E$_0$-semigroups and product systems of W$^*$-bimodules

Yusuke Sawada

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arXiv · 1904.09454

E$_0$-semigroups and product systems of W$^*$-bimodules

Yusuke Sawada

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Abstract

Product systems have been originally introduced to classify E$_0$-semigroups on type I factors by Arveson. We develop the classification theory of E$_0$-semigroups on a general von Neumann algebra and the dilation theory of CP$_0$-semigroups in terms of W$^*$-bimodules. For this, we provide a notion of product system of W$^*$-bimodules. This is a W$^*$-bimodule version of Arveson's and Bhat-Skeide's product systems. There exists a one-to-one correspondence between CP$_0$-semigroups and units of product systems of W$^*$-bimodules. The correspondence implies a construction of a dilation of a given CP$_0$-semigroup, a classification of E$_0$-semigroups on a von Neumann algebra up to cocycle equivalence and a relationship between Bhat-Skeide's and Muhly-Solel's constructions of minimal dilations of CP$_0$-semigroups.

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Yusuke Sawada. 2019-04-20. E$_0$-semigroups and product systems of W$^*$-bimodules. https://arxiv.org/abs/1904.09454

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