Multipliers on matrix ordered operator spaces and some K-groups

Matrix ordered operator spaces are ‘non-commutative Banach spaces equipped with a non-commutative order’. Examples include C*-algebras as well as their duals. In this article, we define and intrinsically characterize the multiplier algebra for this class of spaces and briefly tackle the problem of extending K-theory to this context.     

Continuous *-homomorphisms of Banach Partial *-algebras

We continue the study of Banach partial *-algebras, in particular the question of the interplay between *-homomorphisms and biweights. Two special types of objects are introduced, namely, relatively bounded biweights and Banach partial *-algebras satisfying a certain Condition (S), which behave in a more regular way. We also present a systematic construction of Banach partial *-algebras of this type and exhibit several examples.     

Manifolds of tripotents in JB*-triples

Abstract. We describe the affine connections, geodesics and symmetries of various Banach manifolds of tripotents in JB*-triples which include the C*-algebras and Hilbert spaces where the nonzero tripotents are respectively the partial isometries and the extreme points of the closed unit ball. Received July 7, 1998; in final form November 16, 1998     

On K-theoretic invariants of semigroup C*-algebras attached to number fields

We show that semigroup C*-algebras attached to ax+b$ax+b$-semigroups over rings of integers determine number fields up to arithmetic equivalence, under the assumption that the number fields have the same number of roots of unity. For finite Galois extensions, this means that the semigroup C*-algebras are isomorphic if and only if the number fields are isomorphic.     

Positive Functionals on some Spaces of Fractions

The aim of this work is to study norm preserving extensions of positive functionals on some spaces of fractions. The main result is stated in the case of unitary complex Banach algebras with involution. Moreover, we deal with C*-algebras and the commutative case as well. An application is also given.     

Inverse closedness ofC *-algebras in Banach algebras

LetA denote a unital Banach algebra, and letB denote aC ^*-algebra which is contained as a unital subalgebra inA. We prove thatB is inverse closed inA if the norms ofA andB coincide. This generalizes well known result about inverse closedness ofC ^*-subalgebras inC ^*-algebras.     

LINEAR MAPPINGS THAT PRESERVE THE DERIVATIONAL STRUCTURE OF C*-ALGEBRAS

Abstract

Given a C*-algebra A and a suitable set of derivations on A, we consider the algebras A _n of n-differentiable elements of A as described in [B], before passing to an analysis of important classes of bounded linear maps between two such spaces. We show that even in this general framework, all the main features of the theory for the case C^(m)(U) → C ^(p) (V) where U and V are open balls in suitable Banach spaces, are preserved (see for example [A-G-L], [Gu-L], [Ja] and [L]). As part of the theory developed we obtain a non-trivial extension of the Kleinecke-Shirokov theorem in the category of C*-algebras to unbounded partially defined *-derivations. This indicates the existence of a single mathematical principle governing both the non-increasibility of differentiability by continuous homomorphisms and the untenability of the Heisenberg Uncertainty Principle for bounded observables.     

Compact quantum groups and group duality

In this paper, we consider the *-representations of compact quantum groups and group duality. The main results in the paper are: (1) there is a one-to-one correspondence between the *-representations of compact quantum groups and *-representations of the dual Banach *-algebra; (2) the category of commutative compact quantum groups (semigroups) is a dual category to the category of compact groups (semigroups); (3) the dual category of the category of locally compact groups (semigroups) is the category of commutative Hopf C^*-algebras with a particular property. Our group duality has the flavor of a Gelfand-Naimark type theorem for compact quantum groups, and for Hopf C^*-algebras.     

Linear Functional Equations in Banach Modules over a C *-Algebra

The paper is a survey on the Hyers–Ulam–Rassias stability of linear functional equations in Banach modules over a C ^*-algebra. Its contents is divided into the following sections: 1. Introduction; 2. Stability of the Cauchy functional equation in Banach modules; 3. Stability of the Jensen functional equation in Banach modules; 4. Stability of the Trif functional equation in Banach modules; 5. Stability of cyclic functional equations in Banach modules over a C ^*-algebra; 6. Stability of cyclic functional equations in Banach algebras and approximate algebra homomorphisms; 7. Stability of algebra *-homomorphisms between Banach *-algebras and applications.     

Extensions of Banach Lie-Poisson spaces

The extension of Banach Lie-Poisson spaces is studied and linked to the extension of a special class of Banach Lie algebras. The case of W∗-algebras is given particular attention. Semidirect products and the extension of the restricted Banach Lie-Poisson space by the Banach Lie-Poisson space of compact operators are given as examples.     

C*-Bundles for Certain Liminal C*-Algebras

It is shown that certain liminal C*-algebras whose limit sets in their primitive ideal space are discrete can be described as algebras of continuous sections of a C*-bundle associated with them. Their multiplier algebras are also described in a similar manner. The class of C*-algebras under discussion includes all the liminal C*-algebras with Hausdorff primitive ideal spaces but also many other liminal algebras. A large sub-class of examples is examined in detail.      

THE UNIQUENESS OF OPERATIONAL QUANTITIES IN VON NEUMANN ALGEBRAS

Abstract

Since 1970 a number of operational quantities, characteristic of either the semi-Fredholm operators or of some “ideal” of compact-like operators, have been introduced in the theory of bounded operators between Banach spaces and applied successfully to for example perturbation theory. More recently such quantities have been introduced even in the abstract setting of Fredholm theory in a von Neumann algebra relative to some closed two-sided ideal. We show that in this fairly general setting there is only one “reasonable” set of such quantities—a result which in its present form is to the best of our knowledge new even in the case of B(H), the algebra of all bounded operators on a Hilbert space H. We accomplish this by first of all introducing the concept of a (reduced) minimum modulus in the setting of C*-algebras and developing the relevant techniques. In the process we generalise a result of Nikaido [N].     

Invariant Subspaces of Positive Quasinilpotent Operators on Ordered Banach Spaces

In this paper we find invariant subspaces of certain positive quasinilpotent operators on Krein spaces and, more generally, on ordered Banach spaces with closed generating cones. In the later case, we use the method of minimal vectors. We present applications to Sobolev spaces, spaces of differentiable functions, and C*-algebras.      

Linear maps between C*-algebras that are *-homomorphisms at a fixed point

Let A and B be C*-algebras. A linear map T : A → B is said to be a *-homomorphism at an element z ∈ A if ab* = z in A implies T (ab*) = T (a)T (b)* = T (z), and c*d = z in A gives T (c*d) = T (c)*T (d) = T (z). Assuming that A is unital, we prove that every linear map T : A → B which is a *-homomorphism at the unit of A is a Jordan *-homomorphism. If A is simple and infinite, then we establish that a linear map T : A → B is a *-homomorphism if and only if T is a *-homomorphism at the unit of A. For a general unital C*-algebra A and a linear map T : A → B, we prove that T is a *-homomorphism if, and only if, T is a *-homomorphism at 0 and at 1. Actually if p is a non-zero projection in A, and T is a ^?-homomorphism at p and at 1 ? p, then we prove that T is a Jordan *-homomorphism. We also study bounded linear maps that are *-homomorphisms at a unitary element in A.     

Chain conditions for C*-algebras coming from Hilbert C*-modules

In this paper we define and study chain conditions for Hilbert C*-modules through their C*-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that these chain conditions are passed from the C*-algebra to its Hilbert module under certain conditions. We also study chain conditions for Hilbert modules coming from inclusion of C*-algebra with a faithful conditional expectation.     

On Quasi-Chebyshevity Subsets of Unital Banach Algebras

In this paper,first,we consider closed convex and bounded subsets of infinite-dimensional unital Banach algebras and show with regard to the general conditions that these sets are not quasi-Chebyshev and pseudo-Chebyshev.Examples of those algebras are given including the algebras of continuous functions on compact sets.We also see some results in C*-algebras and Hilbert C*-modules.Next,by considering some conditions,we study Chebyshev of subalgebras in C*-algebras.     

Duality and operator algebras II: operator algebras as Banach algebras

We answer, by counterexample, several questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator spaces, as opposed to that of Banach spaces and Banach algebras. In particular, the ‘nonselfadjoint analogue’ of a w*-algebra resides naturally in the category of dual operator spaces, as opposed to dual Banach spaces. We also show that an automatic w*-continuity result in the preceding paper of the authors, is sharp.     

Classes of Operator-Smooth Functions - II. Operator-Differentiable Functions

This paper studies the spaces of Gateaux and Frechet Operator Differentiable functions of a real variable and their link with the space of Operator Lipschitz functions. Apart from the standard operator norm on B(H), we consider a rich variety of spaces of Operator Differentiable and Operator Lipschitz functions with respect to symmetric operator norms. Our approach is aimed at the investigation of the interrelation and hierarchy of these spaces and of the intrinsic properties of Operator Differentiable functions. We apply the obtained results to the study of the functions acting on the domains of closed *-derivations of C*-algebras and prove that Operator Differentiable functions act on all such domains.We also obtain the following modification of this result: any continuously differentiable, Operator Lipschitz function acts on the domains of all weakly closed *-derivations of C*-algebras.     

Operator space projective tensor product of $C^*$-algebras

For a finite dimensional -algebra A and any -algebra B, we determine a constant of equivalence of operator space projective norm and the Banach space projective norm on . We also discuss the *-Banach algebra . Received May 12, 1999; in final form September 8, 1999 / Published online April 12, 2001