Stable outer conjugacy and strong Morita equivalence of group actions on pro-C *-algebras

We show that two continuous inverse limit actions α and β of a locally compact group G on two pro-C ^*-algebras A and B are stably outer conjugate if and only if there is a full Hilbert A-module E and a continuous action u of G on E such that E and E ^*(the dual module of E) are countably generated in M(E)(the multiplier module of E), respectively M(E ^*) and the pair (E, u) implements a strong Morita equivalence between α and β. This is a generalization of a result of F. Combes [Proc. London Math. Soc. 49(1984), 289–306].      

C*-Isomorphisms Associated with Two Projections on a Hilbert C*-Module*

Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies C^*-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple (P, Q, H) is said to be matched if H is a Hilbert C^*-module, P and Q are projections on H such that their infimum P ∧ Q exists as an element of L(H), where L(H) denotes the set of all adjointable operators on H. The C^*-subalgebras of L(H) generated by elements in {P - P ∧ Q, Q - P ∧ Q, I} and {P, Q, P ∧ Q, I} are denoted by i(P, Q, H) and o(P, Q, H), respectively. It is proved that each faithful representation (π, X) of o(P, Q, H) can induce a faithful representation (π, X e) of i(P, Q, H) such that e π(P - P ∧ Q) = π(P) - π(P) ∧ π(Q),eπ(Q - P ∧ Q) = π(Q) - π(P) ∧ π(Q).When (P, Q) is semi-harmonious, that is, R(P + Q) and R(2I - P - Q) are both orthogonally complemented in H, it is shown that i(P, Q, H) and i(I - Q, I - P, H) are unitarily equivalent via a unitary operator in L(H). A counterexample is constructed, which shows that the same may be not true when (P, Q) fails to be semi-harmonious. Likewise, a counterexample is constructed such that (P, Q) is semi-harmonious, whereas (P, I - Q) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert C^*-modules are also provided.     

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.     

Complete positivity,tensor products and C*-nuclearity for inverse limits of C*-algebras

The paper aims at developing a theory of nuclear (in the topological algebraic sense) pro-C^*-algebras (which are inverse limits of C^*-algebras) by investigating completely positive maps and tensor products. By using the structure of matrix algebras over a pro-C^*-algebra, it is shown that a unital continuous linear map between pro-C^*-algebrasA andB is completely positive iff by restriction, it defines a completely positive map between the C^*-algebrasb(A) andb(B) consisting of all bounded elements ofA andB. In the metrizable case,A andB are homeomorphically isomorphic iff they are matricially order isomorphic. The injective pro-C^*-topology α and the projective pro-C^*-topology v on A⊗B are shown to be minimal and maximal pro-C^*-topologies; and α coincides with the topology of biequicontinous convergence iff eitherA orB is abelian. A nuclear pro-C^*-algebraA is one that satisfies, for any pro-C^*-algebra (or a C^*-algebra)B, any of the equivalent requirements; (i) α =v onA ⊗B (ii)A is inverse limit of nuclear C^*-algebras (iii) there is only one admissible pro-C^*-topologyon A⊗B (iv) the bounded partb(A) ofA is a nuclear C⊗-algebra (v) any continuous complete state map A→B^* can be approximated in simple weak^* convergence by certain finite rank complete state maps. This is used to investigate permanence properties of nuclear pro-C^*-algebras pertaining to subalgebras, quotients and projective and inductive limits. A nuclearity criterion for multiplier algebras (in particular, the multiplier algebra of Pedersen ideal of a C^*-algebra) is developed and the connection of this C^*-algebraic nuclearity with Grothendieck’s linear topological nuclearity is examined. A σ-C^*-algebraA is a nuclear space iff it is an inverse limit of finite dimensional C^*-algebras; and if abelian, thenA is isomorphic to the algebra (pointwise operations) of all scalar sequences.     

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.     

On partial actions and groupoids

We prove that, as in the case of global actions, any partial action gives rise to a groupoid provided with a Haar system, whose -algebra agrees with the crossed product by the partial action.

     

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.     

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.     

On weighted L p integrability and approximation of trigonometric series

We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first results on the relations among the weighted best approximation and the coefficients of trigonometric series. Supported in part by Hangzhou Normal University.     

Extension algebras of Cuntz algebra

In this paper, we construct representatives for all equivalence classes of the unital essential extension algebras of Cuntz algebra by the C^*-algebras of compact operators on a separable infinite-dimensional Hilbert space. We also compute their K-groups and semigroups and classify these extension algebras up to isomorphism by their semigroups.     

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.     

Symmetric K-Theory Spectra of C*-Categories

We define K-theory groups and symmetric K-theory spectra associated to ₂-graded C ^*-categories and show that the exterior product of K-theory groups can be expressed in terms of the smash product of symmetric spectra.     

Weak exactness for dual operator spaces

We introduce a new tensor product and study the weak^∗ condition C^′, which is also called weak^∗ exactness, for dual operator spaces. Our definition of weak^∗ condition C^′ is equivalent to Kirchberg's notion of weak exactness in the case of von Neumann algebras. We also study the connection of weak^∗ exact W^∗-TROs with their linking von Neumann algebras and study the structure of exact (respectively, nuclear) W^∗-TROs.     

Rkk (X)-nucléarité (d‘après; G. Skandalis)

We define a notion of nuclearity for C(X)-C^*-algebras yielding the same results of semi-exactness, for Kasparovs bifunctor RKK(X), as those obtained by Skandalis in the case X=. We connect explicitly this new notion of nuclearity with those already known.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

NonstableK-Theory for operator algebras

We show that the homotopy groups of the group of quasi-unitaries inC ^*-algebras form a homology theory on the category of allC ^*-algebras which becomes topologicalK-theory when stabilized. We then show how this functorial setting, in particular the half-exactness of the involved functors, helps to calculate the homotopy groups of the group of unitaries in a series ofC ^*-algebras. The calculations include the case of all AbelianC ^*-algebras and allC ^*-algebras of the formAB, whereA is one of the Cuntz algebras O_n n=2, 3, ..., an infinite dimensional simpleAF-algebra, the stable multiplier or corona algebra of a-unitalC ^*-algebra, a properly infinite von Neumann algebra, or one of the projectionless simpleC ^*-algebras constructed by Blackadar.     

Convex -closures versus convex norm-closures in dual Banach spaces

A subset Y of a dual Banach space X^* is said to have the property (P) if for every weak^*-compact subset H of Y. The purpose of this paper is to give a characterization of the property (P) for subsets of a dual Banach space X^*, and to study the behavior of the property (P) with respect to additions, unions, products, whether the closed linear hull has the property (P) when Y does, etc. We show that the property (P) is stable under all these operations in the class of weak^* -analytic subsets of X^*.     

K-theory for rickart C*-algebras

We give an explicit index map for any properly infinite closed ideal of a Rickart C ^*-algebra. This generalizes Olsen's work on von Neumann algebras. We use our results to compute the topological and the algebraic K ₁-groups of any quotient algebra of a Rickart C ^*-algebra.     

A bott periodicity map for crossed products of C*-algebras by discrete groups

A generalization is given of the canonical map from a discrete group into K ₁ of the group C ^*-algebra. Our map also generalizes Rieffel's construction of a projection in an irrational rotation C ^*-algebra.     

Real <Emphasis Type="Italic">AW</Emphasis><Superscript>*</Superscript>-Algebras of type I

Let R be a real AW ^*-algebra, and suppose that its complexification M = R + iR is also a (complex) AW ^*-algebra. We prove that R is of type I if and only if so is M.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 79–81, 2004Original Russian Text Copyright © by Sh. A. Ayupov