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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