The Baum-Connes Conjecture for $KK$-theory, Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, Available on CJO 2010 doi:10.1017/is010003012jkt114

Abstract: We define and compare two bivariant generalizations of the topological $K$-group $K^\top(G)$ for a topological group $G$. We consider the Baum-Connes conjecture in this context and study its relation to the usual Baum-Connes conjecture.

Multiplicativity of the JLO character, Journal of Noncommutative Geometry, 5 (2011), Issue 3, pp. 387–399.

Abstract: We prove that the Jaffe-Lesniewski-Osterwalder character is compatible with the $A_{\infty}$-structure of Getzler and Jones.

Pseudo-differential Operators and Regularity of Spectral Triples, Fields Communications Series, Volume 61 -- Perspectives on Noncommutative Geometry, 2011.

Abstract: We introduce a notion of an algebra of generalized pseudo-differential operators and prove that a spectral triple is regular if and only if it admits an algebra of generalized pseudo-differential operators. We also provide a self-contained proof of the fact that the product of regular spectral triples is regular.

Restriction maps in equivariant $KK$-theory, to appear in the Journal of K-theory.

Abstract: We extend McClure's results on the restriction maps in equivariant $K$-theory to bivariant $K$-theory: Let $G$ be a compact Lie group and $A$ and $B$ be $G$-$C^*$-algebras. Suppose that $KK^{H}_{n}(A, B)$ is a finitely generated $R(G)$-module for every $H \le G$ closed and $n \in \Z$. Then, if $KK^{F}_{*}(A, B) = 0$ for all $F \le G$ {\em finite cyclic}, then $KK^{G}_{*}(A, B) = 0$.

Homotopical Algebra for C^*-algebras.

Abstract: Category of fibrant objects is a convenient framework to do homotopy theory, introduced and developed by Ken Brown. In this paper, we apply it to the category of C^{*}-algebras. In particular, we get a unified treatment of (ordinary) homotopy theory for C^{*}-algebras, KK-theory and E-theory, as all of these can be expressed as the homotopy category of a category of fibrant objects.

Unsuspended Connective $E$-Theory.

Abstract: We prove connective versions of results by Shulman [Shu10] and Dadarlat-Loring [DL94]. As a corollary, we see that two separable $C^*$-algebras of the form $C_0(X) \otimes A$, where $X$ is a based, connected, finite CW-complex and $A$ is a unital properly infinite algebra, are $\bu$-equivalent if and only if they are asymptotic matrix homotopy equivalent.

Counterexample to the K\"{u}nneth theorem in $K$-theory

Abstract: In this note, we give counterexamples to the K\"{u}nneth theorems for the minimal and the maximal tensor products, thus answering some of the questions raised by Blackadar in \cite[23.13.2]{MR1656031}. Our main tools are the mapping cone construction and the Puppe exact sequence.