On approximation properties of Pimsner algebras and crossed products by Hilbert bimodules

Abstract

Let X be a Hilbert bimodule over a C*-algebra A and $O_X=A{⋊}_X\mathbb{Z}$. Using a finite section method we construct a sequence of completely positive contractions factoring through matrix algebras over A which act on $s_{\xi }{s}_{\eta }^{\ast }$ as Schur multipliers converging to the identity. This shows immediately that for a finitely generated X the algebra $O_X$ inherits any standard approximation property such as nuclearity, exactness, CBAP or OAP from A. We generalise this to certain general Pimsner algebras by proving semi-splitness of the Toeplitz extension under certain conditions and discuss some examples.

arXiv comment11 pages, v2 has a modified title, abstract and introduction. The paper will appear in the Rocky Mountain Journal of Mathematics