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The Proximinality of the Centre of aC*-Algebra
Abstract
It is shown that ifAis a unitalC*-algebra thenZ(A), the centre ofA, is a proximinal subspace. In other words, for eacha∈Athere existsz∈Z(A) such that ‖a−z‖ is equal to the distance fromatoZ(A).
... One of the early results in this direction is due to Mazur: [4] and Blatter [1]. There is also a generalization for vector spaces of 'continuous functions' on 'noncommutative spaces' in terms of C*-algebras [8]. For more details see [7, page 15]. ...
The proximinality of certain subspaces of spaces of bounded affine functions is proved. The results presented here are some linear versions of an old result due to Mazur. For the proofs we use some sandwich theorems of Fenchel's duality theory.
... In this section, we obtain elements in Z(A) by using Michael's selection theorem, rather than the Katětov-Tong theorem (cf. [52], [78]) ...
It is shown that a unital C*-algebra A has the Dixmier property if and only if it is weakly central and satisfies certain tracial conditions. This generalises the Haagerup-Zsido theorem for simple C*-algebras. We also study a uniform version of the Dixmier property, as satisfied for example by von Neumann algebras and the reduced C*-algebras of Powers groups, but not by all C*-algebras with the Dixmier property, and we obtain necessary and sufficient conditions for a simple unital C*-algebra with unique tracial state to have this uniform property. We give further examples of C*-algebras with the uniform Dixmier property, namely all C*-algebras with the Dixmier property and finite radius of comparison-by-traces. Finally, we determine the distance between two Dixmier sets, in an arbitrary unital C*-algebra, by a formula involving tracial data and algebraic numerical ranges.
... Remarks. (i) A subspace X of a Banach space Y is said to be proximinal if every element of Y attains its distance to X. Ideals in C * -algebras are proximinal ([1], 4.3), and so too is the centre of a unital C * -algebra ( [20]). This makes it natural to wonder if J A is proximinal in A ⊗ h A. ...
Let A be a C -algebra with an identity and let Z be the canon- ical map from A Z A, the central Haagerup tensor product of A, to CB(A), the algebra of completely bounded operators on A. It is shown that if ev- ery Glimm ideal of A is primal then Z is an isometry. This covers unital quasi-standard C -algebras and quotients of AW -algebras.
... But only an affirmative answer in the non-separable case would cover, extend and unify the results that every derivation of a simple C*-algebra is inner in the multiplier algebra [13] and that all derivations of von Neumann algebras [6], [12] and AW*-algebras [10] are inner. This quest becomes even more attractive by the recent results in [9] and [14] implying that, if a derivation δ on A is inner in the multiplier algebra, then there is a local multiplier a of A implementing δ such that δ = 2 a . ...
A condition on a derivation of an arbitrary C*-algebra is presented entailing that it is implemented as an inner derivation by a local multiplier. It is an outstanding open question whether every derivation of a C*-algebra A can be implemented as an inner derivation by a local multiplier, that is, an element in the direct limit of the multiplier algebras of the closed essential ideals of A.A n armative answer was given by Elliott (4) for AF-algebras, and by Pedersen (11) for general separable C*-algebras. In fact, it suces to assume that every closed essential ideal of A is -unital; hence Pedersen's result entails Sakai's theorem that every derivation of a simple unital C*-algebra is inner. But only an armative answer in the non-separable case would cover, extend and unify the results that every derivation of a simple C*-algebra is inner in the multiplier algebra (13) and that all derivations of von Neumann algebras (6), (12) and AW*-algebras (10) are inner. This quest becomes even more attractive by the recent results in (9) and (14) implying that, if a derivation on A is inner in the multiplier algebra, then there is a local multiplier a of A implementing such that kk =2 k a k . No progress on the above question seems to have been made since it was raised in (11) (see also (4)). The purpose of this note is to present a criterion on a given derivation of a (possibly non-separable) C*-algebra A implying that is inner in the local multiplier algebra Mloc(A). Though this criterion, inspired by Herstein's work (5), is rather algebraic in nature, it is hoped that some approximate version may eventually yield a positive solution of the general problem.
... We were able to show that this is true for local multiplier algebras and, more generally, for boundedly centrally closed C*-algebras (see below for the definition). Moreover, Somerset proved that the distance dist(a, Z(A)) from any element a to the centre is always attained, regardless of the nature of the C*-algebra A [19]. ...
We discuss some basic features of the local multiplier algebra of a C*-algebra, the analytic analogue of the well-known Kharchenko-Martindale symmetric ring of quotients, and also the more recent
maximal C*-algebra of quotients, which is the analytic companion to the Utumi-Lanning maximal symmetric ring of quotients, together
with some of the applications to operator theory on C*-algebras. The emphasis lies in illustrating the interrelations between noncommutative ring theory and functional analysis.
... The local multiplier algebra of a C*-algebra A, that is, the direct limit of the multiplier algebras of the closed essential ideals of A, has proved to be a useful device in operator theory on C*-algebras [1,2,4,11,12,15]. For example, if δ a is an inner derivation on A implemented by an element a in the multiplier algebra M (A)o fA, there is always a local multiplier a ′ of A such that δ a = δ a ′ and δ a =2 a ′ ,a sa consequence of the results in [10] and [14]. These applications stimulated the investigation of the structure of the local multiplier algebra M loc (A), for a comprehensive account see [3]. ...
We construct an AF-algebra A such that its local multiplier algebra Mloc(A) does not agree with Mloc(Mloc(A)), thus answering a question raised by G.K. Pedersen in 1978.
... Subsequent work has generalised this equality to various classes of C * -algebras but [22, 3.2, 3.3] implies a characterisation of those A where equality always holds (those where all Glimm ideals of M(A) are 3-primal). Moreover in case this condition is not true, then there is a ∈ M(A) with δ a ≤ √ 3 inf z∈Z(M (A)) a − z (and further related work is to be found in [4,21,22,23,6]). An example of [7] shows that the condition on Glimm ideals of M(A) is difficult to relate to the structure of the primitive ideal space of A, so that the results are perhaps most satisfactory in the unital case where M(A) = A. ...
We present a formula for the norm of an elementary operator on a C*-algebra that seems to be new. The formula involves (matrix) numerical ranges and a kind of geometrical mean for positive matrices, the tracial geometric mean, which seems not to have been studied previously and has interesting properties. In addition, we characterise compactness of elementary operators.
Among the outstanding problems in the theory of elementary operators on Banach algebras is the task to find a formula which describes the norm of an elementary operator in terms of the norms of its coefficients. Here we report on the state-of-the-art of the knowledge on this problem along the lines of our talk at the Functional Analysis Valencia 2000 Conference in July 2000.
Abstact
A necessary and sufficient condition is given for a separable C *-algebra to be *-isomorphic to a maximal full algebra of cross-sections over a base space such that the fibre algebras are primitive throughout a dense subset. The condition is that the relation of inseparability for pairs of points in the primitive ideal space should be an open equivalence relation.
In [ 5 ] C. Akemann and G. Pedersen defined four concepts of semicontinuity for elements of A **, the enveloping W *-algebra of a C *-algebra A . For three of these the associated classes of lower semicontinuous elements are , and (notation explained in Section 2), and we will call these the classes of strongly lsc , middle lsc , and weakly lsc elements, respectively. There are three corresponding concepts of continuity: The strongly continuous elements are the elements of A itself, the middle continuous elements are the multipliers of A , and the weakly continuous elements are the quasi-multipliers of A . It is natural to ask the following questions, each of which is three-fold.
(Q1) Is every lsc element the limit of a monotone increasing net of continuous elements?
(Q2) Is every positive lsc element the limit of an increasing net of positive continuous elements?
(Q3) If h ≧ k , where h is lsc and k is usc, does there exist a continuous x such that h ≧ x ≧ k ?
Let A be a C*-algebra. For a ∈ A let D(a, A) denote the inner derivation induced by a, regarded as a bounded operator on A, and let d(a, Z(A)) denote the distance of a from Z(A), the centre of A. Let K(A) be the smallest number in [0, ∞] such that d(a, Z(A))≤ K(A)∥D(a, A)∥ for all a ∈ A. It is shown that if A is non-commutative and has an identity then either K(A) = , or K(A) = 1 / √3, or K(A) ≥ 1. Necessary and sufficient conditions for these three possibilities are given in terms of the primitive and primal ideals of A. If A is a quotient of an AW*-algebra then K(A) ≤ . Helly's Theorem is used to show that if A is a weakly central C*-algebra then K(A) ≤ 1.
For a C∗-algebra A let M(A) denote the two-sided multipliers of A in its enveloping von Neumann algebra. A complete description of M(A) is given in the case where the spectrum of A is Hausdorff. The formula M(A ⊗αB) = M(A) ⊗αM(B) is discussed and examples are given where is non-simple even though A is simple and separable. As a generalization of Tietze's Extension Theorem it is shown that a multiplier of a quotient of A is the image of an element from M(A), if A is separable. Finally, deriving algebras and thin operators and their relations to multipliers are discussed.
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