A $C^\ast$-Cover Lattice Dichotomy

$C^\ast$-covers of operator algebras, first studied by Arveson about 50 years ago, remain a vibrant area of research. Recently, Adam Humeniuk and Christopher Ramsey studied the lattice of $C^\ast$-covers of operator algebras, focusing on its structure and on whether this lattice uniquely determines an operator algebra up to completely isometric isomorphisms.

In my talk, I will give an introduction to operator algebras and their $C^\ast$-covers, and answer the question of whether there exists nontrivial operator algebras with a one point lattice. Additionally, I will characterize the possible cardinalities of the lattice of $C^\ast$-covers. This is a joint work with Adam Humeniuk and Christopher Ramsey.