Australian Category Seminar

In this talk, I will introduce $M^*$-categories—an abstraction capturing both algebraic and analytic aspects of the theory of Hilbert spaces. Examples include the categories of self-dual Hilbert modules over a W*-algebra and the category of unitary representations of a group. If $\mathcal{C}$ is an $M^*$-category, then each homset $\mathcal{C}(A,A)$ is canonically a monotone complete partially ordered $\ast$-ring and each homset $\mathcal{C}(A,X)$ is canonically an orthogonally complete inner product $\mathcal{C}(A,A)$-module. I will introduce these analytic completeness properties and explain how they arise from categorical limits.