Venue: Department of Mathematics, Lecture Hall I

von Neumann algebras are non commutative analogue of measure spaces. The study of maximal abelian subalgebras (masas) in finite von Neumann algebras is classical to the subject from its birth and is closely tied up with Ergodic theory. Dixmier introduced two types of masas, namely, regular (Cartan) and singular. The philosophies of these two kinds of masas until recently were regarded as being different from each other. After an introduction on the subject, we justify that the existing theories can be unified. Using techniques from Free Probability and playing with suitable amenable groups we exhibit: For each subset S of $\\mathbb{N}$ (could be empty), there exist uncountably many pairwise non conjugate (by automorphism) singular masas in the free group factors for each of which $S\\cup {\\infty\\}$ arises as its Pukanzsky invariant (multiplicity function). If time permits, some other issues related to mixing, coarse bimodules, and Banach’s problem on simple Lebesgue spectrum will be addressed.

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Last updated: 06 Mar 2020