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.     

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. $C^*(\mathcal{A}_{sa})= \mathcal{A}+i\mathcal{A}$ .     

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 \({\Omega}\).     

Property T＊＊ for C＊-algebras

Extending the notion of property T of finite von Neumann algebras to general von Neumann algebras, we define and study in this paper property T** for (possibly non-unital) C* -algebras. We obtain several results of property T** parallel to those of property T for unital C* -algebras. Moreover, we show that a discrete group Γ has property T if and only if the group C* -algebra Cr* (Γ) (or equivalently, the reduced group C* -algebra Cr* (Γ)) has property T**. We also show that the compact operators K(l2 ) has property T** but c0 does not have property T**.     

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.     

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.     

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     

Twisted bounded-dilation group C*-algebras as C*-metric algebras

We construct a class of C~*-metric algebras. We prove that for a discrete group Γ with a 2-cocycle σ,the closure of the seminorm ||[M?, ·]|| on Cc(Γ, σ) is a Leibniz Lip-norm on the twisted reduced group C~*-algebra C*r(Γ, σ) for the pointwise multiplication operator M?on ?2(Γ), induced by a proper length function ? on Γ with the property of bounded θ-dilation. Moreover, the compact quantum metric space structures depend only on the cohomology class of 2-cocycles in the Lipschitz isometric sense.     

Comparison Properties for Asymptotically Tracially Approximation C*-algebras

Acta Mathematica Sinica, English Series - We show that the following properties of the C*-algebras in a class $$\cal{P}$$ are inherited by simple unital C*-algebras in the class of asymptotically...     

C*-algebras associated with interval maps

For each piecewise monotonic map of , we associate a pair of C*-algebras and and calculate their K-groups. The algebra is an AI-algebra. We characterize when and are simple. In those cases, has a unique trace, and is purely infinite with a unique KMS state. In the case that is Markov, these algebras include the Cuntz-Krieger algebras , and the associated AF-algebras . Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and -transformations. For the case of interval exchange maps and of -transformations, the C*-algebra coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani, respectively.

     

Projections in Free Product C*-algebras

Consider the reduced free product of C*-algebras, , with respect to states and that are faithful. If and are traces, if the so-called Avitzour conditions are satisied, (i.e. A ₁ and A ₂ are not "too small" in a specific sense) and if A ₁ and A ₂ are nuclear, then it is shown that the positive cone, K ₀(A)⁺, of the K ₀-group of A consists of those elements for which or . Thus, the ordered group K ₀(A) is weakly unperforated.?If, on the other hand, or is not a trace and if a certain condition weaker than the Avitzour conditions holds, then A is properly infinite. Submitted: January 1997, Final version: April 1997     

Lifting positive elements in C*-algebras

Supported by an NSERC grant, a NATO International Collaboration grant, and an E.W.R.Steacie Fellowship     

C *-algebras generated by partial isometries

We study the interconnection between directed graphs and operators on a Hilbert space. The intuition supporting this link is the following feature shared by partial isometries (as operators on a Hilbert space) on the one hand and edges in directed graphs on the other. A partial isometry a is an operator in a Hilbert space H, i.e., a:H→H which maps a (closed) subspace in H isometrically onto a generally different subspace. The respective subspaces are called the initial space and the final space of a. Denoting the corresponding (orthogonal) projections by p _i and p _f , note that a partial isometry a may be thought of as an edge from one vertex to another (which are not necessarily distinct) in a directed graph. And conversely, every directed graph has such a representation. Since neither the partial isometries nor the directed edges in a fixed model allow unrestricted composition, the algebraic construct which is useful is that of a groupoid. In this paper we develop this as a representation theory, and we explore the connection between realizations in the context of C ^*-algebras. The building blocks in our theory are certain matricial C ^*-algebras which we define. We then prove how they serve to localize our global representations.     

Nuclear C*-algebras and related questions

Results are given characterizing the class of nuclear C^*-algebras from various points of view, and a number of consequences of the nuclearity condition are given, and properties of C^*-algebras and W^*-algebras related to nuclearity are discussed.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 26, pp. 107–126, 1985.