The Rohlin property for automorphisms of the Jiang-Su algebra

For projectionless C^∗-algebras absorbing the Jiang-Su algebra tensorially, we study a kind of the Rohlin property for automorphisms. We show that the crossed products obtained by automorphisms with this Rohlin property also absorb the Jiang-Su algebra tensorially under a mild technical condition on the C^∗-algebras. In particular, for the Jiang-Su algebra we show the uniqueness up to outer conjugacy of the automorphism with this Rohlin property.     

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**.     

Crossed products by finite group actions with certain tracial Rokhlin property

We introduce a special tracial Rokhlin property for unital C~*-algebras. Let A be a unital tracial rank zero C~*-algebra(or tracial rank no more than one C~*-algebra). Suppose that α : G → Aut(A) is an action of a finite group G on A, which has this special tracial Rokhlin property, and suppose that A is a α-simple C~*-algebra. Then, the crossed product C~*-algebra C~*(G, A, α) has tracia rank zero(or has tracial rank no more than one). In fact,we get a more general results.     

La C*-algèbre d'une quasi-représentation

We show that the C^* -algebra of the regular representation of a discrete group G onto a subset Σ of G is the reduced C^* -algebra of an r-discrete groupoid whose space of units is totally disconnected and contains Σ as a dense subset. The C^*-algebra of quasicrystals, some Cuntz-Krieger and crossed product algebras, and Wiener-Hopf algebras are particular cases of this construction     

On a Class of Graded Ideals of Semigroup <Emphasis Type="Italic">C</Emphasis>*-Algebras

We present general results about graded C*-algebras and continue the previously initiated research of the C*-algebra generated by the left regular representation of an abelian semigroup. We study the invariant ideals of this C*-algebra invariant with respect to the representation of a compact group G in the automorphism group of this algebra. We prove that the invariance of the ideal is equivalent to the fact that this ideal is graded C*-algebra, that there is a maximum of all invariant ideals, and it is the commutator ideal. Separately we study a class of graded primitive ideals generated by a single projector.     

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.     

Unbounded derivations

In this note we discuss various extensions of a normal 1 derivation of a uniformly hyperfinite C¹-algebra. Various approximation theorems are employed to show when said extensions generate automorphism groups of the C¹-algebra. We characterize the “maximal” extension of Sakai and Powers as a graph limit and show when this extension is the closure of the given derivation. We also discuss an identity obeyed by the resolvent of a derivation.     

Spectral invariance of smooth crossed products, and rapid decay locally compact groups

We show that all rapid-decay locally compact groups are unimodular and that the set of rapid-decay functions on a locally compact rapidly decaying group forms a dense and spectral invariant Fréchet *-subalgebra of the reduced group C ^*-algebra. In general, the set of rapid-decay functions on a locally compact strongly rapid-decay group with values in a commutative C ^*-algebra forms a dense and spectral invariant Fréchet *-subalgebra of the twisted crossed product C ^*-algebra. The spectral invariance property implies that the K-theories of both algebras are naturally isomorphic under inclusion.This project is supported in part by the National Science Foundation Grant #DMS 92-04005.     

Reflexivity of the automorphism and isometry groups of C*-algebras in BDF theory

We prove that the automorphism and isometry groups of any extension of the C*-algebra C (H)\cal C (\cal H) of all compact operators by a separable commutative C*-algebra are algebraically reflexive. Concerning the possibly most important extensions by the algebra C(\Bbb T)C(\Bbb T) of all continuous complex valued functions on the perimeter of the unit disc, we show that these groups are topologically nonreflexive.     

Quotient-bounded elements in locally convex algebras. II

Consideration of quotient-bounded elements in a locally convexGB ^*-algebra leads to the study of properGB ^*-algebras viz those that admit nontrivial quotient-bounded elements. The construction and structure of such algebras are discussed. A representation theorem for a properGB ^*-algebra representing it as an algebra of unbounded Hilbert space operators is obtained in a form that unifies the well-known Gelfand-Naimark representation theorem forC ^*-algebra and two other representation theorems forb ^*-algebras (also calledlmc ^*-algebras), one representinga b ^*-algebra as an algebra of quotient bounded operators and the other as a weakly unbounded operator algebra. A number of examples are discussed to illustrate quotient-bounded operators. An algebra of unbounded operators constructed out of noncommutativeL ^p-spaces on a regular probability gauge space and the convolution algebra of periodic distributions are analyzed in detail; whereas unbounded Hilbert algebras andL ^w-integral of a measurable field ofC ^*-algebras are discussed briefly.     

Haar measures on HopfC *-algebras

In this paper, we first use Markov-Kakutani's fixed point theorem to prove the existence and uniqueness of Haar measures on cocommutative HopfC ^*-algebras. Also we show that in the commutative case, there exists a natural one-to-one correspondence between the Haar measure on a given HopfC ^*-algebra and Haar measures on the associated semigroup. Finally, we show that for HopfC ^*-algebras with Peter-Weyl property, they have Haar measures.Work supported in part by the NSF.     

The automorphism group of the Toeplitz algebra onH 2(T2)

The automorphism group of the C^* -algebra generated by the Toeplitz operatos with symbols being continuous functions on the bicircle is characterized completely. The investigation is based on the analysis of the behaviour of an automorphism of the Toeplitz algebra on itsC* -ideal chains, and the state of the closed ideals in .     

On the topology of the Kasparov groups and its applications

In this paper we establish a direct connection between stable approximate unitary equivalence for *-homomorphisms and the topology of the KK-groups which avoids entirely C^*-algebra extension theory and does not require nuclearity assumptions. To this purpose we show that a topology on the Kasparov groups can be defined in terms of approximate unitary equivalence for Cuntz pairs and that this topology coincides with both Pimsner's topology and the Brown-Salinas topology. We study the generalized Rørdam group , and prove that if a separable exact residually finite dimensional C^*-algebra satisfies the universal coefficient theorem in KK-theory, then it embeds in the UHF algebra of type 2^∞. In particular such an embedding exists for the C^*-algebra of a second countable amenable locally compact maximally almost periodic group.     

TheK-theory of AF embeddings of the rational rotation algebras

Our main result is the construction of an embedding ofC(T²) into an approximately finite-dimensionalC ^*-algebra which induces an injection onK ₀(C(T²)). The existence of this embedding implies that Cech cohomology cannot be extended to a stable, continuous homology theory forC ^*-algebras which admits a well-behaved Chern character. Homotopy properties ofC ^*-algebras are also considered. For example, we show that the second homotopy functor forC ^*-algebras is discontinuous. Similar embeddings are constructed for all the rational rotation algebras, with the consequence that none of the rational rotation algebras satisfies the homotopy property called semiprojectivity.     

On the classification of nonsimple graph C *-algebras

We prove that a graph C ^*-algebra with exactly one proper nontrivial ideal is classified up to stable isomorphism by its associated six-term exact sequence in K-theory. We prove that a similar classification also holds for a graph C ^*-algebra with a largest proper ideal that is an AF-algebra. Our results are based on a general method developed by the first named author with Restorff and Ruiz. As a key step in the argument, we show how to produce stability for certain full hereditary subalgebras associated to such graph C ^*-algebras. We further prove that, except under trivial circumstances, a unique proper nontrivial ideal in a graph C ^*-algebra is stable.     

2-Local automorphisms of operator algebras

A not necessarily continuous, linear or multiplicative function θ from an algebra A into itself is called a 2-local automorphism if θ agrees with an automorphism of A at each pair of points in A. In this paper, we study when a 2-local automorphism of a C^∗-algebra, or a standard operator algebra on a locally convex space, is an automorphism.     

Fixed points of involutive automorphisms of the Bruhat order

Applying a classical theorem of Smith, we show that the poset property of being Gorenstein^* over Z₂ is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every interval in the Bruhat order on (twisted) involutions in an arbitrary Coxeter group has this property, and we find the rank function. This implies results conjectured by F. Incitti. We also show that the Bruhat order on the fixed points of an involutive automorphism induced by a Coxeter graph automorphism is isomorphic to the Bruhat order on the fixed subgroup viewed as a Coxeter group in its own right.