Operator algebras seminar

Speaker: Gabor Szabo (KU Leuven)

Title: Property (SI) for C*-dynamical systems

Abstract: Starting around 2009, Matui and Sato initiated a highly
successful series of collaborations centered on the concept of property
(SI), which was earlier defined by Sato. In this context "SI" means
"small isometries", and the main thrust of the property is that a
tracial comparison property can often not only be expected for a given
C*-algebra, but even its central sequences. Their work is mostly known
for their breakthroughs in the Toms-Winter conjecture, but the original
motivation and framework was about applications to C*-dynamics rather
than the structure of C*-algebras. Indeed, Matui and Sato also proposed
a very fruitful approach to verify that certain kinds of dynamical
systems on simple C*-algebras yielded Z-stable crossed products, or
rather that the dynamical systems were themselves Z-stable in an
equivariant sense. Until recently the main limitation of property (SI)
was that the framework only made sense if the underlying C*-algebra is
unital or at least algebraically simple, which is a hard restriction in
the context of C*-dynamics. In this talk I will demonstrate how to
extend the framework of (equivariant) property (SI) to C*-algebras that
can have unbounded traces. I will then discuss why it automatically
holds whenever one might expect it, and how to apply it to make progress
on the Z-stability problem for C*-dynamical systems, or (if time
permits) other interesting structural questions.