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Register a new userNon-stable K-theory and extremally rich C*-algebras
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We consider the properties weak cancellation, K_1-surjectivity, good index theory, and K_1-injectivity for the class of extremally rich C*-algebras, and for the smaller class of isometrically rich C*-algebras. We establish all four properties for isometrically rich C*-algebras and for extremally rich C*-algebras that are either purely infinite or of real rank zero, K_1-injectivity in the real rank zero case following from a prior result of H. Lin. We also show that weak cancellation implies the other properties for extremally rich C*-algebras and that the class of extremally rich C*-algebras with weak cancellation is closed under extensions. Moreover, we consider analogous properties which replace the group K_1(A) with the extremal K-set K_e(A) as well as t
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We develop methods for computing graded K-theory of C*-algebras as defined in terms of Kasparov theory. We establish grad
We initiate the study of real $C^*$-algebras associated to higher-rank graphs $Lambda$, with a focus on their $K$-theory. Following Kasparov and Evans, we identify a spectral sequence which computes the $mathcal{CR}$ $K$-theory of $C^*_{mathbb R} (Lambda, gamma)$ for any involution $gamma$ on $Lambda$, and show that the $E^2$ page of this spectral sequence can be straightforwardly computed from the combinatorial data of the $k$-graph $Lambda$ and the involution $gamma$. We provide a complete description of $K^{CR}(C^*_{mathbb R}(Lambda, gamma))$ for several examples of higher-rank graphs $Lambda$ with involution.
We establish exact sequences in $KK$-theory for graded relative Cuntz-Pimsner algebras associated to nondegenerate $C^*$-correspondences. We use this to calculate the graded $K$-theory and $K$-homology of relative Cuntz-Krieger algebras of directed graphs for gradings induced by ${0,1}$-valued labellings of their edge sets.
Exploiting the graph product structure and results concerning amalgamated free products of C*-algebras we provide an explicit computation of the K-theoretic invariants of right-angled Hecke C*-algebras, including concrete algebraic representants of a basis in K-theory. On the way, we show that these Hecke algebras are KK-equivalent with their undeformed counterparts and satisfy the UCT. Our results are applied to study the isomorphism problem for Hecke C*-algebras, highlighting the limits of K-theoretic classification, both for varying Coxeter type as well as for fixed Coxeter type.
Let M be a smooth Fredholm manifold modeled on a separable infinite-dimensional Euclidean space E with Riemannian metric g. Given an (augmented) Fredholm filtration F of M by finite-dimensional submanifolds (M_n), we associate to the triple (M, g, F) a non-commutative direct limit C*-algebra A(M, g, F) = lim A(M_n) that can play the role of the algebra of functions vanishing at infinity on the non-locally compact space M. The C*-algebra A(E), as constructed by Higson-Kasparov-Trout for their Bott periodicity theorem for infinite dimensional Euclidean spaces, is isomorphic to our construction when M = E. If M has an oriented Spin_q-structure (1 <= q <=infty), then the K-theory of this C*-algebra is the same (with dimension shift) as the topological K-theory of M defined by Mukherjea. Furthermore, there is a Poincare duality isomorphism of this K-theory of M with the compactly supported K-homology of M, just as in the finite-dimensional spin setting.
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