Ben Stanley
The Ohio University-Ohio State University ring theory seminar series presents Ben Stanley discussing “Gauging the amenability of bases of countable dimensional algebras” on Friday, Oct. 19, at 4:45 p.m. in Cockins Hall 240, OSU-Columbus.
Stanley is a Teaching Assistant and graduate student in Mathematics at Ohio University.
Abstract: Let $A$ be a countably infinite dimensional $K$-algebra, where $K$ is a field and let $\mathcal{B}$ be a basis for $A$. In recent publications, the basis $\mathcal{B}$ has been called amenable when $K^\mathcal{B}$ (the direct product indexed by $\mathcal{B}$ of copies of the field $K$) can be made into an $A$-module in a natural way. More precisely, $\mathcal{B}$ is (left) amenable if for every $ a \in A$ the matrix $[l_a]_{\mathcal{B}}$ that represents left multiplication by $a$ with respect to $\mathcal{B}$ is row and column finite. More generally, for any basis $\mathcal{C}$, let $dom(\mathcal{C}) = \{ a \in A|\text { } [l_a]_{\mathcal{C}} \text { is row and column finite}\}$. Then $dom(\mathcal{C})$ is a subalgebra of $A$ called its {\it amenability domain}. The collection of amenability domains of bases of the algebra $A$ is said to be its {\it amenability profile}.
We study amenability profiles and determine that, in addition to $A$ itself, it also always contains $F$. A basis with amenability domain $F$ is said to be {\it contrarian}. We will explore the connections between the amenability profile of an algebra and the lattice of all of its subalgebras. An algebra will be said to have {\it no discernment} if its profile equals $\{ F, A \}$. At the opposite end, the algebra has {\it full discernment} if its profile equals the lattice of all subalgebras of $A$. We will consider the feasibility of both these possibilities in the context of the algebra $F[x]$ and of the {\it graph algebras} recently introduced by Pınar Aydoğdu, Sergio Lopez-Permouth and Rebin Muhammad. \\ \\
This is a preliminary report on joint work in progress with Sergio R. Lopez-Permouth.
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