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SUMMARY:New Examples of Strict Comparison in C*-Algebras - Gregory Patchel
 l (University of Oxford)
DTSTART:20251006T143000Z
DTEND:20251006T153000Z
UID:TALK236221@talks.cam.ac.uk
DESCRIPTION:One of the most fundamental ways to compare matrices is via th
 eir 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 sube
 quivalence. C*-algebras for which the dimension functions recover Cuntz su
 bequivalence are said to have strict comparison. Strict comparison is know
 n to have applications to classification of *-homomorphisms of C*-algebras
 \, including existence and uniqueness of embeddings of the Jiang-Su algebr
 a\; to the calculation of the Cuntz semigroup\; and to a breakthrough in t
 he C*-algebraic analogue of Tarski's problem determining non-elementary eq
 uivalence of the reduced group C*-algebras of free groups\, contrasting Se
 la's results to elementary equivalence of free groups. In the nuclear sett
 ing\, strict comparison is equivalent to tensorial absorption of the Jiang
 -Su algebra (see Matsui-Soto 2012).&nbsp\; 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-Haag
 erup and Dykema-R&oslash\;rdam showed that comparison phenomena can occur 
 in free products\, but for the free group on two generators strict compari
 son of the reduced group C*-algebra remained a long-standing open problem.
  In our work (joint with Tattwamasi Amrutam\, David Gao\, and Srivatsav Ku
 nnawalkam Elayavalli) we show that the reduced group C*-algebra of the fre
 e group on two generators has strict comparison. Our methods are very gene
 ral and lead to proving strict comparison (and the stronger property of se
 lflessness\, due to Robert) for a huge family of groups.
LOCATION:Seminar Room 1\, Newton Institute
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