Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: E$_0$-semigroups and product systems of W$^*$-bimodules   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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The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: solv-int
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The categories of this paper are: astro-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: math-ph
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The categories of this paper are: stat
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ex
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The categories of this paper are: supr-con
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin