Groups and Operator Algebras Seminar

Speaker: Gregory Patchell (UC San Diego)

Title: Strict Comparison in C*-algebras

Abstract: One of the most fundamental ways to compare matrices is via their rank. For two matrices X and Y, rank(X) is less than or equal to rank(Y) if and only if there are matrices S and T such that X = SYT. The rank can be generalized to C*-algebras using dimension functions and the latter algebraic condition can be generalized to a condition known as Cuntz subequivalence. C*-algebras for which the dimension functions recover Cuntz subequivalence are said to have strict comparison. Strict comparison is known to have applications to classification of *-homomorphisms of C*-algebras, including existence and uniqueness of embeddings of the Jiang-Su algebra; and to the calculation of the Cuntz semigroup. In the nuclear setting, strict comparison is equivalent to tensorial absorption of the Jiang-Su algebra (see Matsui-Sato 2012). However, previously there was a severe lack of non-nuclear examples of strict comparison in the setting of reduced group C*-algebras. In 1998 Dykema-Rørdam showed that infinite reduced free products have strict comparison, but even for the free group on two generators strict comparison of the reduced group C*-algebra was a long-standing open problem. In our work (joint with Tattwamasi Amrutam, David Gao, and Srivatsav Kunnawalkam Elayavalli) we show that the reduced group C*-algebra of the free group on two generators has strict comparison. Our methods are very general and lead to proving strict comparison (and the stronger property of selflessness, due to Robert) for all acylindrically hyperbolic groups with rapid decay.