The lattice of C*-covers of an operator algebra

Abstract

In this article, it is shown that the lattice of C ∗ -covers of an operator algebra does not contain enough information to distinguish operator algebras up to completely isometric isomorphism. In addition, four natural equivalences of the lattice of C ∗ -covers are developed and proven to be distinct. The lattice of C ∗ -covers of direct sums and tensor products are studied. Along the way key examples are found of operator algebras, each of which generates exactly n C ∗ -algebras up to ∗-isomorphism, and a simple operator algebra that is not similar to a C ∗ -algebra.