Page 182 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 182
OPERATOR ALGEBRAS (MS-14)
Noncommutative ergodic theory of higher-rank lattices
Cyril Houdayer, cyril.houdayer@universite-paris-saclay.fr
Université Paris-Saclay, France
I will survey recent results in the study of dynamical properties of the space of positive def-
inite functions and characters of higher-rank lattices. I will explain that for a large class of
higher-rank lattices, all their Uniformly Recurrent Subgroups (URS) are finite and all their
weakly mixing unitary representations weakly contain the left regular representation. These re-
sults strengthen celebrated results by Margulis (1978), Stuck-Zimmer (1992) and Nevo-Zimmer
(2000). The key novelty in our work is a structure theorem for equivariant normal unital com-
pletely positive maps between von Neumann algebras and function spaces associated with Pois-
son boundaries.
Based on joint works with R. Boutonnet and U. Bader, R. Boutonnet, J. Peterson.
Boundary quotient C*-algebras for product systems
Evgenios Kakariadis, evgenios.kakariadis@ncl.ac.uk
Newcastle University, United Kingdom
A boundary quotient is a terminal object for a class of representations. In most of the cases
boundary quotients are manifestations of corona sets and they are indispensable for our under-
standing beyond the “classical” universe of finite dimensional algebras and their norm-limits.
C*-terminal objects are central elements of the theory, while they provide key C*-constructs
in the study of geometric and topological objects. This interplay goes as back as the classifica-
tion of factors by Murray and von Neumann in the 1930’s, and has been a continuous source of
inspiration for further developments. Applications include for example detecting phase transi-
tions of C*-invariants.
On the other hand the Shilov and the Choquet boundaries of (nonselfadjoint) function al-
gebras have been the subject of intense research since the 1950’s providing fruitful interac-
tions with convexity and approximation theory. Their noncommutative analogues have been
a groundbreaking foresight of Arveson’s seminal work in the 1960’s, and beyond. Applica-
tions include interactions with group theory, noncommutative geometry, and noncommutative
convexity, to mention only but a few.
In this talk we will show how a combination of the selfadjoint and the nonselfadjoint view-
points gives the existence of the boundary quotient C*-algebra for product systems. This class
has been under consideration for the past 30 years and models a great number of C*-constructs,
including graphs (of rank 1 or higher), C*-dynamics of several flavors (reversible and irre-
versible), semigroup C*-algebras, and Nica covariant representations of compactly aligned
product systems.
The talk is based on joint works with Dor-On, Katsoulis, Laca and Li.
180
Noncommutative ergodic theory of higher-rank lattices
Cyril Houdayer, cyril.houdayer@universite-paris-saclay.fr
Université Paris-Saclay, France
I will survey recent results in the study of dynamical properties of the space of positive def-
inite functions and characters of higher-rank lattices. I will explain that for a large class of
higher-rank lattices, all their Uniformly Recurrent Subgroups (URS) are finite and all their
weakly mixing unitary representations weakly contain the left regular representation. These re-
sults strengthen celebrated results by Margulis (1978), Stuck-Zimmer (1992) and Nevo-Zimmer
(2000). The key novelty in our work is a structure theorem for equivariant normal unital com-
pletely positive maps between von Neumann algebras and function spaces associated with Pois-
son boundaries.
Based on joint works with R. Boutonnet and U. Bader, R. Boutonnet, J. Peterson.
Boundary quotient C*-algebras for product systems
Evgenios Kakariadis, evgenios.kakariadis@ncl.ac.uk
Newcastle University, United Kingdom
A boundary quotient is a terminal object for a class of representations. In most of the cases
boundary quotients are manifestations of corona sets and they are indispensable for our under-
standing beyond the “classical” universe of finite dimensional algebras and their norm-limits.
C*-terminal objects are central elements of the theory, while they provide key C*-constructs
in the study of geometric and topological objects. This interplay goes as back as the classifica-
tion of factors by Murray and von Neumann in the 1930’s, and has been a continuous source of
inspiration for further developments. Applications include for example detecting phase transi-
tions of C*-invariants.
On the other hand the Shilov and the Choquet boundaries of (nonselfadjoint) function al-
gebras have been the subject of intense research since the 1950’s providing fruitful interac-
tions with convexity and approximation theory. Their noncommutative analogues have been
a groundbreaking foresight of Arveson’s seminal work in the 1960’s, and beyond. Applica-
tions include interactions with group theory, noncommutative geometry, and noncommutative
convexity, to mention only but a few.
In this talk we will show how a combination of the selfadjoint and the nonselfadjoint view-
points gives the existence of the boundary quotient C*-algebra for product systems. This class
has been under consideration for the past 30 years and models a great number of C*-constructs,
including graphs (of rank 1 or higher), C*-dynamics of several flavors (reversible and irre-
versible), semigroup C*-algebras, and Nica covariant representations of compactly aligned
product systems.
The talk is based on joint works with Dor-On, Katsoulis, Laca and Li.
180
