On the classification of certain real rank zero C*-algebras

In this paper,we give a classification of real rank zero C^*-algebras that can be expressed as inductive limits of a sequence of a subclass of Elliott-Thomsen algebras C.     

On the stable rank of simple C*-algebras

It is shown that there exists a simple, finite C*-algebra whose topological (or Bass) stable rank is any given natural number or .

     

Projectionless C*-algebras

We give a sufficient condition for a unital C*-algebra to have no nontrivial projections, and we apply this result to known examples and to free products. We also show how questions of existence of projections relate to the norm-connectedness of certain sets of operators.

     

Ring C*-algebras

We associate reduced and full C*-algebras to arbitrary rings and study the inner structure of these ring C*-algebras. As a result, we obtain conditions for them to be purely infinite and simple. We also discuss several examples. Originally, our motivation comes from algebraic number theory.     

AW*-algebras Which are Enveloping C*-algebras of JC-algebras

In the given article, enveloping C*-algebras of AJW-algebras are considered. Conditions are given, when the enveloping C*-algebra of an AJW-algebra is an AW*-algebra, and corresponding theorems are proved. In particular, we proved that if $\mathcal{A}$ is a real AW*-algebra, $\mathcal{A}_{sa}$ is the JC-algebra of all self-adjoint elements of $\mathcal{A}$ , $\mathcal{A}+i\mathcal{A}$ is an AW*-algebra and $\mathcal{A}\cap i\mathcal{A} = \{0\}$ then the enveloping C*-algebra $C^*(\mathcal{A}_{sa})$ of the JC-algebra $\mathcal{A}_{sa}$ is an AW*-algebra. Moreover, if $\mathcal{A}+i\mathcal{A}$ does not have nonzero direct summands of type I₂, then $C^*(\mathcal{A}_{sa})$ coincides with the algebra $\mathcal{A}+i\mathcal{A}$ , i.e. $C^*(\mathcal{A}_{sa})= \mathcal{A}+i\mathcal{A}$ .     

Noncommutative Jordan C*-algebras

We introduce noncommutative JB*-algebras which generalize both B*-algebras and JB*-algebras and set up the bases for a representation theory of noncommutative JB*-algebras. To this end we define noncommutative JB*-factors and study the factor representations of a noncommutative JB*-algebra. The particular case of alternative B*-factors is discussed in detail and a Gelfand-Naimark theorem for alternative B*-algebras is given.     

Separably injective C*-algebras

We show that a C*-algebra is a 1-separably injective Banach space if and only if it is linearly isometric to the Banach space \({C_0(\Omega)}\) of complex continuous functions vanishing at infinity on a substonean locally compact Hausdorff space \({\Omega}\).     

Topologically injective C*-algebras

A criterion for the topological injectivity of an AW*-algebra as a right Banach module over itself is given. A necessary condition for a C* -algebra to be topologically injective is obtained.     

On action of diffeomorphisms of C*-algebras on derivations

In this paper we consider automorphisms of the domains of closed *-derivations of C*-algebras and show that they extend to automorphisms of C*-algebras, so we call them diffeomorphisms. The diffeomorphisms generate transformations of the sets of closed *-derivations of C*-algebras. In this paper we study the subgroups of diffeomorphisms that define “bounded” shifts of derivations and the subgroups of the stabilizers of derivations.     

Non-Hausdorff symmetries of C*-algebras

Symmetry groups or groupoids of C*-algebras associated to non-Hausdorff spaces are often non-Hausdorff as well. We describe such symmetries using crossed modules of groupoids. We define actions of crossed modules on C*-algebras and crossed products for such actions, and justify these definitions with some basic general results and examples.     

On a class of stably finite nuclear C*-algebras

In this note we show that a separable C^*-algebra is nuclear and has a quasidiagonal extension by (the ideal of compact operators on an infinite-dimensional separable Hilbert space) if and only if it is anuclear finite algebra (NF-algebra) in the sense of Blackadar and Kirchberg, and deduce that every nuclear C^*-subalgebra of aNF-algebra isNF. We show that strongNF-algebras satisfy a Følner type condition.     

On convolution products and automorphisms in Hopf C*-algebras

We obtain two characterizations of the bi-inner Hopf *-automorphisms of a finite-dimensional Hopf C*-algebra, by means of an analysis of the structure of convolution products in this class of Hopf C*-algebra.     

On a class of smooth Frechet subalgebras of C * -algebras

The paper contributes to understanding the differential structure in a C ^*-algebra. Refining the Banach $(D_p^*)$ -algebras investigated by Kissin and Shulman as noncommutative analogues of the algebra C ^p [a,b] of p-times continuously differentiable functions, we investigate a Frechet $(D_\infty^*)$ -subalgebra $\ensuremath{{\mathcal B}}$ of a C ^*-algebra as a noncommutative analogue of the algebra C ^?∞?[a,b] of smooth functions. Regularity properties like spectral invariance, closure under functional calculi and domain invariance of homomorphisms are derived expressing $\ensuremath{{\mathcal B}}$ as an inverse limit over n of Banach $(D^*_n)$ -algebras. Several examples of such smooth algebras are exhibited.     

Jordan axioms for C*-algebras

A complex Banach spaceA which is also an associative algebra provided with a conjugate linear vector space involution * satisfying (a ²)^*=(a ^*)², aa ^* a=a³ and ab+ba2ab for alla, b inA is shown to be a C^*-algebra. The assumptions onA can be expressed in terms of the Jordan algebra obtained by symmetrization of the product ofA and are satisfied by any C^*-algebra. Thus we obtain a purely Jordan characterization of C^*-algebras.     

Inner ideals in C*-algebras

Research supported by a grant from the Schweizerische Nationalfonds/Fonds national suisse     

Infinite simple C*-algebras and Reduced cross products of abelian C*-algebras and free groups

Summary Let Γ=〈g ₁〉*〈g ₂〉*...*〈g _n 〉*... be a free product of cyclic groups with generators {g _i }, andC _r ^* (Γ,℘ _Λ) be the C*-algebra generated by the reduced group C*-algebraC _r ^* Γ and a set of projectionsP _gL associated with a subset Λ of {g _i }. We prove the following: (1)C _r ^* (Γ,℘ _Λ) is *-isomorphic to the reduced cross product for certain Hausdorff compact spaceX _Λ constructed from Γ and its boundary ∂Γ. (2)C _r ^* (Γ,℘ _Λ) is either a purely infinite, simple C*-algebra or an extension of a purely infinite, simple C*-altebra, depending on the pair (Γ, Λ). (3)C _r ^* (Г,℘ _Λ) is nuclear if and only if the subgroup Γ_Λ generated by {g _i }/Λ is amenable. Partially supported by RMC grant 45/290/603 from the University of Newcastle Partially supported by NSF grant DMS-9225076 and a Taft travel grant from the University of Cincinnati