Real C*-Algebras, United K-Theory, and the Künneth Formula

We define united K-theory for real C^*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors – real K-theory, complex K-theory, and self-conjugate K-theory – and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Künneth-type, nonsplitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C^*-algebras A and B which holds as long as the complexification of A is in the bootstrap category . Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C^*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product is not determined solely by the greatest common divisor of K and l. Hence, we have examples of nonisomorphic, simple, purely infinite, real C^*-algebras whose complexifications are isomorphic.     

The range of united K-theory

We prove that the united K-theory functor is a surjective functor from the category of real simple separable purely infinite C^∗-algebras to the category of countable acyclic CRT-modules. As a consequence, we show that every complex Kirchberg algebra satisfying the universal coefficient theorem is the complexification of a real C^∗-algebra.     

Embedding an AH-Algebra into a Simple C*-Algebra with Prescribed KK-Data

Let X be a connected finite CW complex. We show that, given a positive homomorphism Hom(K _*(C(X)), K _*(A)) with [1 _C(X)][1 _A ], where A is a unital separable simple C ^*-algebra with real rank zero, stable rank one and weakly unperforated K ₀(A), there exists a homomorphism h: C(X)A such that h induces . We also prove a structure result for unital separable simple C ^*-algebras A with real rank zero, stable rank one and weakly unperforated K ₀(A), namely, there exists a simple AH-algebra of real rank zero contained in A which determines the K-theory of A.     

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.     

The Milnor–Chow homomorphism revisited

The aim of this note is to give a simplified proof of the surjectivity of the natural Milnor–Chow homomorphism between Milnor K-theory and higher Chow groups for essentially smooth (semi-)local k-algebras A with infinite residue fields. It implies the exactness of the Gersten resolution for Milnor K-theory at the generic point. Our method uses the Bloch–Levine moving technique and some properties of the Milnor K-theory norm for fields. Furthermore we give new applications. Supported by Studienstiftung des deutschen Volkes and Deutsche Forschungsgemeinschaft.     

Abstract Bott periodicity inKK-theory

We show that the universalC*-algebras KqA and K²A are homotopy equivalent and define abstract analogues of the Bott elements inKK-theory.     

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.     

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.     

A Fredholm operator approach to Morita equivalence

GivenC*-algebrasA andB and an imprimitivityA-B-bimoduleX, we construct an explicit isomorphismX _*:K _i (A)K _i (B), whereK _i denotes the complexK-theory functors fori=0, 1. Our techniques do not require separability nor the existence of countable approximate identities. We thus extend to generalC*-algebras the result of Brown, Green, and Rieffel according to which, strongly Morita equivalentC*-algebras have isomorphicK-groups. The method employed includes a study of Fredholm operators on Hilbert modules.On leave from the University of São Paulo, Brazil.     

On equivariant Kasparov theory and Spanier-Whitehead duality

Suppose thatG is a second countable compact Lie group and thatA andB are commutativeG-C*-algebras. Then the Kasparov groupKK _* ^G (A, B) is a bifunctor onG-spaces. It is computed here in terms of equivariant stable homotopy theory. This result is a consequence of a more general study of equivariant Spanier-Whitehead duality and uses in an essential way the extension of the Kasparov machinery to the setting of -G-C*-algebras. As a consequence, we show that if (X, x ₀) is a based separable compact metricG-ENR (such as a smooth compactG-manifold) and (Y, y ₀) is a based countableG-CW-complex then there is a natural isomorphism

Universal coefficient theorems for the stable Ext-groups

Let A be a unital separable nuclear C^∗-algebra and let B be a stable C^∗-algebra. Using K-theory and KK-theory we establish universal coefficient theorems for the stable Ext-groups of unital extensions of A by B when A and B have certain properties, which generalize a result of L. Brown and M. Dadarlat for the strong Ext-groups. The class of extensions being studied are also enlarged.     

Symmetrization of Cuntz’ Picture for the Kasparov <Emphasis Type="Italic">KK</Emphasis>-Bifunctor

Given C^*-algebras A and B, we generalize the notion of a quasi-homomorphism from A to B in the sense of Cuntz by considering quasi-homomorphisms from some C^*-algebra C to B such that C surjects onto A and the two maps forming the quasi-homomorphism agree on the kernel of this surjection. Under an additional assumption, the group of homotopy classes of such generalized quasi-homomorphisms coincides with KK(A, B). This makes the definition of the Kasparov bifunctor slightly more symmetric and provides more flexibility in constructing elements of KK-groups. These generalized quasi-homomorphisms can be viewed as pairs of maps directly from A (instead of various C’s), but these maps need not be *-homomorphisms.     

Operator space tensor products of <Emphasis Type="Italic">C</Emphasis>*-algebras

For C*-algebras A and B, the identity map from into A _λ B is shown to be injective. Next, we deduce that the center of the completion of the tensor product A⊗B of two C*-algebras A and B with centers Z(A) and Z(B) under operator space projective norm is equal to . A characterization of isometric automorphisms of and A _h B is also obtained. Dedicated to Ajit Iqbal Singh on her 65th birthday.     

Semi-Invertible Extensions and Asymptotic Homomorphisms

We consider the semi-group Ext(A, B) of extensions of a separable C ^*-algebra A by a stable C ^*-algebra B modulo unitary equivalence and modulo asymptotically split extensions. This semi-group contains the group Ext^–1/2(A, B) of invertible elements (i.e. of semi-invertible extensions). We show that the functor Ext^–1/2(A, B) is homotopy invariant and that it coincides with the functor of homotopy classes of asymptotic homomorphisms from C A to M(B) that map S A C( ) A into 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.     

The Generalized Chern Character and Lefschetz Numbers in W*-Modules

We define N-theory as being an analogue of K-theory on the category of von Neumann algebras such that K ₀(A)N ₀(A) for any von Neumann algebra A. Moreover, it turns out to be possible to construct the extension of the Chern character to some homomorphism from N ₀(A) to an even Banach cyclic homology of A. Also, we define generalized Lefschetz numbers for an arbitrary unitary endomorphism U of an A-elliptic complex. We study them in the situation when U is an element of a representation of some compact Lie group.     

Decomposition rank and absorbing extensions of type I algebras

We prove the following: Let A and B be separable C^*-algebras. Suppose that B is a type I C^*-algebra such that

(i)

B has only infinite dimensional irreducible *-representations, and

(ii)

B has finite decomposition rank.

If

0→B→C→A→0     

Twisted K-theory of differentiable stacks

In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an S¹-gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure are derived. Our approach provides a uniform framework for studying various twisted K-theories including the usual twisted K-theory of topological spaces, twisted equivariant K-theory, and the twisted K-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted K-groups can be expressed by so-called “twisted vector bundles”.Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of K-theory (KK-theory) of C^∗-algebras.