The Proof of Inseparable Prime AW^＊—Algebras Being Primitive C^＊—Algebras

The relation between the inseparable prime C^*-algebras and primitive C^*-algebras is studied,and we prove that prime AW^*-algebras are all primitive C^*-algebras.     

State spaces of <Emphasis Type="Italic">JB</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-triples<Superscript>*</Superscript>

An atomic decomposition is proved for Banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process. This is then applied to yield, under appropriate assumptions, geometric characterizations, up to isometry, of the unit ball of the dual space of a JB^*-triple, and up to complete isometry, of one-sided ideals in C^*-algebras.Mathematics Subject Classification (2000):17C65, 46L07Both authors are supported by NSF grant DMS-0101153     

Mapping Surgery to Analysis I: Analytic Signatures

We develop the theory of analytically controlled Poincaré complexes over C ^*-algebras. We associate a signature in C ^*-algebra K-theory to such a complex, and we show that it is invariant under bordism and homotopy. The authors were supported in part by NSF Grant DMS-0100464.     

Representations and positive functionals

We discuss the representation theory of both the locally convex and non-locally convex topological^*-algebras. First we discuss the^*-representation of topological^*-algebras by operators on a Hilbert space. Then we study those topological^*-algebras so that every^*-representation of which on a Hilbert space is necessarily continuous. It is well-known that each^*-representation of aB ^*-algebra on a Hilbert space is continuous. We show that this is true for a large class of^*-algebras more general thanB ^*-algebras, including certain non-locally convex^*-algebras. Finally, we investigate the conditions under which a positive functional on a topological^*-algebra is representable.The research of the first-named author was partially supported by an NSERC grant. This work was done by the second-named author when he was a post-doctoral fellow at McMaster University.     

Identifying assembly maps in K- and L-theory

In this paper we prove the equivalence of various algebraically or geometrically defined assembly maps used in formulating the main conjectures in K- and L-theory, and C^*-theory.Partially supported by NSERC grant A4000 and NSF grant DMS 9104026. The authors also wish to thank the SFB für Geometrie und Analysis, Münster, and the Max Planck Institut für Mathematik, Bonn, for their hospitality and support.Received: 7 May 2001     

KK-Theories for topological algebras

We propose a unified construction of generalizedKK-functors on various categories of topological algebras ranging between the two extreme cases: the categories of discrete algebras and the category ofC ^*-algebras. Usual functorial properties of these functors are derived.Supported in part by an NSF grant.     

Perturbations of C*-Algebraic Invariants

Kadison and Kastler introduced a metric on the set of all C*-algebras on a fixed Hilbert space. In this paper structural properties of C*-algebras which are close in this metric are examined. Our main result is that the property of having a positive answer to Kadison’s similarity problem transfers to close C*-algebras. In establishing this result we answer questions about closeness of commutants and tensor products when one algebra satisfies the similarity property. We also examine K-theory and traces of close C*-algebras, showing that sufficiently close algebras have isomorphic Elliott invariants when one algebra has the similarity property.     

A geometrical property ofC(K) spaces

We introduce a geometrical property of norm one complemented subspaces ofC(K) spaces which is useful for computing lower bounds on the norms of projections onto subspaces ofC(K) spaces. Loosely speaking, in the dual of such a space ifx* is a w* limit of a net (x _a ^* ) andx*=x*₁+x*₂ with ‖x*‖=‖x*₁‖ + ‖x*₂‖, then we measure how efficiently thex _a ^* 's can be split into two nets converging tox*₁ andx*₂, respectively. As applications of this idea we prove that if for everyε>0,X is a norm (1+ε) complemented subspace of aC(K) space, then it is norm one complemented in someC(K) space, and we give a simpler proof that a slight modification of anl ₁-predual constructed by Benyamini and Lindenstrauss is not complemented in anyC(K) space. Research partially supported by a grant of the U.S.-Israel Binational Science Foundation. Research of the first-named author is supported in part by NSF grant DMS-8602395. Research of the second-named author was partially supported by the Fund for the Promotion of Research at the Technion, and by the Technion VPR-New York Metropolitan Research Fund.     

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.     

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 Atiyah-Segal completion theorem for C*-algebras

We generalize the Atiyah-Segal completion theorem to C ^*-algebras as follows. Let A be a C ^*-algebra with a continuous action of the compact Lie group G. If K _* ^G (A) is finitely generated as an R(G)-module, or under other suitable restrictions, then the I(G)-adic completion K _* ^G (A) is isomorphic to RK _*([A C(EG)]^G), where RK _* is representable K-theory for - C ^*-algebras and EG is a classifying space for G. As a corollary, we show that if and are homotopic actions of G, and if K _*(C ^* (G,A,)) and K _*(C ^* (G,A,)) are finitely generated, then K _*(C ^*(G,A,))K_*(C ^* (G,A,)). We give examples to show that this isomorphism fails without the completions. However, we prove that this isomorphism does hold without the completions if the homotopy is required to be norm continuous.This work was partially supported by an NSF Graduate Fellowship and by an NSF Postdoctoral Fellowship.     

Representable K-Theory for σ-C*-algebras

We define a version of K-theory on the category of -C ^*-algebras (countable inverse limits of C ^*-algebras). Our theory is homotopy invariant, has long exact sequences and a Milnor sequence, and satisfies Bott periodicity. On C ^*-algebras it gives the ordinary K-theory, and on the space of continuous functions on a countable direct limit X of compact Hausdorff spaces, it gives the representable K-theory of X. (We do not claim that our theory is in general a representable functor.) We also define an equivariant version, and discuss several related groups.Partially supported by a National Science Foundation Postdoctoral Fellowship.     

Fell topology on the hyperspace of a non-Hausdorff space

Fell topology is very widely used today, even in metric spaces; but J. Fell introduced it in a non-Hausdorff context in the connection with the theory of C ^*-algebras. In spite of this, it has been studied only on the hyperspace of a Hausdorff space, except for the first results due to Fell himself. The present paper aims to fill this gap, in particular extending some results of H. Poppe and of G. Beer to the general case. Project 10251002 supported by National Natural Science Foundation of China.     

Strong Morita equivalence of higher-dimensional noncommutative tori. II

We show that two C*-algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K ₀-groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C*-algebras of arbitrary finitely generated abelian groups. This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada, held by George A. Elliott.     

Continuity of positive functionals on topological*-algebras

To investigate the continuity of positive functionals on certain topological^*-algebras, we consider general locally convex and non-locally convex topological^*-algebras which need not have identity or continuity of involution. We generalize a number of known results.The research of the first-named author was partially supported by an NSERC grant. The research of the second-named author was carried on when the was a post-doctoral fellow at McMaster University.     

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.     

Improving Pogorelov’s isometric embedding counterexample

We construct a C ^2,1 metric of non-negative Gauss curvature with no C ² local isometric embedding in Dedicate to Professor Wei-Yue Ding on his sixtieth birthday. Y.Y. was partially supported by an NSF grant, and a Sloan Research Fellowship.     

On the asymptotic tensor C*-norm

We introduce a new asymptotic one-sided and symmetric tensor norm, the latter of which can be considered as the minimal tensor norm on the category of separable C^*-algebras with homotopy classes of asymptotic homomorphisms as morphisms. We show that the one-sided asymptotic tensor norm differs in general from both the minimal and the maximal tensor norms and discuss its relation to semi-invertibility of C^*-extensions. Received: 23 September 2004; revised: 30 May 2005