [14, page 540], and [15, Theorem 1]). Let denotes the norm closure of , which is the smallest norm closed inner ideal of containing (cf. [16, pages 19-20]). From [10, Lemma 3.2], we see that is norm closed in . Hence, since is von Neumann regular in (cf. [10, page 572]). Thus, is positive in by [16, Proposition 2.1].
Remark 5. The range tripotent of is defined as the least tripotent for which is a positive element in the -algebra . If and are two tripotents in such that is positive invertible in both the Peirce spaces and , then ; for details, see [17, Lemma 3.3] and [16, Proposition 2.1].
Theorem 6. Let be a -triple and . Then the range tripotent is a unique extreme point of such that is positive invertible in . Moreover, there exists a unique satisfying , , , and .
Proof. Since being BP-quasi invertible is a von Neumann regular element, Theorem 4 gives the existence of a unique tripotent in such that is positive invertible in . By [10, Lemma 3.2], there exists a unique generalized inverse satisfying all the conditions of the theorem; in fact, this generalized inverse is called the Moore-Penrose inverse, usually denoted by . Since , = = for all . So that = . Hence, = = . In view of [18, Lemma 2.1] and [10, page 573], we have since is the unique von Neumann regular element in . Therefore, , where stands for the Peirce projection associated with the tripotent onto the subtriple (cf. ). In [19, page 192], authors show that is unitary in , hence . Therefore, by [20, Lemma 4]. Thus, . We conclude that .
Remark 7. The unique BP-quasi inverse of any BP-quasi invertible matrix in the -triple , satisfying the conditions of Theorem 6, is precisely the Moore-Penrose inverse (cf. [10, Lemma 3.2]), which can be calculated according to the rank of the matrix (cf. ). If then is invertible in , and the Moore-Penrose inverse of is given by . For example, let . Then is BP-quasi invertible and its unique BP-quasi inverse, via Theorem 6, is given by . Since (the unit in ), for all and so . Thus, .
Next result establishes an important topological property of the BP-quasi invertible elements in a -triple.
Theorem 8. For any -triple , the set is open in the norm topology.
Proof. Assume where for some nonzero , and denotes the Moore-Penrose type inverse of (cf. [4, 10]). Consequently, , and so = = = . Thus, , which means . Recall that (see  and [19, page 192]). Also, since implies that , the nonzero element . Moreover, as a projection, is the identity operator on , so since . Thus, and . Conversely, if is BP-quasi invertible with the unique BP-quasi inverse and , then = = + + + + = + + + . Since and [12, Theorem 5.4 (9)], , and we get the value for the right hand side. Hence is BP-quasi invertible. The elements with spectrum form an open subset of . It follows that is an open subset of . Thus, is an open set.
Now, we give the following improvement of [22, Theorem 4.12].
Theorem 9. Let be an invertible element in a unital -algebra . Then there exists unique such that is positive and invertible in .
Proof. Since , there is a unique element with and where “” denotes the Jordan binary product and is the unit in . Since and that any -algebra is a -triple under the triple product , so, by Theorem 6, there exists a unique (viz, the range tripotent ) such that is positive and invertible in with the inverse of the same in (cf. [15, Theorem 6]). Moreover, , and so . Hence, the Peirce decomposition of reduces to . Of course, the product and the involution defined on both the -algebra and the unitary isotope of are the same. Further, = = = = and = = , so that . Hence, with inverse ; that is, . We conclude that and coincide as -algebras.
The following result shows that positive invertibility of an element in the Peirce 1-space for some extreme point of the unit ball is sufficient for the BP-quasi invertibility of .
Theorem 10. Let be a -triple and . Then is not positive invertible in the Peirce 1-space , for all .
Proof. Suppose is positive invertible in for some . Then there exists a unique inverse satisfying and . Since , is von Neumann regular with generalized inverse satisfying and , where . Being positive, is self-adjoint in ; that is, . However, the invertibility is invariant under the involution (see above). Therefore, with inverse such that = = = by Theorem 2. This together with the von Neumann regularity of in gives = = . Thus, the element is von Neumann regular in .
Now, by Theorem 4, there exists a (unique) tripotent such that is positive invertible in . Hence, by Remark 5 and our supposition on , we get the coincidence .
Next, by Theorem 6, is the unique generalized inverse of satisfying , , and . Hence, . This means , 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 in a -triple is BP-quasi invertible if and only if is positive and invertible in the Peirce 1-space for some extreme point .
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 -triple. We extend [7, Theorem 2.2] for -triples as follows.
Theorem 12. Let be a -triple with and (the open unit ball of ). Then for any positive integer , with for each .
Proof. Since , is a -algebra with unit . Since in , and so are invertible elements in (cf. [22, Lemma 2.1(iii)]). Clearly, . Hence, by [7, Lemma 2.1], there exist unitaries with . Moreover, by [20, Lemma 4]. The required result now follows by induction on .
Definition 13. For any element in a -triple , the number is defined by . If has no such decomposition, then .
Next result describes a basic connection between and the distance to the set .
Theorem 14. Let be a -triple, and with satisfying for some . Then . On the other hand, if , then .
Proof. Clearly, . Replacing by in Theorem 12, we get with for . Thus, . On the other hand, suppose . Then for some with s in . Then . Further, = . Hence, = + + = since . Thus, = .
For any , the set includes the set of all unitaries in , as seen above from [20, Lemma 4]; the following example shows that the inclusion generally is proper.
Example 15. Let be the -triple , and let denotes the matrix . Then and . So, is a norm one tripotent matrix. Moreover, for any matrix , we have . Also note that , where denotes the complex conjugate of . So that, . Hence, . Thus, .
Next, consider the Peirce spaces of , relative to the extreme point matrix , given by , and . Clearly, both the balls and are nontrivial. It is easily seen that the matrix .
From [6, 7], we know different proofs of strict analogue of the Russo-Dye theorem for general (unital) -algebras. Unfortunately, there exists no exact analogue of the Russo-Dye theorem for nonunital -triples. The following result is an appropriate analogue for an arbitrary -triple, where and denote the convex hull of and its norm closure, respectively.
Theorem 16 (A Russo-Dye Theorem type for -triples). Let be a -triple, . (i)For any with for some , there exists for all such that . (ii).(iii).
Proof. (i) Since , we have , so that . Also, note that ; recall that is in the -algebra . Hence, by taking the same as in Theorem 12 while , we get for some extremes s in .
(ii) Suppose . Then for some positive integer . Therefore, by the part (i).
(iii) Using the part (ii), we get .
Remark 17. If a -triple has a BP-quasi invertible element then is a nonempty set by Theorem 6. Hence, the above theorem holds in this case.
Corollary 18. For any extreme point of the closed unit ball in a -triple , . If is a unitary element in , then . Moreover, = = = .
Proof. We know from [20, Lemma 4] that . So, by the Russo-Dye Theorem for -algebras [7, Theorem 2.3], . By Example 15, there exists a -triple with such that . Hence, we have .
If , then (the identity operator on ) (cf. ). So, which is a unital -algebra, and hence is a -algebra. Then, by [7, Theorem 2.3]. Hence, .
As mentioned above, every unitary element in a -triple satisfies , and so (cf. [10, page 582]). Hence, is an extreme point of by [23, Lemma 3.2 and Proposition 3.5]. Thus, for any , = = . Therefore, = = = .
Corollary 19. Each element of the Peirce 1-space in a -triple with is a positive multiple of the sum of three extreme points of .
Proof. Let and . Let . Then . Hence, by Theorem 16(i), there exist three extreme points in such that . Thus, .
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 -triple.
Theorem 20. In any -triple , .
Proof. Let . Then, there is a unique such that is positive and invertible in by Theorem 6. In particular, is self-adjoint in the -algebra . This together with gives the existence of unitaries satisfying by [24, Theorem 2.11]. The result now follows from [20, Lemma 4].
The unit ball of a -triple often has extreme points in abundance, as in the case when is a Hilbert space ; however, the set may be empty as in the -triple of compact operators on [25, page 151].
Theorem 21. Let be a -triple and . Let with and . Then .
Let be a -triple and . We define , where in which the positive integer satisfies the condition . We observe the following immediate connection of these constructs with the number .
Lemma 22. Let be a -triple. Let and (the set of positive integers). Then(i).(ii).(iii) .
Theorem 23. Let be a -triple, and .(a) Let for some and some . Let such that for all and . Then there exist satisfying . Moreover, .(b) On the other hand, if holds, then for each , there is such that .
Corollary 24. For any -triple , if . In particular, for each , is either empty or equal to or for some .
6. Distance to the Extreme Points
In this section, we prove some results on distances from an element of a -triple to the set and to the set . We define the function by .
Lemma 25. Let be a -triple with nonempty and . Then(i) for all ;(ii);(iii) for all ;(iv) if is a -algebra, then .
Proof. (i) For any complex number , if and only if . So, the part (i) is clear for all nonzero complex numbers. If , then , one may consider the BP-quasi invertible element with and positive integer .
(ii) Since for any and any positive integer , ; hence, = . It follows that .
(iii) Since the set is nonempty, and since any extreme point is BP-quasi invertible, . Now, by definition of , for each , there exists an element such that . Hence, for all , so that . Interchanging and , we get . Thus, for all .
(iv) Now, let the be a -algebra. If , then by [4, Theorem 3.2], and hence . Next, if is not BP-quasi invertible, then = = = = since the involution “” is an isometry.
provides information about as follows.
Theorem 26. Let be a -triple, and . Then ≥ .
Proof. Since , . We assume , then the fact that is the unit of the -algebra gives the invertibility of in , and so is positive and invertible in the unitary isotope of induced by certain unitary element in ; hence, by [20, Lemma 4]. For any , we have and on since is unitary in , so that by Theorem 2. Hence, . For the reverse inclusion, recall that , and so (see the above proof of Theorem 4); similarly, , and so for some . Hence, for any fixed , there exists with . Thus, . Therefore, as sets. Moreover, we observe that and by Theorem 2. We conclude that and coincide as -triples and is positive invertible in the Peirce 1-space . So, by Theorem 10; a contradiction to the hypothesis. Thus, .
By setting , we get since . Then . However, the open unit ball in about is included in (cf. [22, Lemma 2.1]). Therefore, . From our above discussion, we also have . Thus, = = = = as ; which means . However, , as seen above. We conclude that .
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 is a -triple, then for all .
Proof. Let . Then by Theorem 6, is positive invertible in the -algebra for some , which is a -algebra with unit . Hence, being positive satisfies by [26, Proposition 3.2]. By the functional calculus for positive elements in -algebras,
Hence, for all . Next, suppose is a sequence in such that for all . By Theorem 6, there exists a unique corresponding to each . Hence, for all . By (2), for all . Hence, for large , we get by using (2). It follows that .
The following result gives some cases in which the inequality appearing in Theorem 26 becomes equality.
Theorem 28. Let be a -triple, , and .(i) If then . In particular, implies .(ii) If , then .
Proof. (i) Since ≤ + ≤ = by Theorem 27. On the other hand, ≥ by Theorem 26. Hence, the part (i) is proved.
(ii) Let with . Then there exists such that . So, . Since , is positive and invertible in for some by Theorem 6; is a -algebra with unit . By the continuous functional calculus, we get since . Therefore, . Hence, . Moreover, by Theorem 26.
The next result gives some conditions sufficient for the existence of extreme point approximants.
Corollary 29. Let be a -triple, and with .(i) If is positive in , then .(ii) If , then there is such that .
Proof. (i) Since being an extreme point of the closed unit ball, is a tripotent in , . Hence, , so that . Now, since is positive in and since , we get by the functional calculus for positive elements that .
(ii) Suppose . By Theorem 6, is positive in for some . Hence, by Part (i).
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.
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