Pure C*-algebras

Hannes Thiel  (Chalmers University & University of Gothenburg)

15:30-16:30£¨July 31th£¨2024    Shanghai Institute for Mathematics and Interdisciplinary Sciences


In his seminal investigation of Z-stability for simple, nuclear C*-algebras, Winter introduced the notion of (m,n)-pureness with m and n quantifying comparison and divisibility properties in the Cuntz semigroup, and he showed that every simple C*-algebra that has locally finite nuclear dimension and that is (m,n)-pure for some m and n is Z-stable. Combined with the result of Roerdam that every Z-stable C*-algebra is pure (that is, (0,0)-pure, which means that its Cuntz semigroup has the strongest comparison and divisibility properties), this provides a situation where (m,n)-pureness implies pureness. In a recent paper with R. Antoine, F. Perera and L. Robert, we removed the assumption of locally finite nuclear dimension and showed that every simple, (m,n)-pure C*-algebra is pure. In this work we generalize the result even further by showing that (m,n)-pureness implies pureness in general. As an application we show that every C*-algebra with the Global Glimm Property and finite nuclear dimension is pure. This is joint work with R. Antoine, F. Perera, and E. Vilalta

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