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

Adam Skalski·Joachim Zacharias
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Abstract

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 $s_ξ s_η^*$ 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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Adam Skalski, Joachim Zacharias. 2008-04-03. On approximation properties of Pimsner algebras and crossed products by Hilbert bimodules. https://arxiv.org/abs/math/0607628

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