Noncommutative Geometry Seminar

Date: October 8, 2021

Time: 2:00PM - 3:00PM

Location: ZOOm

Speaker: Benjamin Steinberg, City College of New York and CUNY Graduate Center

Title: Cartan pairs of algebras

Abstract: In the seventies, Feldman and Moore studied Cartan pairs of von Neumann algebras. These pairs consist of an algebra A and a maximal commutative subalgebra B with B sitting “nicely” inside of A. They showed that all such pairs of algebras come from twisted groupoid algebras of quite special groupoids (in the measure theoretic category) and their commutative subalgebras of functions on the unit space, and that moreover the groupoid and twist were uniquely determined (up to equivalence). Kumjian and Renault developed the C*-algebra theory of Cartan pairs. Again, in this setting all Cartan pairs arise as twisted groupoid algebras, this time of effective etale groupoids, and again the groupoid and twist are unique (up to equivalence). In recent years, Matsumoto and Matui exploited that for directed graphs satisfying Condition (L), the corresponding graph C*-algebra and its commutative subalgebra of functions on the path space of the graph form a Cartan pair to give C*-algebraic characterizations of continuous orbit equivalence and flow equivalence of shifts of finite type. The key point was translating these dynamical conditions into groupoid language. Since the Leavitt path algebra associated to a graph is the “Steinberg” algebra of the same groupoid, this led people to wonder about whether these dynamical invariants can be read off the pair consisting of the Leavitt path algebra and its subalgebra of locally constant maps on the path space. The answer is yes and it turns out in the algebraic setting, one doesn’t even need Condition (L). Initially work was focused on recovering an ample groupoid from the pair consisting of its “Steinberg” algebra and the algebra of locally constant functions on the unit space. But no abstract theory of Cartan pairs existed and twists had not yet been considered. Our work develops the complete picture. This is joint work with Becky Armstrong, Gilles G. de Castro, Lisa Orloff Clark, Kristin Courtney, Ying-Fen Lin, Kathryn McCormick, Jacqui Ramagge and Aidan Sims

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