##### Department of Mathematics,

University of California San Diego

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### Algebra Seminar

## Manny Reyes

#### Bowdoin College

## Diagonalizable algebras of operators on infinite-dimensional vector spaces

##### Abstract:

Given a vector space $V$ over a field $K$, let $\mathrm{End}(V)$ denote the algebra of linear endomorphisms of $V$. If $V$ is finite-dimensional, then it is well-known that the diagonalizable subalgebras of $\mathrm{End}(V)$ are characterized by their internal algebraic structure: they are the subalgebras isomorphic to $K^n$ for some natural number $n$. In case $V$ is infinite dimensional, the diagonalizable subalgebras of $\mathrm{End}(V)$ cannot be characterized purely by their internal algebraic structure: one can find diagonalizable and non-diagonalizable subalgebras that are isomorphic. I will explain how to characterize the diagonalizable subalgebras of $\mathrm{End}(V)$ as \emph{topological} algebras, using a natural topology inherited from $\mathrm{End}(V)$. I also hope to show how this characterization relates to an infinite-dimensional Wedderburn-Artin theorem that characterizes ``topologically semisimple'' algebras.

Host: Dan Rogalski

### April 13, 2015

### 3:00 PM

### AP&M 7218

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