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# Journal of Operator Theory

Volume 64, Issue 1, Summer 2010  pp. 3-17.

Morita type equivalences and reflexive algebras

Authors G.K. Eleftherakis
Author institution: Department of Mathematics, University of Athens, Panepistimiopolis, 15784, Greece

Summary:  Two unital dual operator algebras $\cl{A}, \cl{B}$ are called $\Delta$-equivalent if there exists an equivalence functor $\cl{F}: \, _{\cl{A}}\mathfrak{M}\rightarrow \, _{\cl{B}}\mathfrak{M}$ which extends" to a $*$-functor implementing an equivalence between the categories $_{\cl{A}}\mathfrak{DM}$ and $_{\cl{B}}\mathfrak{DM}.$ Here $_{\cl{A}}\mathfrak{M}$ denotes the category of normal representations of $\cl{A}$ and $_{\cl{A}}\mathfrak{DM}$ denotes the category with the same objects as $_{\cl{A}}\mathfrak{M}$ and $\Delta (\cl{A})$-module maps as morphisms ($\Delta (\cl{A})=\cl{A}\cap \cl{A}^*$). We prove that any such functor maps completely isometric representations to completely isometric representations, respects" the lattices of the algebras and maps reflexive algebras to reflexive algebras. We present applications to the class of CSL algebras.

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