Totally Disconnected Locally Compact Groups And Operator .
Totally disconnected locally compact groupsand operator algebrasGeorge WillisThe University of NewcastleJuly 27th, 2017
Totally disconnected locally compact groupsThe locally compact group G is totally disconnected if the onlyconnected components in G are singletons.TheoremLet G be a locally compact group. Then the connectedcomponent of the identity, N, is a closed normal subgroup of Gand G/N is a t.d.l.c. group.Theorem (van Dantzig, 1930’s)Suppose that G is a t.d.l.c. group and let O 3 1 beneighbourhood of the identity. Then there is a compact opensubgroup V O.
Compact open subgroups are commensuratedLet U be a compact open subgroup of G. Then[U : U xUx 1 ] and [xUx 1 : U xUx 1 ] for every x G,i.e., U is a commensurated subgroup of G.On the other hand, if G is any group and H is a commensuratede and asubgroup of G, then there is a t.d.l.c. group Gehomomorphism ϕ : G G such that ϕ(H) is a compact openesubgroup of G.e is the relative profinite completion of the pair (G, H),Gsee C. D. Reid and P. R. Wesolek, Homomorphisms into totally disconnected, locally compact groups with dense image,arXiv:1509.00156v1 and references therein.
Weight functions and the scale functionFor a fixed compact open U G define weight functionwU (x) [xUx 1 : U xUx 1 ],(x G).Then wU (xy ) wU (x)wU (y ) for all x, y G, that is, wU issubmultiplicative.The scale of x G is the positive integers(x) min {wU (x) U G is compact and open} ,(x G).Say that U is minimising if the minimum is attained at U.It may be shown that, for every U G compact and open,1s(x) lim wU (x n ) n .n
A weighted convolution algebraGiven U G compact and open, let ZL1 (G, wU ) f L(G) f (x)wU (x) dx .GThen L1 (G, wU ) is a Banach algebra under convolution.IFor U, V G compact and open, there is B 1 such thatB 1 wV wU BwV .Hence L1 (G, wU ) does not depend on U.IwU is bounded there is V / G compact and open.Is(x) is the spectral radius of the operator on L1 (G, wU ) oftranslation by x.
A weighted convolution algebra 2The convolution algebra L1 (G, wU ) is just natural for the t.d.l.c.group G as is L1 (G).ProblemHow do properties of the Banach algebra L1 (G, wU ) reflect thestructure of the totally disconnected, locally compact group G?IWeighted convolution algebras L1 (G, w) often havenon-trivial cohomology (point derivations in thecommutative case).IUnlike L1 (G) and C*-algebras, L1 (G, wU ) does not have aunique natural norm.
A characterisation of minimising subgroupsTheoremLet x G and U G be compact and open. Put\\U x k Ux k and U x k Ux k .k 0k 0Then U is minimising for x if and only ifTA U U U , andSTB U : k Z x k U x k is closed.In this case, s(x) [xU x 1 : U ].A compact open subgroup satisfying TA and TB is tidy for x.
The tree representation theoremThe group V n hxi is an HNN-extension and so Bass-Serretheory implies the following.TheoremSuppose that U is tidy for x G. Then V n hxi is a closedsubgroup of G. There is a regular tree Tq 1 , where q s(x),and a homomorphism ρ : V n hxi Aut(Tq 1 ) such that:Iρ(V n hxi) is a closed subgroup of Aut(Tq 1 ) fixing anend, ω, of the tree;Iker ρ is the largest compact normal subgroup of V n hxi;andIρ(x) is a hyperbolic element of Aut(Tq 1 ) which translatesby distance 1 and has ω is its attracting end.Closed subgroups of Aut(Tq 1 ) fixing and end of the tree arekey ingredients in the structure theory of t.d.l.c. groupscorresponding to the (ax b)-group and R n R in Lie theory.
Representations and C*-algebras of ρ(V n hxi)ProblemWhat are the unitary representations of the closed subgroupsof Aut(Tq 1 ) which fix an end of Tq 1 ?(There are uncountably many such groups.)ProblemInduce representations of ρ(V n hxi) to representations of G(for certain groups G).ProblemHow much information about ρ(V n hxi) is retained byC ρ(V n hxi)?
Contraction groups 1Theorem (The Mautner phenomenon)Let σ : G U(H) be a unitary representation of the locallycompact group G and suppose that σ(x)ξ ξ for some x Gand non-zero ξ V . Then σ(h)ξ ξ for every h G such thatx n hx n 1 as n .The set con(x) {h G x n hx n 1 as n } is thecontraction subgroup for x.Thus the Mautner phenomenon says that, if ξ is fixed by x, thenξ is fixed by every h con(x).
Contraction groups 2Theorem (Baumgartner & W.)Suppose that V is tidy for x G. Then C : con(x 1 ) is aco-compact normal subgroup of V invariant underconjugation x and s(x C ) s(x).Theorem (Glöckner & W.)Suppose that H : con(x) is closed. ThenIH T D, where T , D are closed x-invariant subgroupsof H such that T is torsion and D is divisible;ID is a direct sum of nilpotent p-adic Lie groups for a finiteset of primes p; andIT has a finite composition series of x-invariant subgroupswherethe compositionfactors are isomorphic toQP( n 0 Fi ) n 0 Fi , for some finite simple group Fi andthe automorphism induced by x is the shift.
Totally disconnected locally compact groups The locally compact group G is totally disconnected if the only connected components in G are singletons. Theorem Let G be a locally compact group. Then the connected component of the identity, N, is a closed normal subgroup of G a
of left cosets G G is a totally disconnected locally compact (t.d.l.c.) group. The study of locally compact groups therefore in principle, although not always in practice, reduces to studying connected locally compact groups and t.d.l.c. groups. The study of locally compact groups begins
In particular, a totally disconnected space is Hausdor . The sets U;V in the de nition are not only open but also closed. We claim that at every point xof a locally compact totally disconnected space Xthere is a local basis consisting of compact open sets. To see this, take an open set V
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percent of disconnected young men and 43 percent of disconnected young women. Wald and Martinez further conclude that the South has more disconnected young adults than the Northeast and West combined, with nearly 61 percent of the nation’s disconnected African-American males living in the r
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Disconnected operation refers to the ability of a distributed system client to operate despite server inaccessibility by emulating services locally. The capability to operate disconnected is already valuable in many systems,
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