Worldwide Operator Algebras Colloquium 

Title: Going with the flow: from type III factors to flows on II1 factors

Abstract: We present a complete classification of continuous flows on the hyperfinite II1 factor up to cocycle conjugacy.

We begin with the history of this classification problem, from the work of Connes and Takesaki in the 1970s, through Kawahigashi’s contributions in the 1980s, to the more recent work of Matsuda–Tomatsu on Rokhlin flows.

We then explain how we came to work on this problem and how we adapted ideas from the theory of type III factors to tackle it. In particular, our bicentralizer approach to flows on II1 factors is directly inspired by progress over the past decade on Connes’ bicentralizer problem for type III1 factors.

Finally, our study of the relative commutant of flows on II1 factors revealed a new resonance mechanism that led us to a complete solution of Connes’ bicentralizer problem.

Title: Wright's Normality Conjecture Via Set Theoretic Forcing 

Abstract: AW*-algebras are C*-algebras whose projection structures behave like those of von Neumann algebras, but which need not arise from weak operator closures. Two open problems have been central to the study of AW*-algebras for decades: the monotone completeness conjecture and the Kaplansky conjecture. In 1980, Wright introduced the notion of a normal AW*-algebra as a weakening of monotone completeness and conjectured that every AW*-algebra is normal. He proved this in the finite case, while Saitô and Wright later proved that every AW*-factor is normal. We discuss our recent positive resolution of the normality conjecture using a set-theoretic forcing technique called Boolean-valued models. Given an AW*-algebra A, this technique allows us to produce a new model of ZFC in which A "looks like" a factor. Since Saitô and Wright showed that all AW*-factors in any model of ZFC are normal, A is normal in this new model. We then establish transfer principles which allow us to conclude that A must have been normal in our original model. We also discuss applications of these techniques to the harder monotone completeness conjecture. No background in set theory/logic will be assumed.

Title: Nonisomorphism of q-Gaussian algebras in infinite variables

Abstract: The q-Gaussian von Neumann algebras, introduced by Bożejko and Speicher, form a family of II_1 factors that deform Voiculescu’s free Gaussian factors. For finitely many variables, Guionnet and Shlyakhtenko showed that small deformations do not change the isomorphism class. We focus on the case of infinitely many variables.

Previous work of Borst, Caspers, Klisse, and Wasilewski showed that the corresponding C*-algebras A_q and A_0 are not isomorphic for q nonzero, using the failure of the Akemann-Ostrand property (AO). Caspers later established the corresponding result for the von Neumann algebras M_q. We show more generally that M_q is isomorphic to M_r if and only if q=r, which also implies the same result for A_q and A_r.

The proof is based on a closer study of the kernel of the Akemann-Ostrand map. We show that this kernel has a natural filtration by ideals. In particular, its least noncompact ideal contains the projection onto the first chaos of the q-Fock space. This property allows us to recover the parameter q intrinsically from the von Neumann algebra and therefore shows the nonisomorphism.

Title: Strict comparison and selflessness.

Abstract:  I will discuss the following theorem and the ideas surrounding it:

An infinite-dimensional unital C*-algebra is selfless if and only if it is simple

and either purely infinite or monotracial with strict comparison relative to its

unique trace.

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