Title: A Topological Characterization of Amenability and Congeniality of Bases of Infinite Dimensional Algebras
Speaker: Benjamin Stanley (Ohio University)
Abstract: The related notions of amenable bases for infinite dimensional algebras and their consequent (so called) Basic modules were introduced in [2], a paper that followed the doctoral dissertation [1] .
Let $A$ be an infinite dimensional $K$-algebra, where $K$ is a field and let $\mathcal{B}$ be a basis for $A$. The situation when $\mathcal{B}$ is such that $K^{\mathcal{B}}$ (the direct product indexed by $\mathcal{B}$ of copies of the field $K$) can be made into an $A$-module in a natural way has been a topic of recent interest. Such a basis is called an amenable basis. Furthermore, while exploring the question of whether two amenable bases yield isomorphic modules, a relation among bases named congeniality was also introduced. Both ideas, amenability and congeniality translate into the requirement that the (infinite) matrices representing certain linear operators with respect to the bases in question be row finite. This seems like a reasonable notion of {\it naturality} since it translates into summability of the columns in the sense of formal calculus. On the other hand, anyone who spends time thinking about these notions is likely to be left with the impression that some kind of convergence is present. Infinite linear combinations can conceivably be considered as limits of some sort for sequences of finite linear combinations. The purpose of this presentation is to respond to that intuition by formalizing it in carefully crafted topological considerations. We characterize amenability and congeniality in terms of the uniform continuity of certain maps with respect to metrics that generate product topologies on spaces of the form $K^{\mathcal{B}}$, where $\mathcal{B}$ is a chosen basis and the copies of $K$ are equipped with the discrete topology . This is a report on joint work with Daniel Bossaller, Sergio López-Permouth and Rebin Muhammad.
- L.M. Al-Essa (2015). Modules over Infinite Dimensional Algebras (Dissertation)
- L.M. Al-Essa, S. López-Permouth, N.M. Muthana, Modules over infinite-dimensional algebras, Linear and Multilinear Algebra, (2017). Available from: http://dx.doi.org/10.1080/03081087.2017.1301365