Title: Matrix Embeddings of Directly Infinite Algebras
Speaker: Daniel Bossaller - John Caroll University, University Heights
Abstract: A 1993 result of Goodearl, Menal and Moncasi showed that every countable-dimensional algebra over a field may be embedded as an algebra in the algebra of row and finite matrices over a field $K$, $B(K)$. In many cases a directly infinite algebra, that is, an algebra where there exist elements $x$ and $y$ such that $xy =1$ but $yx \neq 1$, will have an ideal $I$ which is isomorphic to $M_\infty(K)$, the algebra of infinite matrices with only finitely many nonzero entries. When this ideal is ubiquitous in the algebra, one can then define an alternate embedding into the row and column finite matrices. This embedding, though, is not unique, and the set of possible embeddings of any single algebra may be vastly different. This talk will develop an equivalence relation on these embeddings, based on the theory of extensions of $C^*$-algebras, and use this equivalence to give a classification of embeddings of the Toeplitz-Jacobson algebra $\langle x, y \st xy = 1\rangle$ into $B(K)$.
