Why MIP* = RE implies not-CEP and Blackadar-Kirchberg's MF problem

Sam Kim

25-Aug-2020, 13:00-14:00 (4 years ago)

Abstract: In this expository talk, we explain a direct route to why the results of Ji, Natarajan, Vidick, Wright, and Yuen give us a II$_1$-factor that cannot embed into a tracial ultrapower of the separable hyperfinite II$_1$-factor $\mathcal{R}$. More specifically, we define a class of finitely generated unital C*-algebras $C^*(\mathcal{G})$ for a fixed parameter $\mathcal{G}$ with the following property: there exist extremal traces $\tau$ on $C^*(\mathcal{G})$ such that the II$_1$-factor $\mathcal{M}_\tau$ generated by $C^*(\mathcal{G})$ in the GNS representation of $\tau$ has the property that there cannot exist a unital *-homomorphism from $\mathcal{M}_\tau$ into a tracial ultrapower of $\mathcal{R}$. We describe ways in which convex geometry over $\mathbb{R}^n$ will give us such parameters $\mathcal{G}$ and extremal traces $\tau$. As a consequence of our construction, we have a separable counter-example of Blackadar-Kirchberg's MF problem, which asks whether every stably finite C*-algebra embeds into a norm ultrapower of the UHF algebra $\mathcal{Q}$. Questions related to the refinement of both the MF conjecture and the refutation of CEP are raised at the end.

functional analysisgroup theoryoperator algebrasquantum algebra

Audience: researchers in the topic


Groups, Operators, and Banach Algebras Webinar

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Organizers: Jared White*, Ulrik Enstad, Bence Horvath
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