Pure C*-algebras
Hannes Thiel (Chalmers University & University of Gothenburg)
15:30-16:30，July 31th，2024 Shanghai Institute for Mathematics and Interdisciplinary Sciences
Abstract:
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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