Title: Topological Entropy and Algebraic Entropy on Locally Compact Abelian Groups

by Anna Giordano Bruno (University of Udine) as part of Topological Groups

Lecture held in Elysium.

Abstract

Since its origin, the algebraic entropy $h_{alg}$ was introduced in connection with the topological entropy $h_{top}$ by means of Pontryagin duality. For a continuous endomorphism $phicolon Gto G$ of a locally compact abelian group $G$, denoting by $widehat G$ the Pontryagin dual of $G$ and by $widehat phicolon Gto G$ the dual endomorphism of $phi$, we prove that $$h_{top}(phi)=h_{alg}(widehatphi)$$ under the assumption that $G$ is compact or that $G$ is totally disconnected. It is known that this equality holds also when $phi$ is a topological automorphism.

Title: The Large-Scale Geometry of Locally Compact Abelian Groups

by Nicolò Zava (University of Udine) as part of Topological Groups

Lecture held in Elysium.

Abstract

Large-scale geometry, also known as coarse geometry, is the branch of mathematics that studies the global, large-scale properties of spaces. This theory is distinguished by its applications which include the Novikov and coarse Baum-Connes conjectures. Since the breakthrough work of Gromov, large-scale geometry has played a prominent role in geometric group theory, in particular, in the study of finitely generated groups and their word metrics. This large-scale approach was successfully extended to all countable groups by Dranishnikov and Smith. A further generalisation introduced by Cornulier and de la Harpe dealt with locally compact σ-compact groups endowed with particular pseudo-metrics.

To study the large-scale geometry of more general groups and topological groups, coarse structures are required. These structures, introduced by Roe, encode global properties of spaces. We also mention the equivalent approach provided by Protasov and Banakh using balleans. Coarse structures compatible with a group structure can be characterised by special ideals of subsets, called group ideals. While the coarse structure induced by the family of all finite subsets is well-suited for abstract groups, the situation is less clear for groups endowed with group topologies, as exemplified by the left coarse structure, introduced by Rosendal, and the compact-group coarse structure, induced by the group ideal of all relatively compact subsets, each suitable in disparate settings.

We present the large-scale geometry of groups via the historically iterative sequence of generalisations, enlisting illustrative examples specific to distinct classes of groups and topological groups. We focus on locally compact abelian groups endowed with compact-group coarse structures. In particular, we discuss the role of Pontryagin duality as a bridge between topological properties and their large-scale counterparts. An overriding theme is an evidence-based tenet that the compact-group coarse structure is the right choice for the category of locally compact abelian groups.

Title: On the Mackey Topology of an Abelian Topological Group

by Lydia Außenhofer (Universität Passau) as part of Topological Groups

Lecture held in Elysium.

Abstract

For a locally convex vector space $(V,tau)$ there exists a finest locally convex vector space topology $mu$ such that the topological dual spaces $(V,tau)’$ and $(V,mu)’$ coincide algebraically. This topology is called the $Mackey$ $topology$. If $(V,tau)$ is a metrizable locally convex vector space, then $tau$ is the Mackey topology.

In 1995 Chasco, Martín Peinador, and Tarieladze asked, “Given a locally quasi-convex group $(G,tau),$ does there exist a finest locally quasi-convex group topology $mu$ on $G$ such that the character groups $(G,tau)^wedge$ and $(G,mu)^wedge$ coincide?”

In this talk we give examples of topological groups which

1. have a Mackey topology,

2. do not have a Mackey topology,

and we characterize those abelian groups which have the property that every metrizable locally quasi-convex group topology is Mackey (i.e., the finest compatible locally quasi-convex group topology).

Title: Simply Given Compact Abelian Groups

by Peter Loth (Sacred Heart University) as part of Topological Groups

Lecture held in Elysium.

Abstract

A compact abelian group is called simply given if its Pontrjagin dual is simply presented. Warfield groups are defined to be direct summands of simply presented abelian groups. They were classified up to isomorphism in terms of cardinal invariants by Warfield in the local case, and by Stanton and Hunter–Richman in the global case. In this talk, we classify up to topological isomorphism the duals of Warfield groups, dualizing Stanton’s invariants. We exhibit an example of a simply given group with nonsplitting identity component.

Title: Topological Groups Seminar One-Week Hiatus

by Break (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract: TBA

Title: Variants of Invariant Means of Amenability

by Ajit Iqbal Singh (Indian National Science Academy) as part of Topological Groups

Lecture held in Elysium.

Abstract

It all started, like many other amazing theories, in nineteen twenty-nine,

With John von Neumann, the greatest of the great.

The question of existence of a finitely additive measure on a group, a mean of a kind,

That is invariant, under any translation, neither gaining nor losing any weight.

Mahlon M. Day, in his zest and jest, giving double importance to semigroups, too,

Took up the study of conditions and properties, and named it amenability.

Erling Folner followed it up, more like a combinatorial maze to go through,

Whether or not translated set meets the original in a sizeable proportionality.

How could functional analysts sit quiet, who measure anything by their own norms,

Lo and behold, it kept coming back to the same concept over and over again.

Group algebras were just as good or bad, approximate conditions did no harms,

With the second duals of lofty Richard Arens, it became deeper, but still a fun-game.

Ever since, with the whole alphabet names, reputed experts or budding and slick,

Considering several set-ups and numerous variants of the invariance.

Actions on Manifolds or operators, dynamical systems nimble or quick,

We will have a look at some old and some new, closely or just from the fence.

Title: Dynamics of Distal Actions on Locally Compact Groups

by Riddhi Shah (Jawaharlal Nehru University, New Delhi, India) as part of Topological Groups

Lecture held in Elysium.

Abstract

Distal maps were introduced by David Hilbert on compact spaces to study non-ergodic maps. A homeomorphism T on a topological space X is said to be distal if the closure of every double T-orbit of (x, y) does not intersect the diagonal in X x X unless x=y. Similarly, a semigroup S of homeomorphisms of X is said to act distally on X if the closure of every S-orbit of (x,y) does not intersect the diagonal unless x=y. We discuss some properties of distal actions of automorphisms on locally compact groups and on homogeneous spaces given by quotients modulo closed invariant subgroups which are either compact or normal. We relate distality to the behaviour of orbits. We also characterise the behaviour of convolution powers of probability measures on the group in terms of the distality of inner automorphisms.

Title: The group algebra of a compact group and Tannaka duality for compact groups

by Karl Hofmann (Technische Universität Darmstadt) as part of Topological Groups

Lecture held in Elysium.

Abstract

In the 4th edition of the text- and handbook “The Structure of Compact Groups”,

de Gruyter, Berlin-Boston, having appeared June 8, 2020, Sidney A. Morris and

I decided to include, among material not contained in earlier editions, the Tannaka-Hochschild Duality Theorem which says that $the$ $category$ $of$ $compact$ $groups$ $is,dual$

$to$ $the$ $category,of,real,reductive$ $Hopf$ $algebras$. In the lecture I hope to explain

why this theorem was not featured in the preceding 3 editions and why we decided

to present it now. Our somewhat novel access led us into a new theory of real

and complex group algebras for compact groups which I shall discuss. Some Hopf

algebra theory appears inevitable. Recent source: K.H.Hofmann and L.Kramer,

$On$ $Weakly,Complete,Group,Algebras$ $of$ $Compact$ $Groups$, J. of Lie Theory $bold{30}$ (2020), 407-424.

Karl H. Hofmann