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

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

2006-07-25

arxiv.org/abs/math/0607628v2

Mathematics

Operator Algebras

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

Scientific paper

Let X be a Hilbert bimodule over a C*-algebra A and $O_X= A \rtimes_X \Z$. Using a finite section method we construct a sequence of completely positive contractions factoring through matrix algebras over A which act on $s_{\xi} s_{\eta}^*$ 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.

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