Compact symmetry groups and generators for sub-Markovian semigroups

We investigate the relations between the (completely bounded) local Coulhon-Varopoulos dimension and the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical (or noncommutative) Lp-spaces associated to finite measure spaces. More precisely, we prove that the completely bounded local Coulhon-Varopoulos dimension d exceeds the spectral dimension, i.e. that the associated Hodge-Dirac operator is d+-summable. We explore different settings to compare these two values: compact Riemannian manifolds, compact Lie groups, sublaplacians, metric measure spaces, noncommutative tori and quantum groups. Specifically, we prove that, while very often equal in smooth compact settings, these dimensions can diverge. Finally, we show that the existence of a symmetric sub-Markovian semigroup on a von Neumann algebra with finite completely bounded local Coulhon-Varopoulos dimension implies that the von Neumann algebra is necessarily injective.

It is shown that every compact group G is a -simplicial space where Q is any field of characteristic zero.As a consequence it follows that G satisfies a variation of the Lefschetz fixed point theorem.It has been known for some time that the Lefschetz fixed point theorem applies to a few spaces other than just ANR spaces, especially if some care is taken to use coefficients in certain fields [2].The case of all compact groups provides a broad class of spaces which may not have local connectivity of any order.It is shown that every compact group G satisfies the Lefschetz fixed point theorem when coefficients for the homology groups are taken in a field of characteristic zero.1. Theorem.Let G be a compact group.Then G is an inverse limit of Lie groups G = proj lim GM and if f : G" -> G is a homomorphism belonging to this inverse system, then there is a commutative diagram of covering compact Lie groups Presented to the Society, January 18,

In this paper we present symbolic criteria for invariant operators on compact topological groups $G$ characterising the Schatten-von Neumann classes $S_{r}(L^{2}(G))$ for all $0<r\leq\infty$. Since it is known that for pseudo-differential operators criteria in terms of kernels may be less effective (Carleman's example), our criteria are given in terms of the operators' symbols defined on the noncommutative analogue of the phase space $G\times\hat{G}$, where $G$ is a compact topological (or Lie) group and $\hat{G}$ is its unitary dual. We also show results concerning general non-invariant operators as well as Schatten properties on Sobolev spaces. A trace formula is derived for operators in the Schatten class $S_{1}(L^{2}(G))$. Examples are given for Bessel potentials associated to sub-Laplacians (sums of squares) on compact Lie groups, as well as for powers of the sub-Laplacian and for other non-elliptic operators on SU(2)$\simeq\mathbb S^3$ and on SO(3).

On square roots of the Haar state on compact quantum groups

In a recent article [K. H. Hofmann and F. G. Russo, ‘The probability that$x$and$y$commute in a compact group’,Math. Proc. Cambridge Phil Soc., to appear] we calculated for a compact group$G$the probability$d(G)$that two randomly selected elements$x, y\in G$satisfy$xy=yx$, and we discussed the remarkable consequences on the structure of$G$which follow from the assumption that$d(G)$is positive. In this note we consider two natural numbers$m$and$n$and the probability$d_{m,n}(G)$that for two randomly selected elements$x, y\in G$the relation$x^my^n=y^nx^m$holds. The situation is more complicated whenever$n,m\gt 1$. If$G$is a compact Lie group and if its identity component$G_0$is abelian, then it follows readily that$d_{m,n}(G)$is positive. We show here that the following condition suffices for the converse to hold in an arbitrary compact group$G$: for any nonopen closed subgroup$H$of$G$, the sets$\{g\in G: g^k\in H\}$for both$k=m$and$k=n$have Haar measure$0$. Indeed, we show that if a compact group$G$satisfies this condition and if$d_{m,n}(G)\gt 0$, then the identity component of$G$is abelian.

There exists an infinite family of p p -compact groups whose Weyl groups correspond to the finite p p -adic pseudoreflection groups G ( q , r , n ) G(q,r,n) of family 2a in the Clark-Ewing list. In this paper we study these p p -compact groups. In particular, we construct an analog of the classical Whitney sum map, a family of monomorphisms and a spherical fibration which produces an analog of the classical J J -homomorphism. Finally, we also describe a faithful complexification homomorphism from these p p -compact groups to the p p -completion of unitary compact Lie groups.

Let p t {p_{t}} , t &gt; 0 t &gt; 0 , be a strongly continuous submarkovian semigroup on a real Hilbert space L 2 ( X , m ) {L^2}(X, m) . The measure m is assumed to be excessive and the L 2 {L^2} generator A is assumed to satisfy an estimate (the sector condition) which permits the application of Dirichlet spaces (not necessarily symmetric). Other submarkovian semigroups P t ∼ P_t^ \sim with the same local generator and cogenerator and relative to which m is again excessive are classified in terms of generators for processes which live on a suitable boundary.

We consider various algebras of functions on a semitopological left-group S = X × G S = X \times G , the direct product of a left-zero semigroup X and a group G. In §1 we examine various analogues to the theorem of Eberlein that a weakly almost periodic function on a locally compact abelian group is uniformly continuous. Several appealing conjectures are shown by example to be false. In the second section we look at compactifications of products S × T S \times T of semitopological semigroups with right identity and left identity, respectively. We show that the almost periodic compactification of the product is the product of the almost periodic compactifications, thus generalizing a result of deLeeuw and Glicksberg. The weakly almost periodic compactification of the product is not the product of the weakly almost periodic compactifications except in restrictive circumstances; for instance, when T is a compact group. Finally, as an application, we define and study analytic weakly almost periodic functions and derive the theorem, analogous to a classical theorem about almost periodic functions, that an analytic function which is weakly almost periodic on a single line is analytic weakly almost periodic on a whole strip.

The Burnside ring A{G) of an arbitrary compact group G is defined in analogy to torn Dieck's definition for a compact Lie group.A topology is introduced on the set of conjugacy classes of closed subgroups, and a theorem of Dress on the topology of the prime spectrum of A(G) for profinite G is improved and extended to this general setting.This paper is condensed from part of my doctoral thesis [5], written under the supervision of Professor T. torn Dieck, to whom I am indebted for encouragement and advice.1.The Burnside ring and its additive structure.Suppose G is a compact group, and define a G-manifold as a differentiable manifold M together with a continuous left G-action such that, for each g G G, the translation g : M - M is a diffeomorphism.Examples are real G-representations and manifolds invariantly imbedded in such.Theorem 1.1.If G is a compact group, M a compact effective G-manifold, then G is a Lie group and M is a differentiable G-manifold.Proof.Since M is compact and locally connected, it has finitely many connected components, say M = Ji"A/,.For 1 < / < n, set G, = {g G GI g(M) = A/,}.This is a subgroup of G, and, since for any w, G M, the map /: G- M via f(g) = g-m is continuous, while G, = f~x(M), G, is clopen in G. Thus H = n"G, is a clopen normal subgroup.Now set H = {h E H\im G M, h-m = m], so that H is a normal subgroup of H and H/H acts effectively on A/,.Then since M is connected, Montgomery-Zippin [7, p. 208, Theorem 2] shows /////, to be a Lie group.Hence H/C\"xHi is again a Lie group.But since G operates effectively.Pi?//, = {h G //|Vw G M, h m = m) = {e}, so that H is a Lie group, and with it G since it is open in G.That M is then a differentiable G-manifold follows from [7, p. 208, Theorem 3].Now for any G-space X, let t(X) = {g G G|Vx G X,g x = x) be that part of G which acts trivially.Then for a G-manifold M, the G-action is derived from an effective G/i(A/)-action via the projection G-G/i(A/).Hence G/t(M) is a Lie group.

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C*-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C 0-semigroups of maps which we characterise. On C*-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.

We say that an element g of a group G is almost right Engel if there is a finite set R(g) such that for every x∈G, there is a positive integer n(x,g) such that […[[g,x],x],…,x]∈R(g) if x is repeated at least n(x,g) times. Thus, g is a right Engel element precisely when we can choose R(g)={1}. We prove that if all elements of a compact (Hausdorff) group G are almost right Engel, then G has a finite normal subgroup N such that G/N is locally nilpotent. If in addition there is a uniform bound ∣R(g)∣⩽m for the orders of the corresponding sets, then the subgroup N can be chosen of order bounded in terms of m. The proofs use the Wilson–Zelmanov theorem saying that profinite Engel groups are locally nilpotent and previous results of the authors about compact groups in which all elements are almost left Engel.

This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a finite group to the compact Lie group. If the compact Lie group is connected, then there exists a unique bi-invariant top-degree form with total integral 1, which simplifies the presentation of averaging. The averaging operator is useful for constructing invariant objects. For example, suppose a compact connected Lie group G acts smoothly on the left on a manifold M. Given any C∞ differential k-form ω‎ on M, by averaging all the left translates of ω‎ over G, one can produce a C∞ invariant k-form on M. As another example, on a G-manifold one can average all translates of a Riemannian metric to produce an invariant Riemann metric.

This paper explores further the connection between Langlands duality and T-duality for compact simple Lie groups, which appeared in work of Daenzer-Van Erp and Bunke-Nikolaus. We show that Langlands duality gives rise to isomorphisms of twisted K-groups, but that these K-groups are trivial except in the simplest case of SU(2) and SO(3). Along the way we compute explicitly the map on $H^3$ induced by a covering of compact simple Lie groups, which is either 1 or 2 depending in a complicated way on the type of the groups involved. We also give a new method for computing twisted K-theory using the Segal spectral sequence, giving simpler computations of certain twisted K-theory groups of compact Lie groups relevant for D-brane charges in WZW theories and rank-level dualities. Finally we study a duality for orientifolds based on complex Lie groups with an involution.

Using the star exponential, a path integral is constructed for arbitrary compact Lie groups. This is a globalization of the results of Alcalde, Cadavid, and Nakashima, in which a path integral was constructed for SU(2) in terms of local coordinates. Also the phase factor encountered by Alcalde, Cadavid, and Nakashima is shown to be zero.

Notes on compact Discrete Analysis 2017:16, 57 pp. This is the second paper in a two-part series. The first paper, [also published in this journal](http://discreteanalysisjournal.com/article/2105-notes-on-nilspaces-algebraic-aspects), developed the algebraic theory of nilspaces, which are an abstraction both of the notion of a nilmanifold, and of that of an abelian group (both with their attendant parallelogram structures). In this paper, the author applies this algebraic theory to study compact which can be viewed as a generalization of the notion of a compact abelian group. The main motivation for this is that such compact nilspaces arise naturally when trying to attack additive combinatorial problems such as the inverse conjecture for the Gowers norms via the methods of nonstandard analysis (or ultraproducts), as was demonstrated in work of Szegedy ([On higher order Fourier analysis](https://arxiv.org/abs/1203.2260)). The algebraic theory in the first part lets one describe $k$-step compact nilspaces as abelian extensions of $k-1$-step with the extension given by a cocycle. Among other things, this can be used to assign to each nilspace a probability measure that is analogous to the Haar probability measure one can assign to a compact group. However, it is not immediately obvious that this cocycle has any good measurability or continuity properties. Much of the delicate analysis in this paper (particularly those concerning the properties of continuous systems of measures) is devoted to ensuring that such good properties can be imposed (locally at least) on the cocycle, perhaps after modifying the cocycle by a suitable coboundary that does not affect the isomorphism class of the extension. Indeed, one can view the paper as developing some basic for although the formidable machinery of modern cohomology is not really deployed here. As with the first paper in this series, the paper closely follows the previous work of Camarena and Szegedy, but with many more details provided in the arguments. (An alternative approach to this theory has also recently been worked out in a series of papers by Gutman, Manners, and Varju.) A full structural description of compact nilspaces seems to be quite difficult, but in this paper two important partial descriptions are obtained. The first is to describe every compact nilspace as an inverse limit of compact nilspaces of rank. This is somewhat analogous to how the Peter-Weyl theorem can be used to describe a compact group as the inverse limit of finite-dimensional compact Lie groups. The second is to show that every connected compact nilspace of finite rank is isomorphic to a nilmanifold. These results were previously obtained by Camarena and Szegedy (and later used by Szegedy to provide a new proof of the inverse conjecture for the Gowers uniformity norms), but the author here reproves these results with much more detailed proofs, in particular treating with some care certain steps that were only briefly discussed in the prior paper of Camarena and Szegedy.