Pimsner algebras and Gysin sequences from principal circle actions

Abstract

A self Morita equivalence over an algebra $B$, given by a $B$-bimodule $E$, is thought of as a line bundle over $B$. The corresponding Pimsner algebra ${\mathcal{O}}_E$ is then the total space algebra of a noncommutative principal circle bundle over $B$. A natural Gysin-like sequence relates the $KK$-theories of ${\mathcal{O}}_E$ and of $B$. Interesting examples come from ${\mathcal{O}}_E$ a quantum lens space over $B$ a quantum weighted projective line (with arbitrary weights). The $KK$-theory of these spaces is explicitly computed and natural generators are exhibited.