Iterating the Pimsner construction

Details Iterating the Pimsner construction Iterating the Pimsner construction

2007-07-11

arxiv.org/abs/0707.1710v1

Mathematics

Operator Algebras

Scientific paper

For $A$ a $C^*$-algebra, $E_1, E_2$ two Hilbert bimodules over $A$, and a fixed isomorphism $\chi : E_1\otimes_AE_2\to E_2\otimes_AE_1$, we consider the problem of computing the $K$-theory of the Cuntz-Pimsner algebra ${\mathcal O}_{E_2\otimes_A{\mathcal O}_{E_1}}$ obtained by extending the scalars and by iterating the Pimsner construction. The motivating examples are a commutative diagram of Douglas and Howe for the Toeplitz operators on the quarter plane, and the Toeplitz extensions associated by Pimsner and Voiculescu to compute the $K$-theory of a crossed product. The applications are for Hilbert bimodules arising from rank two graphs and from commuting endomorphisms of abelian $C^*$-algebras.

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