Operator algebras with contractive approximate identities

Details Operator algebras with contractive approximate identities Operator algebras with contractive approximate identities

2010-11-16

arxiv.org/abs/1011.3797v2

Mathematics

Operator Algebras

Revision of 29 pages, some conjectures in paper solved

Scientific paper

We give several applications of a recent theorem of the second author, which solved a conjecture of the first author with Hay and Neal, concerning contractive approximate identities; and another of Hay from the theory of noncommutative peak sets, thereby putting the latter theory on a much firmer foundation. From this theorem it emerges there is a surprising amount of positivity present in any operator algebras with contractive approximate identity. We exploit this to generalize several results previously available only for $C^*$-algebras, and we give many other applications.

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