Diagonal dimension and intermediate sub-C*-algebras

Abstract: The study of C*-diagonals goes back to Kumjian’s work in the 1980s (building on Feldman and Moore’s analysis of Cartan subalgebras in von Neumann algebras and Renault’s work on Cartan sub-C*-algebras) and has lately been receiving renewed interest due to connections with classification theory of C*-algebras. Recently K. Li, H. Liao and W. Winter developed the theory of diagonal dimension for C*-pairs: inspired by Winter’s nuclear dimension (which captures a noncommutative analogue of covering dimension and plays a central role in classification theory), diagonal dimension records the fine structure of a C*-diagonal’s intrinsic dynamics, has natural permanence properties, and connects with established dimension theories in topological dynamics and groupoid theory.

In this talk, we discuss a relevant permanence property: finite diagonal dimension passes to intermediate subalgebras. Consequently, all such intermediate sub-C*-algebras have finite nuclear dimension in their own right, in contrast with the behavior of an arbitrary sub-C*-algebra that in general need not even be nuclear. We discuss the estimate on the diagonal dimension of intermediate subalgebras, some of the key ideas underlying its proof, and some related open questions.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

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