Discrete product systems of Hilbert bimodules.
Let \(A\) be a separable \(C^*\)-algebra. A Hilbert bimodule over \(A\) is a right Hilbert \(A\)-module \(X\) together with a \(*\)-homomorphism \(\varphi:A\to {\mathcal L}(X)\) which is used to define a left action of \(A\) on \(X\) via \(a.x:=\varphi(a)x\) for all \(a\in A\) and \(x\in X\). Here \({\mathcal L}(X)\) is the \(C^*\)-algebra of all adjointable operators on \(X\). Applying the idea of Arveson's continuous tensor product systems over \((0,\infty)\) [\textit{W. Arveson}, ``Continuous analogues of Fock space'' (Mem. Am. Math. Soc. 409) (1989; Zbl 0697.46035)], the author defines the notion of a product system of Hilbert bimodules and then a generalized Cuntz--Pimsner algebra \({\mathcal O}_X\). He associates with each twisted semigroup dynamical system \((A,P,\beta,\omega)\) a product system \(X=X(A,P,\beta,\omega)\) whose Cuntz--Pimsner algebra is isomorphic to the twisted semigroup crossed product \(A\rtimes_{\beta,\omega}P\). He also generalizes the concept of twisted crossed product. Among many interesting results, the author characterizes the faithful representations of \(B_P\rtimes_{\tau,X}P\) under specific assumptions by following the strategy used in Section 5 of \textit{N. J. Fowler} and \textit{I. Raeburn} [J. Funct. Anal. 155, 171--204 (1998; Zbl 0914.46054)].
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