*-Doubles and embedding of associative algebras in B(H)
Abstract
We study the *-double functor between the categories of associative and involutive algebras. It is proved that an associative algebra is isomorphic to a subalgebra of a C*-algebra if and only if its *-double is *-isomorphic to a *-subalgebra of a C*-algebra. Some applications in the theory of operator algebras are presented. In particular each operator algebra is shown to be completely boundedly isomorphic to an operator algebra B with the greatest C*-subalgebra consisting of the multiples of the unit and such that each element in B is determined by its module up to a scalar multiple. We also study the maximal subalgebras of an operator algebra A which are mapped into C*-algebras under completely bounded faithful representations of A.
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