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; to the calculation of the Cuntz semigroup; and to a breakthrough in the C-algebraic analogue of Tarski’s problem determining non-elementary equivalence of the reduced group C-algebras of free groups, contrasting Sela’s results to elementary equivalence of free groups. In the nuclear setting, strict comparison is equivalent to tensorial absorption of the Jiang-Su algebra (see Matsui-Soto 2012).  However, previously there was a severe lack of non-nuclear examples of strict comparison in the setting of reduced group C-algebras. Work in the 1990s of Anderson-Blackadar-Haagerup and Dykema-Rørdam showed that comparison phenomena can occur in free products, but for the free group on two generators strict comparison of the reduced group C-algebra remained 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 a huge family of groups.