Automatic continuity of biorthogonality preservers between compact C-algebras and von Neumann algebras

We prove that every biorthogonality preserving linear surjection between two dual or compact C^?-algebras or between two von Neumann algebras is automatically continuous.     

The relative commutant of separable C-algebras of real rank zero

We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C^∗-algebra in its ultrapower depend on the choice of the ultrafilter?     

A Galois correspondence for compact group actions on C-algebras

In this paper, we prove a Galois correspondence for compact group actions on C^?-algebras in the presence of a commuting minimal action. Namely, we show that there is a one-to-one correspondence between the C^?-subalgebras that are globally invariant under the compact action and the commuting minimal action, that in addition contain the fixed point algebra of the compact action and the closed, normal subgroups of the compact group.     

Isomorphism of Hilbert modules over stably finite C-algebras

It is shown that if A is a stably finite C^∗-algebra and E is a countably generated Hilbert A-module, then E gives rise to a compact element of the Cuntz semigroup if and only if E is algebraically finitely generated and projective. It follows that if E and F are equivalent in the sense of Coward, Elliott and Ivanescu (CEI) and E is algebraically finitely generated and projective, then E and F are isomorphic. In contrast to this, we exhibit two CEI-equivalent Hilbert modules over a stably finite C^∗-algebra that are not isomorphic.     

Haagerup property for C-algebras

We define the Haagerup property for C^?-algebras A and extend this to a notion of relative Haagerup property for the inclusion B⊆A, where B is a C^?-subalgebra of A. Let Γ be a discrete group and Λ a normal subgroup of Γ, we show that the inclusion A_?α,rΛ⊆A_?α,rΓ has the relative Haagerup property if and only if the quotient group Γ/Λ has the Haagerup property. In particular, the inclusion has the relative Haagerup property if and only if Γ/Λ has the Haagerup property; has the Haagerup property if and only if Γ has the Haagerup property. We also characterize the Haagerup property for Γ in terms of its Fourier algebra A(Γ).     

On weakly central C-algebras

A unital C^∗-algebra A is weakly central if and only if for every x∈A there exists a sequence of elementary unital completely positive maps αn on A such that the sequence (αn(x)) converges to a central element.     

Metrics with Positive Scalar Curvature at Infinity and Localization Algebra

In this paper, the authors give a new proof of Block and Weinberger’s Bochner vanishing theorem built on direct computations in the K-theory of the localization algebra.     

Geometric structure of dimension functions of certain continuous fields

For a continuous field of C^?-algebras A, we give a criterion to ensure that the stable rank of A is one. In the particular case of a trivial field this leads to a characterization of stable rank one, completing accomplishments by Nagisa, Osaka and Phillips. Further, for certain continuous fields of C^?-algebras, we study when the Cuntz semigroup satisfies the Riesz interpolation property, and we also analyze the structure of its functionals. As an application, we obtain a positive answer to a conjecture posed by Blackadar and Handelman in a variety of situations.     

Topological free entropy dimension in unital C-algebras

The notion of topological free entropy dimension of n-tuple of elements in a unital C^∗ algebra was introduced by Voiculescu. In the paper, we compute topological free entropy dimension of one self-adjoint element and topological free orbit dimension of one self-adjoint element in a unital C^∗ algebra. We also calculate the values of topological free entropy dimensions of any families of self-adjoint generators of some unital C^∗ algebras, including irrational rotation C^∗ algebra, UHF algebra, and minimal tensor product of two reduced C^∗ algebras of free groups.     

Classification of Simple C-Algebras: Inductive Limits of Matrix Algebras Over One-Dimensional Spaces

In this article, we will give a complete classification of simple C^*-algebras which can be written as inductive limits of algebras of the form A_n=⊕_i=1^k_nM_[n,i](C(X_n,i)), where X_n,i are arbitrary variable one-dimensional compact metrizable spaces. The results unify and generalize the previous results for the case X_n,i=S¹ and for the case of X_n,i being trees. We obtain our classification results by reducing the case of general one-dimensional spaces to the case of circles. The techniques in this paper play important roles in the study of the case of higher-dimensional spaces.     

On the Homotopy Type of the Unitary Group and the Grassmann Space of Purely Infinite Simple C*-Algebras

We completely determine the homotopy groups _n (.) of the unitary group and the space of projections of purely infinite simple C ^*-algebras in terms of K-theory. We also prove that the unitary group of a purely infinite simple C ^*-algebra A is a contractible topological space if and only if K0(A) = K1(A) = {0}, and again if and only if the unitary group of the associated generalized Calkin algebra L(HA) / K(HA) is contractible. The well-known Kuiper's theorem is extended to a new class of C ^*-algebras.     

On classifying monotone complete algebras of operators

We give a classification of “small” monotone complete C ^*-algebras by order properties. We construct a corresponding semigroup. This classification filters out von Neumann algebras; they are mapped to the zero of the classifying semigroup. We show that there are 2 ^c distinct equivalence classes (where c is the cardinality of the continuum). This remains true when the classification is restricted to special classes of monotone complete C ^*-algebras e.g. factors, injective factors, injective operator systems and commutative algebras which are subalgebras of ℓ^∞. Some examples and applications are given.      

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

Classification of extensions of classifiable -algebras

For a certain class of extensions of C^*-algebras in which B and A belong to classifiable classes of C^*-algebras, we show that the functor which sends to its associated six term exact sequence in K-theory and the positive cones of K₀(B) and K₀(A) is a classification functor. We give two independent applications addressing the classification of a class of C^*-algebras arising from substitutional shift spaces on one hand and of graph algebras on the other. The former application leads to the answer of a question of Carlsen and the first named author concerning the completeness of stabilized Matsumoto algebras as an invariant of flow equivalence. The latter leads to the first classification result for nonsimple graph C^*-algebras.     

The Connes-Higson construction is an isomorphism

Let A be a separable C∗-algebra and B a stable C∗-algebra containing a strictly positive element. We show that the group Ext^−1/2(SA,B) of unitary equivalence classes of extensions of SA by B, modulo the extensions which are asymptotically split, coincides with the group of homotopy classes of such extensions. This is done by proving that the Connes-Higson construction gives rise to an isomorphism between Ext^−1/2(SA,B) and the E-theory group E(A,B) of homotopy classes of asymptotic homomorphisms from S²A to B.     

A new look at KK-theory

We describe the Kasparov group KK(A, B) as the set of homotopy classes of homomorphisms from an algebra qA associated with A into K B. The algebra qA consists of K-theory differential forms over A. Its construction is dual to that of M ₂(A). The analysis of qA and of its interplay with M ₂(A) gives the basic results of KK-theory.Partially supported by NSF.     

K-Homology Relative to Semisplit Ideals

Given a C^*-algebra A, and an ideal J of A, we define a relative group K^⁰(A,J) in terms of a relative universal C^*-algebra for the pair (A,J). We show that the natural restriction map K^⁰(A,J) K⁰(J) is an isomorphism, and that, if J is a semisplit ideal of A, the Baum–Douglas–Taylor relative K-homology is recovered. This provides a generalization of one of the main results of the Baum–Douglas–Taylor theory.     

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

EP modular operators and their products

We study first EP modular operators on Hilbert C^?-modules and then we provide necessary and sufficient conditions for the product of two EP modular operators to be EP. These enable us to extend some results of Koliha (2000) [13] for an arbitrary C^?-algebra and the C^?-algebras of compact operators.