The Lattice of C*-covers of an operator algebra

In this paper 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 an operator algebra that generates exactly n C*-algebras up to -isomorphism and a simple operator algebra that is not similar to a C-algebra.