A Projective C*-Algebra Related to K-Theory

TL;DR

This paper investigates the properties of the C*-algebra qC, focusing on its role in K-theory, the exponential map, and semiprojectivity, providing new insights and proofs in the theory of projective C*-algebras.

Contribution

It introduces a universal projective C*-algebra P related to qC, offers a shorter proof of qC's semiprojectivity, and explores the exponential map in K-theory.

Findings

Realization of boundary maps as unitaries in I

Universal projective C*-algebra P for qC relations

Shorter proof of qC's semiprojectivity

Abstract

The C*-algebra qC is the smallest of the C*-algebras qA introduced by Cuntz in the context of KK-theory. An important property of qC is the natural isomorphism of K0 of D with classes of homomorphism from qC to matrix algebras over D. Our main result concerns the exponential (boundary) map from K0 of a quotient B to K1 of an ideal I. We show if a K0 element is realized as a homomorphism from qC to B then its boundary is realized as a unitary in the unitization of I. The picture we obtain of the exponential map is based on a projective C*-algebra P that is universal for a set of relations slightly weaker than the relations that define qC. A new, shorter proof of the semiprojectivity of qC is described. Smoothing questions related the relations for qC are addressed.