Local Compact Groups Research Articles

Subgroup collections controlling the homotopy type of a p -local compact group

Let (S,F,L) be a p-local compact group. We prove that the (uncompleted) homotopy type of the nerve of the linking system L is determined by the collection of subgroups of S that are F-centric and F-radical. This generalizes work of Broto, Grodal, Levi, Oliver, and the second author, who prove an analogous result for p-local finite groups.

Symmetric operator amenability of operator algebras

Using the notion of a symmetric virtual diagonal for a Banach algebra, we prove that a Banach algebra is symmetrically amenable if its second dual is symmetrically amenable. We introduce symmetric operator amenability in the category of completely contractive Banach algebras as an operator algebra analogue of symmetric amenability of Banach algebras. We give some equivalent formulations of symmetric operator amenability of completely contractive Banach algebras and investigate some hereditary properties of symmetric operator amenable algebras. We show that amenability of locally compact groups is equivalent to symmetric operator amenability of its Fourier algebra. Finally, we discuss about Jordan derivation on symmetrically operator amenable algebras.

The weak compactification of locally compact groups

We further investigate the weak topology generated by the irreducible unitary representations of a group G. A deep result due to Ernest [13] and Hughes [22] asserts that every weakly compact subset of a locally compact (LC) group G is compact in the LC-topology, generalizing thereby a previous result of Glicksberg [19] for abelian locally compact (LCA) groups. Here, we first survey some recent findings on the weak topology and establish some new results about the preservation of several compact-like properties when going from the weak topology to the original topology of LC groups. Among others, we deal with the preservation of countable compactness, pseudocompactness and functional boundedness.

Johnson pseudo-Connes amenability of dual Banach algebras

We introduce the notion of Johnson pseudo-Connes amenability for dual Banach algebras. We study the relation between this new notion with the various notions of Connes amenability like Connes amenability, approximate Connes amenability and pseudo Connes amenability. We also investigate some hereditary properties of this new notion. We prove that for a locally compact group G,M(G) is Johnson pseudo-Connes amenable if and only if G is amenable. Also we show that for every non-empty set I,MI(C) under this new notion is forced to have a finite index. Finally, we provide some examples of certain dual Banach algebras and we study their Johnson pseudo-Connes amenability.

Some new examples of simple $p$-local compact groups

In this paper we present new examples of simple p-local compact groupsfor all odd primes. We also develop the necessary tools to show saturation, simpleness, and the non-realizability as p-compact groups or compact Lie groups, which can be applied in a more general framework.

Groups of unstable Adams operations on p–local compact groups

A $p$-local compact group is an algebraic object modelled on the homotopy theory associated with $p$-completed classifying spaces of compact Lie groups and p-compact groups. In particular $p$-local compact groups give a unified framework in which one may study $p$-completed classifying spaces from an algebraic and homotopy theoretic point of view. Like connected compact Lie groups and p-compact groups, $p$-local compact groups admit unstable Adams operations - self equivalences that are characterised by their cohomological effect. Unstable Adams operations on $p$-local compact groups were constructed in a previous paper by F. Junod and the authors. In the current paper we study groups of unstable operations from a geometric and algebraic point of view. We give a precise description of the relationship between algebraic and geometric operations, and show that under some conditions unstable Adams operations are determined by their degree. We also examine a particularly well behaved subgroup of operations.

Finite approximations of p–local compact groups

We show that the classifying space of a $p$-local compact group is approximated by a telescope of classifying spaces of $p$-local finite groups. This result has numerous implications, like a Stable Elements Theorem for $p$-local compact groups, or the description of the homotopy type of certain mapping spaces.

Pro-Lie Groups: A Survey with Open Problems

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity component and, thus, in particular, each compact and each connected locally-compact group; it also includes all locally-compact Abelian groups. This paper provides an overview of the structure theory and the Lie theory of pro-Lie groups, including results more recent than those in the authors’ reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly-complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function that links the two. (A topological vector space is weakly complete if it is isomorphic to a power RX of an arbitrary set of copies of R. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected to pro-Lie groups.

The structure of p-local compact groups of rank 1

Abstract Let p be a fixed prime number. The main purpose of this paper is to introduce irreducible p-local compact groups and to describe all p-local compact groups of rank 1 in terms of this concept.

ACTIVITY IN GALACTIC NUCLEI OF COMPACT GROUP GALAXIES IN THE LOCAL UNIVERSE

We study the nuclear activity of galaxies in local compact groups. We use a spectroscopic sample of 238 galaxies in 58 compact groups from the Sloan Digital Sky Survey data release 7 to estimate the fraction of AGN-host galaxies in compact groups, and to compare it with those in cluster and field regions. We use emission-line ratio diagrams to identify AGN-host galaxies, and find that the AGN fraction of compact group galaxies is 17-42% depending on the AGN classification method. The AGN fraction in compact groups is not the highest among the galaxy environments. This trend remains even if we use several subsamples segregated by galaxy morphology and optical luminosity. The AGN fraction for early-type galaxies decreases with increasing galaxy number density, but the fraction for late-type galaxies changes little. We find no mid-infrared detected AGN-host galaxies in our sample of compact groups using Wide-field Infrared Survey Explorer data. These results suggest that the nuclear activity of compact group galaxies (mostly early types) is not strong because of lack of gas supply even though they may experience frequent galaxy-galaxy interactions and mergers that could trigger nuclear activity.

Cohomology of affine Deligne–Lusztig varieties for [formula omitted

In this paper we study affine Deligne–Lusztig varieties Xw(b) for GL2 and their étale coverings. At first, we compute them explicitly, then we determine associated representations of a certain locally-compact group, the group of rational points of the σ-stabilizer of b, in their étale cohomology. Further, we study these representations by determining morphisms into the irreducible representations of the given group. In particular all cuspidal representations of level 0 of GL2 of a local field and of its inner form, which is the group of units of a quaternion algebra, occur in the cohomology.

Unstable Adams operations acting onp–local compact groups and fixed points

Nevertheless, when passing from a finite setting to an infinite one, some of the techniques used in the former case are not available any more. As a result, some of the more important results in [5] were not extended to p–local compact groups, and, roughly speaking, p–local compact groups are not yet as well understood as p–local finite groups. It is then the aim of this paper to shed some light on the new theory introduced in [7].

Bayesian and Maximum Likelihood Solutions: An Asymptotic Comparison Related to Cost Function

Problem statement: Wald showed that the minimax solution is the Bayes ian solution with respect to the law a priori the worst. We try to establish a similar result by comparing the Bayesian solution and the solution of maximum likel ihood when the parameter space is a compact metrizable group. Approach: we take as a priori law Haar measure because we re duce the problem by invariance. We construct a sequence of cost func tions for which we obtain a sequence of solutions Bayesian which converges to the solution of the maximum likelihood. Results: We show that both solutions are asymptotically equal. Conclusion/Recommendation : The generalization when the parameter space is a local compact group.

Unstable Adams operations onp–local compact groups

A p-local compact group is an algebraic object modelled on the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups. In the study of these objects unstable Adams operations, are of fundamental importance. In this paper we define unstable Adams operations within the theory of p-local compact groups, and show that such operations exist under rather mild conditions. More precisely, we prove that for a given p-local compact group G and a sufficiently large positive integer $m$, there exists an injective group homomorphism from the group of p-adic units which are congruent to 1 modulo p^m to the group of unstable Adams operations on G

Multipliers of commutative Banach algebras, power boundedness and Fourier-Stieltjes algebras

Let A be a semisimple and regular commutative Banach algebra with bounded approximate identity. We study multipliers of A, in particular power bounded ones, and the associated ideals of A and A-invariant projections of the dual space of A. Samples of the results are general versions of the classical theorems of Choquet�Deny and of Foguel about measures on locally compact abelian groups. The results are linked to sets of synthesis in the Gelfand spectrum of A, and the main applications are concerned with Fourier and Fourier�Stieltjes algebras of locally compact groups.

Lévy Khinchin Formula on Commutative Hypercomplex System

Abstract. A commutative hypercomplex system L 1 (Q,m) is, roughly speaking, a spacewhich is defined by a structure measure (c(A,B,r),(A,B ∈ β(Q)). Such space has beenstudied by Berezanskii and Krein. Our main purpose is to establish a generalization ofconvolution semigroups and to discuss the role of the L´evy measure in the L´evy-Khinchinrepresentation in terms of continuous negative definite functions on the dual hypercomplexsystem. 1. IntroductionThe integral representation of negative definite functions is known in the liter-ature as the L´evy-Khinchin formula. This was established for G= Rin the late1930’s by L´evy and Khinchin. It had been extended to Lie groups by Hunt [9] andby Parthasarathy et al [13] to locally compact abelian groups with a countable case.In 1969 Harzallah [7] gave a representation formula for an arbitrary locally compactabelian group. Hazod [8] obtained a L´evy-Khinchin formula for an arbitrary locallycompact group. The general L´evy-Khinchin formula and the special case, where theinvolution is identical are due to Berg [4]. Lasser [12] deduced the L´evy-Khinchinformula for commutative hypergroups. Now these contribution may be viewed asa L´evy-Khinchin formula for negative definite functions defined on commutativehypercomplex systems.Let Q be a complete separable locally compact metric space of pointsp,q,r··· ,β(Q) be the σ-algebra of Borel subsets, and β

Discrete models for thep–local homotopy theory of compact Lie groups andp–compact groups

In our earlier paper [7], we defined and studied a certain class of spaces which in many ways behave like p ‐completed classifying spaces of finite groups. These spaces occur as “classifying spaces” of certain algebraic objects called p ‐local finite groups. The purpose of this paper is to generalize the concept of p ‐local finite groups to what we call p ‐local compact groups. The motivation for introducing this family comes from the observation that p ‐completed classifying spaces of finite and compact Lie groups, as well as classifying spaces of p ‐compact groups (see Dwyer and Wilkerson [11]), share many similar homotopy theoretic properties, but earlier studies of these properties usually required different techniques for each case. Moreover, while p ‐completed classifying spaces of finite and, more generally, compact Lie groups arise from the algebraic and geometric structure of the groups in question, p ‐compact groups are purely homotopy theoretic objects. Unfortunately, many of the techniques used in the study of p ‐compact groups fail for p ‐completed classifying spaces of general compact Lie groups. With the approach presented here, we propose a framework general enough to include p ‐completed classifying spaces of arbitrary compact Lie groups as well as p ‐compact groups. The new idea here is to replace fusion systems over finite p ‐groups, as handled in [7], by fusion systems over discrete p ‐toral groups. A discrete p ‐toral group is a group which contains a discrete p ‐torus (a group of the form .Z=p 1 / r for finite r 0) as a normal subgroup of p ‐power index. A p ‐local compact group consists of a triple

Quasiregular representations of the infinite-dimensional nilpotent group

In the present work an analog of the quasiregular representation which is well known for locally-compact groups is constructed for the nilpotent infinite-dimensional group B 0 N and a criterion for its irreducibility is presented. This construction uses the infinite tensor product of arbitrary Gaussian measures in the spaces R m with m > 1 extending in a rather subtle way previous work of the second author for the infinite tensor product of one-dimensional Gaussian measures.

On the Automorphically Stable Distributions on Abelian Groups

On a local compact Abelian group X, we consider {G-auto}-morphically stable distributions, where~G~is a subgroup of a group~$Aut(X)$. It is shown that if~$\mu$ is G-automorphically stable, then 1)~$\mu$ is either absolutely continuous, singular, or discrete with respect to the Haar measure of the group~X; 2)~if~$\mu$ is discrete, then~$\mu$~is a shift of the Haar distribution of a finite G-characteristic subgroup of the group~X; 3)~if~G consists of elements of finite order, then~$\mu$~is a shift of the Haar distribution of a compact G-automorphically stable subgroup of the group~X.

Remarks on Pseudocharacters and the Real Continuous Bounded Cohomology of Connected Locally Compact Groups

Results on continuous pseudocharacters on locally compactgroups are presented and their applications to the description oftheir second continuous bounded cohomology groups areindicated. In particular, it is proved that the second realcontinuous bounded cohomology group of a connected locallycompact group is finite-dimensional.