We show that in order to prove that all locally compact groups with exact

reduced group C∗-algebras are exact in the dynamical sense (i.e., KW-exact), it suffices to

show this for totally disconnected locally compact groups

Cave, Chris

Zacharias, Joachim

English

arXiv

Cave, C. (Chris)

Zacharias, J. (Joachim)

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Kirchberg–Wassermann exactness vs exactness: reduction to the unimodular totally disconnected case

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Kirchberg-Wassermann exactness vs exactness: reduction to the unimodular totally disconnected case

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Kirchberg–Wassermann exactness vs exactness:reduction to the unimodular totally disconnected case

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      Cave, C. and Zacharias, J. (2018) Kirchberg-Wassermann exactness vs exactness: reduction to the unimodular totally disconnected case. Münster Journal of Mathematics.     There may be differences between this version and the published version. You are advised to consult the publisher’s version if you wish to cite from it.    http://eprints.gla.ac.uk/168809/              Deposited on: 13 September 2018                       Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk KIRCHBERG-WASSERMANN EXACTNESS VSEXACTNESS: REDUCTION TO THE UNIMODULARTOTALLY DISCONNECTED CASECHRIS CAVE AND JOACHIM ZACHARIASAbstract. We show that in order to prove that all locally compactgroups with exact reduced group C*-algebras are exact in the dynamicalsense (i.e. KW-exact) it suffices to show this for totally disconnectedlocally compact groups.1. IntroductionThere are two natural notions of exactness for locally compact groupswhich to our knowledge were first mentioned by Kirchberg in [8]. A weakone, called here C∗-exactness which says that the reduced group algebra isan exact C∗-algebra, and a strong one called KW-exactness, which assertsthat given any exact sequence of dynamical systems over the group thecorresponding sequence of reduced crossed products is exact. The strongerexactness property can thus be regarded as a dynamical form of exactness.Here KW stands for Kirchberg and Wassermann who introduced and studiedthese notions in [9] and [10]. Since the crossed product by trivial actionsis just the tensor product by the reduced group algebra it is evident thatKW-exactness implies C∗-exactness. As announced in [8] and later provedin [9], the two concepts are equivalent for discrete groups but whether thesame equivalence holds true in the case of general locally compact groupshas been an open problem ever since.Note that there are numerous other concepts related to exactness such asamenability at infinity or the non-existence of non-compact ghost operators,which have been studied and put forward in the past decades ([1, 12, 14]).There has been considerable recent progress in the understanding of theseconditions showing that they are equivalent to KW-exactness among sec-ond countable groups ([3]). In view of those developments the question ofequivalence of KW-exactness and C∗-exactness appears more pressing thanever.In this note we do not answer this question but reduce the problem tothe case when the group is unimodular and totally disconnected. Thus ifC∗-exact but non KW-exact groups exist then there must also exist totallydisconnected unimodular such groups. This had already been suspected byexperts (see the introduction of [1]).The first author is supported by the Danish National Research Foundation through theCentre for Symmetry and Deformation (DNRF92).12 CHRIS CAVE AND JOACHIM ZACHARIASAcknowledgementsWe would like to thank Kang Li and Sven Raum for multiple stimulatingdiscussions.2. Preliminaries2.1. C∗-exactness and KW-exactness. As is well-known a C∗-algebra Ais exact if for any exact sequence0→ I → E → Q→ 0of C∗-algebras the sequence of minimal tensor products0→ A⊗ I → A⊗ E → A⊗Q→ 0is exact. Kirchberg and partly Wassermann proved that this property isequivalent to nuclear embeddability and passes to subalgebras and quotients(c.f. [4, Chapter 10]). Exactness of the second sequence can only fail in themiddle. That is the kernel of the map onto A⊗Q is strictly larger than A⊗I.It is easy to check directly from the definition that a minimal tensor productA ⊗ B of two C∗-algebras is exact iff A and B are exact ([4, Proposition10.2.7]).Definition 2.1. Let G be a locally compact group. Then G is said to beC∗-exact if C∗r (G) is an exact C∗-algebra.If A is a C∗-algebra and G is a locally compact group acting on A byα : G → Aut(A) then the action α is called continuous if for all a ∈ A, themap g 7→ αg(a) is norm continuous.Definition 2.2. Let G be a locally compact group. G is said to be KW-exact (KW for Kirchberg and Wassermann) if for all C∗-algebras A and allcontinuous actions α : G→ Aut(A) and for all closed two-sided ideals I EAsuch that αg(I) = I for all g ∈ G, the sequence0→ I oα,r G→ Aoα,r G→ A/I oα,r G→ 0is exact.By recent results in [12, 3] it is now known that it suffices, for secondcountable groups, to check exactness of only one such sequence. That is, alocally compact second countable group G is KW-exact iff0→ C0(G)oL,r G→ C lub (G)oL,r G→ (C lub (G)/C0(G))oL,r G→ 0is exact, where L is the left translation action on the C∗-algebra of boundedleft uniformly continuous functions C lub (G) on G.As already mentioned, since A⊗C∗r (G) ∼= Aoτ,rG where τ : G→ Aut(A)is the trivial action, we have:Proposition 2.3. If G is KW-exact then it is C∗-exact.KW-exactness satisfies the following permanence properties.Proposition 2.4. Let G be a locally compact group.(1) If G is amenable then G is KW-exact [10, Proposition 6.1].(2) If G is connected then G is KW-exact [10, Theorem 6.8].REDUCTION TO T.D.L.C. GROUPS 3(3) Let N E G be a closed normal subgroup. If N and G/N are KW-exact then G is KW-exact [10, Theorem 5.1].Given a subgroup H of a locally compact group G, elements in H andC∗r (H) only act as multipliers on C∗r (G). However if H is open in G then itis easy to see that C∗r (H) ⊂ C∗r (G). Since exactness passes to subalgebraswe get.Proposition 2.5. If G is a locally compact C∗-exact group and H ≤ G isan open subgroup then C∗r (H) ↪→ C∗r (G) is an injective ∗-homomorphismand so C∗r (H) is also exact.2.2. Structure of locally compact groups. The following propositionfollows from the closure properties of the class of amenable locally compactgroups. We indicate the proof for the reader’s convenience.Proposition 2.6 ([15, Proposition 4.1.12]). Every locally compact group Ghas a unique maximal amenable closed normal subgroup.Proof. Since unions of directed systems of amenable subgroups of G areamenable one only needs to show that given two closed normal amenablesubgroups H1 and H2 the closed subgroup H generated by them is amenable.Now the semidirect product H1 o H2 is amenable and H is the closure ofthe continuous image of H1oH2. This implies that H is also amenable. Definition 2.7. Let G be a locally compact group. Then the amenableradical, denoted by Rad(G) is the unique maximal amenable closed normalsubgroup of G.We have the following characterisation of totally disconnected locally com-pact groups which is a classical result by van Danzig.Theorem 2.8 ([13]). Let G be a locally compact group. Then G is totallydisconnected if and only if it admits a neighbourhood basis of the identityconsisting of compact open subgroups.We use the following structure theorem of locally compact groups whichis deduced from a solution to Hilbert’s fifth problem [11, Theorem 4.6].Recall that a subgroup H ≤ G is characteristic if it is preserved under everyautomorphism in Aut(G).Theorem 2.9 ([5, Theorem 3.3.3], [6, Theorem 23]). Let G be any locallycompact group. The quotient group G/Rad(G) has a finite index open char-acteristic subgroup which splits as a direct product S ×D where S is a con-nected semi-simple Lie group and D is totally disconnected locally compact.3. Reduction to the unimodular totally disconnected locallycompact caseThe aim of this section is to prove the following theorem.Theorem 3.1. If KW-exactness and C∗-exactness are equivalent for all uni-modular totally disconnected locally compact groups then they are equivalentfor all locally compact groups.4 CHRIS CAVE AND JOACHIM ZACHARIAS3.1. Induced representations and weak containment.3.1.1. Induced representations. Let G be a locally compact group and H ≤G a closed subgroup. For a Borel measure ν on G/H and g ∈ G, denote νgto be the measure defined as νg(E) = ν(gE) for all Borel sets E ⊂ G/H. Aregular Borel measure ν is quasi-invariant if νg ∼ ν for all g ∈ G where ∼denotes mutual absolute continuity of measures.Let µH be a Haar measure on H and define a mapping TH : Cc(G) →Cc(G/H) whereTH(f)(xH) =∫Hf(xh) dµH(h).This map is surjective [2, Lemma B.1.2].Lemma 3.2 ([2, Lemma B.1.3]). Let ρ : G→ R>0 be a continuous functionon G. Then the following are equivalent(1) for all g ∈ G and h ∈ H one hasρ(gh) =∆H(h)∆G(h)ρ(g);(2) The functional λρ : Cc(G/H)→ C defined byλρ ◦ TH(f) =∫Gf(g)ρ(g) dµGis well-defined and positive.If the above conditions hold then the associated regular Borel measure µρ toλρ under the Riesz representation is quasi-invariant with Radon–Nikodymderivatived(µρ)ydµρ(xH) =ρ(yx)ρ(x)∀x, y ∈ G.Such a function ρ : G→ R>0 is called a rho-function for the pair (G,H).For every pair (G,H) there always exists a rho-function for (G,H). Indeedif f ∈ Cc(G)+ thenρf (x) =∫H∆G(h)∆H(h)−1f(xh)dµH(h)is a continuous rho-function. Thus there always exists a quasi-invariantregular Borel measure on G/H. In fact every quasi-invariant regular Borelmeasure is associated to a rho-function for (G,H) [2, Theorem B.1.4]. WhenH is a closed normal subgroup then one takes ρ = 1 and the associatedquasi-invariant regular Borel measure is the usual Haar measure on G/H.Let pi : H → U(H) be a unitary representation and fix a quasi-invariantmeasure µ on G/H. Define a new Hilbert spaceH(pi) ={ξ : G→ H∣∣∣∣∣ ξ(xh) = pi(h−1)ξ(x) and∫G/H‖ξ(x)‖2 dµ(x) <∞}with inner product given by〈ξ, η〉 =∫G/H〈ξ(x), η(x)〉 dµ(x)REDUCTION TO T.D.L.C. GROUPS 5The representation of G induced from pi or simply the induced representationis the representation indGHpi : G→ U(H(pi)) given byindGHpi(x)ξ(y) =(dµx−1dµ(yH))1/2ξ(x−1y) ∀ξ ∈ H(pi) ∀x, y ∈ G.Given another quasi-invariant measure on G/H the same construction givesa unitarily equivalent representation so we can call indGHpi the induced rep-resentation of pi to G without trepidation [2, Proposition E.1.5].When H = { e } then indGH(1H) = λG. More generally, the representationindGH(1H) is called the quasi-regular representation of G/H. If H is normalin G then indGH(1H) is unitarily equivalent to λG/H ◦ q where q : G→ G/His the natural surjection and λG/H is the left regular representation of G/H[7, Proposition 2.38].3.1.2. Weak containment. The following definition for weak containment ofrepresentations is not standard; however, it is sufficient for our applications.Definition 3.3 ([2, Theorem F.4.4]). Let pi and ρ be unitary representationsof G. Then pi is weakly contained in ρ, denoted by pi ≺ ρ, if ‖pi(f)‖ ≤ ‖ρ(f)‖for all f ∈ L1(G).We have the following properties of weak containment and induced rep-resentations.Proposition 3.4. Let G be a locally compact group and H ≤ G a closedsubgroup. Then(1) indGH(λH) is unitarily equivalent to λG [7, Corollary 2.52];(2) G is amenable if and only if 1G ≺ λG [2, Theorem G.3.2];(3) if σ and ρ are unitary representations on H and σ ≺ ρ then indGH(σ) ≺indGH(ρ) [2, Theorem F.3.5].We believe the following is well known but we provide a proof as we couldnot find a reference.Lemma 3.5. Let G be a locally compact group and suppose H ≤ G is aclosed normal amenable subgroup. If G is C∗-exact then G/H is C∗-exact.Proof. As H is amenable it follows that 1H ≺ λH and so indGH(1H) ≺ λG.However indGH(1H) is unitarily equivalent to λG/H ◦ q where q : G→ G/H isthe quotient map and λG/H is the left regular representation of G/H. HenceλG/H ◦ q ≺ λG and so it remains to show that the natural map TH definedbyTH : Cc(G)→ Cc(G/H), TH(f)(gH) =∫Hf(gh) dµH(h)extends to a surjective ∗-homomorphism from C∗r (G) → C∗r (G/H). So letλG/H ◦ q : Cc(G)→ B(L2(G/H)) be the natural extension of λG/H ◦ q. ThatisλG/H ◦ q(f) =∫Gf(g)λG/H(gH) dgfor all f ∈ Cc(G). We will show that λG/H ◦ q(f) = λG/H(TH(f)) for allf ∈ Cc(G). Then it will follow that‖λG/H(TH(f))‖ = ‖λG/H ◦ q(f)‖ ≤ ‖λG(f)‖6 CHRIS CAVE AND JOACHIM ZACHARIASas λG/H ◦ q ≺ λG and so TH extends to a surjective ∗-homomorphism fromC∗r (G)→ C∗r (G/H). So for all f ∈ Cc(G) and ξ ∈ Cc(G/H) and yH ∈ G/Hwe haveλG/H ◦ q(f)ξ(yH) =∫Gf(g)ξ(g−1yH) dg =∫G/H∫Hf(xh)ξ(h−1x−1yH) dhdx=∫G/H∫Hf(xh)ξ(x−1yH) dhdxwhere the second equality follows from Weil’s integration formula [7, Corol-lary 1.21] and the final equality follows from normality of H. NowλG/H(TH(f))ξ(yH) =∫G/HTHf(xH)λG/H(xH)ξ(yH) d(xH)=∫G/H∫Hf(xh)ξ(x−1yH) dhdx.Hence λG/H ◦ q(f) = λG/H(TH(f)) for all f ∈ Cc(G) and so TH extends toa surjection. As C∗r (G) is exact it follows that C∗r (G/H) is also exact asexactness passes to quotients. 3.2. Reduction to totally disconnected case.Lemma 3.6. If KW-exactness and C∗-exactness are equivalent in the classof unimodular totally disconnected locally compact groups then they are equiv-alent in the class of totally disconnected locally compact groupsProof. Let G be a totally disconnected locally compact group and supposeG is C∗-exact. Let G0 = ker(∆). In particular G0 is a closed normalunimodular subgroup of G. As G is totally disconnected, there exists acompact open subgroup K ≤ G. Compact groups are unimodular it followsthat ∆|K = 1. Hence K ≤ G0 and so µG(G0) ≥ µG(K) > 0 and so G0 isopen. Thus as G is C∗-exact it follows that G0 is C∗-exact.By assumption this implies that G0 is KW-exact and as G/G0 is abelianit follows that G is also KW-exact. We are now ready to prove the main result of this section.Proof of Theorem 3.1. Let G be a C∗-exact locally compact group. LetRad(G) be the amenable radical of G. Then by Lemma 3.5 it follows thatG/Rad(G) is C∗-exact. By Theorem 2.9 there exists an open normal finiteindex subgroup N ≤ G/Rad(G) such that N ∼= S×D where S is a connectedsemisimple Lie group and D is totally disconnected. We have the tensordecomposition where C∗r (N) ∼= C∗r (S)⊗C∗r (D). As C∗r (N) is exact it followsthat C∗r (D) is also exact, as we pointed out just before Def.2.1.By assumption and by Lemma 3.6 this implies that D is KW-exact. Asconnected locally compact groups are KW-exact (Proposition 2.4.(2)) andKW-exactness is preserved under extensions by closed normal subgroups(Proposition 2.4.(3)) it follows that N is KW-exact. We know N is open soin particular it is closed in G/Rad(G). Further N is cocompact in G/Rad(G)so G/Rad(G) is KW-exact. As Rad(G) is a closed normal and amenablesubgroup of G and KW-exactness is preserved under extensions it followsthat G is KW-exact (Proposition 2.4.(1,3)). REDUCTION TO T.D.L.C. GROUPS 7References[1] Claire Anantharaman-Delaroche. Amenability and exactness for dynamical systemsand their C∗-algebras. Trans. Amer. Math. Soc., 354(10):4153–4178 (electronic),2002.[2] Bachir Bekka, Pierre de la Harpe, and Alain Valette. Kazhdan’s property (T), vol-ume 11 of New Mathematical Monographs. Cambridge University Press, Cambridge,2008.[3] Jacek Brodzki, Chris Cave, and Kang Li. Exactness of locally compact groups. Adv.Math., 312:209–233, 2017.[4] Nathanial P. Brown and Narutaka Ozawa. C∗-algebras and finite-dimensional approx-imations, volume 88 of Graduate Studies in Mathematics. American MathematicalSociety, Providence, RI, 2008.[5] M. Burger and N. Monod. Continuous bounded cohomology and applications to rigid-ity theory. Geom. Funct. Anal., 12(2):219–280, 2002.[6] Maxime Gheysens and Nicolas Monod. Fixed points for bounded orbits in Hilbertspaces. Ann. Sci. E´c. Norm. Supe´r. (4), 50(1):131–156, 2017.[7] Eberhard Kaniuth and Keith F. Taylor. Induced representations of locally compactgroups, volume 197 of Cambridge Tracts in Mathematics. Cambridge University Press,Cambridge, 2013.[8] Eberhard Kirchberg. Exact C∗-algebras, tensor products, and the classification ofpurely infinite algebras. In Proceedings of the International Congress of Mathemati-cians, Vol. 1, 2 (Zu¨rich, 1994), pages 943–954. Birkha¨user, Basel, 1995.[9] Eberhard Kirchberg and Simon Wassermann. Exact groups and continuous bundlesof C∗-algebras. Math. Ann., 315(2):169–203, 1999.[10] Eberhard Kirchberg and Simon Wassermann. Permanence properties of C∗-exactgroups. Doc. Math., 4:513–558 (electronic), 1999.[11] Deane Montgomery and Leo Zippin. Topological transformation groups. IntersciencePublishers, New York-London, 1955.[12] John Roe and Rufus Willett. Ghostbusting and property A. J. Funct. Anal.,266(3):1674–1684, 2014.[13] D. Van Dantzig. Zur topologischen Algebra. III. Brouwersche und Cantorsche Grup-pen. Compositio Math., 3:408–426, 1936.[14] Rufus Willett. Some notes on property A. In Limits of graphs in group theory andcomputer science, pages 191–281. EPFL Press, Lausanne, 2009.[15] Robert J. Zimmer. Ergodic theory and semisimple groups, volume 81 of Monographsin Mathematics. Birkha¨user Verlag, Basel, 1984.Department of Mathematical Sciences, University of Copenhagen,Universitetsparken 5, DK-2100 Copenhagen Ø, DenmarkE-mail address: chris.cave@math.ku.dkSchool of Mathematics and Statistics, University of Glasgow,University Place, Glasgow G12 8QQ, ScotlandE-mail address: joachim.zacharias@glasgow.ac.uk

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Kirchberg-Wassermann exactness vs exactness: reduction to the unimodular totally disconnected case

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