*-Doubles and embedding of associative algebras in B(H)

Details *-Doubles and embedding of associative algebras in B(H) *-Doubles and embedding of associative algebras in B(H)

2007-11-18

arxiv.org/abs/0711.2802v2

Indiana University Math. J. 57 (2008) No. 7, pp. 3443-3462

Mathematics

Operator Algebras

Scientific paper

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 $C\sp*$-algebra if and only if its *-double is *-isomorphic to a *-subalgebra of a $C\sp*$-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 $C\sp*$-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 $C\sp*$-algebras under completely bounded faithful representations of $A$.

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