Bundles over Quantum Sphere and Noncommutative Index Theorem

The Noncommutative Index Theorem is used to prove that the Chern numbers of quantum Hopf line bundles over the standard Podle quantum sphere equal the winding numbers of the representations defining these bundles. This result gives an estimate of the positive cone of the algebraic K₀ of the standard quantum sphere.     

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.     

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.     

Rieffel projections in certain transformation group C*-algebras

LetA be the transformation groupC ^*-algebra associated with an arbitrary orientation-preserving homeomorphism of . ThisC ^*-algebra contains an infinite family of projections, called Rieffel projections, each of which generates theK ₀-groupK ₀(A). Although these projections must beK-theoretically equivalent, it is easy to see that most are not Murray-von Neumann equivalent. The mystery of how large the matrix algebra must be to implement theK-theory equivalence, is solved by explicitly constructing the equivalence in the smallest possible algebra:A with unit adjoined.Partially supported by NSF Grant DMS 8901923.     

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

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     

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.     

The Adjoints of Dilation Operators on H-Cones

If S is an H-cone and P:SS is a localizable dilation operator on S (i.e., P is additive increasing, contractive, continuous in order from below and s(Ps+t–Pt+Pf)S, s,tS,f(S–S)₊), then it is proved that its adjoint P ^*:S ^*S ^* (i.e., P ^*=P) is also a localizable dilation operator. This is an improvement of a result obtained by G. Mokobodzki in the frame of excessive functions.     

The Novikov Conjecture for Low Degree Cohomology Classes

We outline a twisted analogue of the Mishchenko–Kasparov approach to prove the Novikov conjecture on the homotopy invariance of the higher signatures. Using our approach, we give a new and simple proof of the homotopy invariance of the higher signatures associated to all cohomology classes of the classifying space that belong to the subring of the cohomology ring of the classifying space that is generated by cohomology classes of degree less than or equal to 2, a result that was first established by Connes and Gromov and Moscovici using other methods. A key new ingredient is the construction of a tautological C* _r (, )-bundle and connection, which can be used to construct a C* _r (, )-index that lies in the Grothendieck group of C* _r (, ), where is a multiplier on the discrete group corresponding to a degree 2 cohomology class. We also utilise a main result of Hilsum and Skandalis to establish our theorem.     

Noncommutative differential geometry on the quantum two sphere of podle?. I: An algebraic viewpoint

We compute the cyclic homology of the coordinate ringA(S _q ² (c, d)) of the quantum two sphereS _q ² (c, d) of Podle. The results are then interpreted from the point of view of Connes' noncommutative index theorem.     

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.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

A q-Analogue of Weber-Schafheitlin Integral of Bessel Functions

In an attempt to find a q-analogue of Weber and Schafheitlin's integral ₀ x ^– J (ax) J (bx) dx which is discontinuous on the diagonal a = b the integral ₀ x ^– J ⁽²⁾ (a(1 – q)x; q)J ⁽¹⁾ (b(1 – q)x; q) dx is evaluated where J ⁽¹⁾ (x; q) and J ⁽²⁾ (x; q) are two of Jackson's three q-Bessel functions. It is found that the question of discontinuity becomes irrelevant in this case. Evaluations of this integral are also made in some interesting special cases. A biorthogonality formula is found as well as a Neumann series expansion for x in terms of J ⁽²⁾ _+1+2n ((1 – q)x; q). Finally, a q-Lommel function is introduced.     

The Trace Conjecture – A Counterexample

In the paper Geometric K-Theory for Lie Groups and Foliations, Baum and Connes conjecture in a remark following Corollary 2 of their famous Isomorphism conjecture that for a finitely generated group with torsion, the trace map tr: K₀(C^*)R maps K₀(C^*) onto the additive subgroup of Q generated by all rational numbers of the form 1/n where n is the order of a finite subgroup of . We construct a counterexample to this conjecture.     

Relative K-homology and K-homology of an ideal

We provide a straightforward proof of one of the main results of the Baum-Douglas K-homology theory: If A is a separable nuclear C ^*-algebra and I an ideal of A, then the natural restriction map K ⁰(A, I) K ⁰(I) is an isomorphism of Abelian groups.     

Hilbertian Versus Hilbert W*-Modules and Applications to L2- and other invariants

Hilbert(ian) A-modules over finite von Neumann algebras with a faithful normal trace state (from global analysis) and Hilbert W*-modules over A (from operator algebra theory) are compared and a categorical equivalence is established. The correspondence between these two structures sheds new light on basic results in L ²-invariant theory providing alternative proofs. We indicate new invariants for finitely generated projective B-modules, where B is a unital C*-algebra (usually the full group C*-algebra C*() of the fundamental group =₁(M) of a manifold M).     

Noncommutative Chern Characters of Compact Lie Group C*-Algebras

For compact Lie groups, the Chern characters K^*(G) Q H^* _DR(G;Q) have been already constructed. In this paper, we construct and study the corresponding noncommutative Chern characters. They are homomorphisms ch_C*: K_*(C^*(G)) from quantum K-groups into entire current periodic cyclic homology groups of group C^*-algebras. We also obtain the corresponding algebraic version ch_alg: K_*(C^*(G)) HP_*(C^*(G)), which can be identified with the classical Chern character K_* (C(T)) HP_* (C(T)), where T is the maximal torus of G.     

Differential structure of certainC(X) xα Z

In this paper, we show that, without involving theC*-bundle theory, elementary differential topology can do the classification of homogeneousC*-crossed productC(X) x Z, whereX is a compact differentiable manifold of low dimension and is a diffeomorphism ofX. The motivation of this work is the Shultz's theorem which states that aC*-algebra can be identified with its pure state space carryingw*-topology and certain geometric structure.     

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.