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Department of Mathematics,
University of California San Diego


Math 243, Functional Analysis

Dr. Brent Nelson

Michigan State University

Von Neumann dimension for faithful normal strictly semifinite weights


The notion of von Neumann dimension for a tracial von Neumann algebra $(M,\tau)$ has been used extensively throughout the theory, particularly in defining numerical invariants from Jones' index of a subfactor to Connes and Shlyakhtenko's $\ell^2$-Betti numbers of von Neumann algebras. The latter relies on work of Lück showing that Murray and von Neumann's original definition could be extended to purely algebraic $M$-modules, and more recently Petersen further extended von Neumann dimension to pairs $(M,\tau)$ where $\tau$ is a faithful normal semifinite tracial weight. In this talk, I will introduce a yet further extension of this theory to pairs $(M,\varphi)$ where $\varphi$ is a faithful normal strictly semifinite weight. Here strict semifiniteness means the restriction of $\varphi$ to the centralizer subalgebra $M^\varphi \subset M$ is still semifinite (note this condition is automatic for faithful normal states), and by work of Takesaki this is equivalent to the existence of a $\varphi$-invariant faithful normal conditional expectation $\mathcal{E}_\varphi\colon M \to M^\varphi$. Consequently, one can consider the Jones basic construction for the inclusion $M^\varphi \subset M$, and this is the key ingredient in our definition of von Neumann dimension for the pair $(M,\varphi)$. I will discuss properties of this dimension and how it can be used to recover the index for certain inclusions of factors. This is based on joint work with Aldo Garcia Guinto and Matthew Lorentz.

Host: Priyanga Ganesan

February 27, 2024

11:00 AM

AP&M 7218 and Zoom (meeting ID:  94246284235)