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The partial-isometric crossed products of \(c_{0}\) by the forward and backward shifts. (English) Zbl 1206.46056

Summary: Let \((A, \alpha )\) be a system consisting of a \(C^*\)-algebra \(A\) and an extendible endomorphism \(\alpha \) on \(A\). We consider the partial-isometric crossed product \(A \times _{\alpha } \mathbb N\) generated by a copy of \(A\) and a power partial isometry. We show that, for an extendible \(\alpha \)-invariant ideal \(I\) in \(A\), the quotient \((A \times _{\alpha } \mathbb N)/(I \times _{\alpha } \mathbb N)\) of partial-isometric crossed products is isomorphic to the partial-isometric crossed product \(A/I \times _{\bar {\alpha }} \mathbb N\) of the quotient algebra. Then, we use this to give concrete descriptions of the partial-isometric crossed products of \(c_{0}\) by the forward shift and the backward shift.

MSC:

46L55 Noncommutative dynamical systems
06F15 Ordered groups
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47L60 Algebras of unbounded operators; partial algebras of operators
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