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Partial-isometric crossed products by semigroups of endomorphisms. (English) Zbl 1078.46047

Let \(\Gamma\) be a totally ordered Abelian group with positive cone \(\Gamma^+\) and consider an action \(\alpha : \Gamma^+ \rightarrow \text{End} A\) of \(\Gamma^+\) by endomorphisms of a \(C^*\)-algebra \(A\). In the paper under review, covariant representations \((\pi ,V)\) of the system \((A,\Gamma^+ ,\alpha )\) in which the endomorphisms \(\alpha_s\) are implemented by partial isometries \(V_s\), and the corresponding crossed-product \(C^*\)-algebra \(A\times_{\alpha}\Gamma^+\) which is generated by a universal covariant representation, are studied.
These partial-isometric crossed products have some interesting properties. For example, every system \((A,\Gamma^+ ,\alpha )\) admits a covariant partial-isometric representation \((\pi ,V)\) in which \(\pi\) is faithful; hence the partial-isometric crossed product contains full information about the system.
The main results of the paper are:
1. Partial-isometric crossed products provide a rich and tractable family of Toeplitz algebras for product systems of Hilbert bimodules.
2. Detailed structure theorems for actions by forward and backward shifts are proven.

MSC:

46L55 Noncommutative dynamical systems
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