Title: On the extent of amenability of bases of infinite dimensional algebras
Speaker: Sergio R. López-Permouth, Ohio University
Abstract: Let $F$ be a field and $A$ an $F$-algebra . For $r \in A$, let $l_r$ denote the left multiplication map $l_r: A \rightarrow A$.
A basis $B$ for $A$ is amenable if the matrix representation with respect to $B$ of every $l_r$ is row finite. An algebra that has an amenable basis is said to be an amenable algebra; countable dimension algebras are amenable. For every countable basis $B= \{ b_i\}_{i=1}^{\infty}$ of $A$, there exists a topology $\tau_{B}$ on $F^{(B)}$ such that $B$ is amenable if and only if, for all $r \in A$, the sequence $\{[rb_i]_{B}\}_{i=1}^{\infty}$ of representations of $\{rb_i\}_{i=1}^{\infty}$ with respect to $B$ converges to $0$ in $\tau_{B}$.
Given an algebra $A$, and a basis $B$ for $A$, the amenability domain of $B$ is the subalgebra of $A$ consisting of all elements $r \in A$ with row finite matrix representation with respect to $B$. A basis having amenability domain $F$ is said to be contrarian; we show that, under some mild additional hypotheses, amenable algebras always have contrarian bases.
The collection of amenability domains of bases of an algebra $A$ is called the amenability profile of $A$. The profile is a measurement of the diversity of the bases of $A$ and serves to sort them according to the extent of their amenability. We consider when profiles are minimal (consisting of only $F$ and $A$) and when they are maximal (consisting of all subalgebras of $A$); the former algebras are said to {\it lack discernment} and the latter to be {\it full rank.} We provide an example of a graph magma algebra without discernment. We show that $F[x]$ does not lack discernment and that graph magma algebras are never full rank.\\
This talk relates to ongoing collaborations with many coauthors including Al-Essa, Ayd\u{o}gdu, D\'iaz Boils, Muhammad, Muthana, and Stanley.