The Shilov boundary of an operator space  and the characterization theorems
Abstract
We study operator spaces, operator algebras, and operator modules, from the point of view of the `noncommutative Shilov boundary'. In this attempt to utilize some `noncommutative Choquet theory', we find that Hilbert C$^*$modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We introduce certain multiplier operator algebras and C$^*$algebras of an operator space, which generalize the algebras of adjointable operators on a C$^*$module, and the `imprimitivity C$^*$algebra'. It also generalizes a classical Banach space notion. This multiplier algebra plays a key role here. As applications of this perspective, we unify, and strengthen several theorems characterizing operator algebras and modules, in a way that seems to give more information than other current proofs. We also include some general notes on the `commutative case' of some of the topics we discuss, coming in part from joint work with Christian Le Merdy, about `function modules'.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 1999
 arXiv:
 arXiv:math/9906083
 Bibcode:
 1999math......6083B
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Functional Analysis
 EPrint:
 This is the final revised version