Page 185 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 185
OPERATOR ALGEBRAS (MS-14)
is that the boundaries can be interpreted as universal objects measuring nonamenability of a
(quantum) group G. I’ll explain how such universal properties can be used to identify non-
commutative Poisson and Furstenberg-Hamana boundaries in some cases, leading to quantum
analogues of classical results of Furstenberg and Moore. (Joint work with Erik Habbestad and
Lucas Hataishi.)
Property (T) for automorphism groups of free groups
Piotr Nowak, pnowak@impan.pl
Institute of Mathematics of the Polish Academy of Sciences, Poland
Kazhdan’s property (T ) is a powerful rigidity property of locally compact groups. It has many
applications, including constructions of expander graphs and of counterexamples to certain ver-
sions of the Baum-Connes conjecture. I will describe a new approach to proving property (T )
that uses optimization methods in the form of semidefinite programming. I will present the
recent proof of property (T ) for Aut(Fn), the automorphism group of the free group Fn on n
generators, for n ≥ 5. These results were obtained jointly with Marek Kaluba, Dawid Kielak
and Narutaka Ozawa.
Higher Kazhdan projections and the Baum-Connes conjecture
Sanaz Pooya, spooya@impan.pl
IMPAN, Poland
Coauthors: Piotr Nowak, Kang Li
The Baum-Connes conjecture, if it holds for a certain group, provides topological tools to com-
pute the K-theory of its reduced group C*-algebra. This conjecture has been confirmed for large
classes of groups, such as amenable groups, but also for some Kazhdan’s property (T) groups.
Property (T) and its strengthening are driving forces in the search for potential counterexamples
to the conjecture. Having property (T) for a group is characterised by the existence of a certain
projection in the universal group C*-algebra of the group, known as the Kazhdan projection.
It is this projection and its analogues in other completions of the group ring, which obstruct
known methods of proof for the Baum-Connes conjecture. In this talk, I will introduce a gen-
eralisation of Kazhdan projections. Employing these projections we provide a link between
surjectivity of the Baum-Connes map and the 2-Betti numbers of the group. A similar relation
can be obtained in the context of the coarse Baum-Connes conjecture. This is based on joint
work with Kang Li and Piotr Nowak.
Irreducible inclusions of simple C*-algebras
Mikael Rordam, rordam@math.ku.dk
University of Copenhagen, Denmark
The literature contains a number interesting examples of inclusions of simple C*-algebras, typ-
ically arising from dynamical systems, with the property that all intermediate C*-algebras are
also simple. One can argue that this property of an inclusion of C*-algebras is the natural
183
is that the boundaries can be interpreted as universal objects measuring nonamenability of a
(quantum) group G. I’ll explain how such universal properties can be used to identify non-
commutative Poisson and Furstenberg-Hamana boundaries in some cases, leading to quantum
analogues of classical results of Furstenberg and Moore. (Joint work with Erik Habbestad and
Lucas Hataishi.)
Property (T) for automorphism groups of free groups
Piotr Nowak, pnowak@impan.pl
Institute of Mathematics of the Polish Academy of Sciences, Poland
Kazhdan’s property (T ) is a powerful rigidity property of locally compact groups. It has many
applications, including constructions of expander graphs and of counterexamples to certain ver-
sions of the Baum-Connes conjecture. I will describe a new approach to proving property (T )
that uses optimization methods in the form of semidefinite programming. I will present the
recent proof of property (T ) for Aut(Fn), the automorphism group of the free group Fn on n
generators, for n ≥ 5. These results were obtained jointly with Marek Kaluba, Dawid Kielak
and Narutaka Ozawa.
Higher Kazhdan projections and the Baum-Connes conjecture
Sanaz Pooya, spooya@impan.pl
IMPAN, Poland
Coauthors: Piotr Nowak, Kang Li
The Baum-Connes conjecture, if it holds for a certain group, provides topological tools to com-
pute the K-theory of its reduced group C*-algebra. This conjecture has been confirmed for large
classes of groups, such as amenable groups, but also for some Kazhdan’s property (T) groups.
Property (T) and its strengthening are driving forces in the search for potential counterexamples
to the conjecture. Having property (T) for a group is characterised by the existence of a certain
projection in the universal group C*-algebra of the group, known as the Kazhdan projection.
It is this projection and its analogues in other completions of the group ring, which obstruct
known methods of proof for the Baum-Connes conjecture. In this talk, I will introduce a gen-
eralisation of Kazhdan projections. Employing these projections we provide a link between
surjectivity of the Baum-Connes map and the 2-Betti numbers of the group. A similar relation
can be obtained in the context of the coarse Baum-Connes conjecture. This is based on joint
work with Kang Li and Piotr Nowak.
Irreducible inclusions of simple C*-algebras
Mikael Rordam, rordam@math.ku.dk
University of Copenhagen, Denmark
The literature contains a number interesting examples of inclusions of simple C*-algebras, typ-
ically arising from dynamical systems, with the property that all intermediate C*-algebras are
also simple. One can argue that this property of an inclusion of C*-algebras is the natural
183
