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THE PARTIAL-ISOMETRIC CROSSED PRODUCTS BY SEMIGROUPS OF ENDOMORPHISMS AS FULL CORNERS
Published online by Cambridge University Press: 01 April 2014
Abstract
Suppose that ${\Gamma }^{+ } $ is the positive cone of a totally ordered abelian group
$\Gamma $, and
$(A, {\Gamma }^{+ } , \alpha )$ is a system consisting of a
${C}^{\ast } $-algebra
$A$, an action
$\alpha $ of
${\Gamma }^{+ } $ by extendible endomorphisms of
$A$. We prove that the partial-isometric crossed product
$A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{piso} } \hspace{0.167em} {\Gamma }^{+ } $ is a full corner in the subalgebra of
$\L ({\ell }^{2} ({\Gamma }^{+ } , A))$, and that if
$\alpha $ is an action by automorphisms of
$A$, then it is the isometric crossed product
$({B}_{{\Gamma }^{+ } } \otimes A)\hspace{0.167em} {\mathop{\times }\nolimits }^{\mathrm{iso} } \hspace{0.167em} {\Gamma }^{+ } $, which is therefore a full corner in the usual crossed product of system by a group of automorphisms. We use these realizations to identify the ideal of
$A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{piso} } \hspace{0.167em} {\Gamma }^{+ } $ such that the quotient is the isometric crossed product
$A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{iso} } \hspace{0.167em} {\Gamma }^{+ } $.
MSC classification
- Type
- Research Article
- Information
- Copyright
- Copyright ©2013 Australian Mathematical Publishing Association Inc.
References
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