Title: Gauging the amenability of bases of countable dimensional algebras
Speaker: Ben Stanley (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 $dom(\mathcal{C}) = \{ a \in A|\text { } [l_a]_{\mathcal{C}}$ is row and column finite$\}$. Then $dom(\mathcal{C})$ is a subalgebra of $A$ called its amenability domain.
The collection of amenability domains of bases of the algebra $A$ is said to be its 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 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 no discernment if its profile equals $\{ F, A \}$. At the opposite end, the algebra has 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 graph algebras recently introduced by Ayduogdu, López-Permouth and Muhammad.
This is a preliminary report on joint work in progress with Sergio R. L\'opez-Permouth.
