Erratum for: Induction for Banach algebras, groupoids and KK ban

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1 Erratum for: Induction for Banach algebras, groupoids and KK ban Walther Paravicini February 17, 2008 Abstract My preprint [Par07a] and its extended version [Par07b] cite an argument from a remote part of my thesis [Par07c], namely Example 5.8. This argument turned out to be invalid, and it is not yet clear whether the results based on it are still correct in full generality. owever, the present note fixes this hole at least in an important case. More precisely, a major claim of the abovementioned preprints was that the Bost Conjecture for a locally compact group G passes down to closed subgroups of G. From the results of the present note, this can be deduced if G is a discrete group or, more generally, if is an open subgroup of G. 1 The problem Let G be a locally compact ausdorff group and let be a closed subgroup. Let B be a nondegenerate -Banach algebra, i.e., let the non-degenerate Banach algebra B carry a strongly continuous -action by isometric algebra isomorphisms. We define Ind G B to be the following G-Banach algebra: The underlying Banach algebra is Ind G B := {f : G B : g G h : fgh 1 ) = hfg) and fg) 0 if g }; the G-action on Ind G B is given by gf)g ) = fg 1 g ) for all g, g G and f Ind G B. As for every G-Banach algebra, we can form the convolution algebra L 1 G, Ind G B). The aim of this note is to compare this algebra to the algebra L1, B). More precisely, we ask the following questions: Question 1.1. Is L 1 G, Ind G B) Morita equivalent to L1, B)? Question 1.2. Do L 1 G, Ind G B) and L1, B) have the same K-theory? As a matter of fact, the Banach algebra L 1, B) is Morita equivalent to the Banach algebra L 1 G G/, Ind G G/ B) as shown in [Par07c], the crucial point being that the groupoids G G/ and are equivalent. By G := G G/ we denote the transformation groupoid of the left action of G on G/, which is the locally compact ausdorff space G = G G/ together with the structure maps r : G G/, g, g ) g and s: G G/, g, g ) g 1 g and µ: G G/ G G, g, g ), g, g )) gg, g ). 1

2 We also recall the definition of the algebra L 1 G, Ind G B): Firstly, the Banach algebra IndG B is not really a Banach algebra, but a G-Banach algebra, i.e., an upper semi-continuous field of Banach algebras over the unit space G/ of G = G G/. One way to obtain it is to equip the Banach algebra Ind G B described above with the C 0G/)-action χf)g) = χg)fg) for all g G, χ C 0 G/) and f Ind G B; one can now identify the G-C 0G/)-Banach algebra Ind G B with a G G/-Banach algebra which we then call Ind G B. Alternatively, we can define the fibres of Ind G B directly: If g G, then the fibre of IndG B over g G/ is given by Ind G B ) g = {f : g B : h : fgh 1 ) = hfg)}. So every fibre of Ind G B is isomorphic to B, but not canonically. The norm of some f Ind G B) g is just fg) B, because the -action on B is isometric. Now the algebra L 1 G, Ind G B) is defined to be the completion of the sections with compact support Γ c G, r Ind G B) of the pulled-back algebra r Ind G B for the norm ξ L 1 G) := sup ξg, g ) dg g G for all ξ Γ c G, r Ind G B). ere we have identified r 1 g ) G G/ with G noting that the measure on r 1 g ) which we did not yet specify, can be chosen to be aar measure on G under this identification). The norm ξg, g ) appearing in the formula is the norm of the fibre of Ind G B over g. Note that Γ c G, r Ind G B) can be identified into C cg, Ind G B) and hence into L1 G, Ind G B) by sending a function ξ to g g ξg, g )g )). owever, the norm of ξ in both pictures is not necessarily the same: in L 1 G, Ind G B) the norm of ξ is ξ L 1 G) = sup g G ξg, g ) dg. We have ξ L 1 G) ξ L 1 G). Since Γ c G, r Ind G B) is dense in L1 G, Ind G B), we obtain a canonical norm-decreasing homomorphism ψ from L 1 G, Ind G B) to L1 G, Ind G B). So we have the following diagram 1) L 1 G, Ind G B) ψ L 1 G, Ind G B) L 1, B). One cannot expect L 1 G, Ind G B) to be Morita equivalent to L1, B) because the canonical Morita equivalence is already a Morita equivalence between the algebras L 1 G, Ind G B) and L1, B), and the L 1 G, Ind G B)-valued inner product on it does not have to be ψl1 G, Ind G B))-valued. It is not clear whether one can modify the canonical Morita equivalence so that the inner product takes its values in L 1 G, Ind G B). So the answer to Question 1.1 is probably negative. Nevertheless, Question 1.2 can still have a positive answer: Because Morita equivalence of nondegenerate Banach algebras induces isomorphisms in K-theory, we have a diagram 2) K L 1 G, Ind G B)) K ψ) K L 1 G, Ind G B)) = K L 1, B) ). So to answer Question 1.2 it seems advisable to ask: Question 1.3. Is the homomorphism K ψ) an isomorphism? 2

3 Example 5.8 of [Par07c] asserts that this is indeed the case, but the argument given there is only valid for compact. Down below we give a positive answer for discrete G/. We conclude this section by considering the case of trivial coefficients, i.e., B = C with the trivial -action. Then Ind G C can be identified with C 0G/). ence L 1 G, Ind G C) is L1 G, C 0 G/)). This is a completion of C c G G/). On the other hand, Ind G C is the trivial continuous field over G/ with fibre C. ence L 1 G, Ind G C) is L1 G), the L 1 -algebra of the groupoid G = G G/. Just as L 1 G, C 0 G/)), it is also a completion of C c G G/) but for a different norm. In this case, the diagrams 1) and 2) reduce to L 1 G, C 0 G/)) ψ L 1 G) L 1 ). and K L 1 G, C 0 G/)) ) K ψ) K L 1 G) ) = K L 1 ) ). 2 A more general setting As above, let G be a locally compact ausdorff group. Let X be a locally compact ausdorff space on which G acts continuously from the left in the first section of this note we have considered the case X = G/, where is a closed subgroup of G). Let G X be the transformation groupoid of this action and let C be a non-degenerate G X-Banach algebra above, we have considered the G G/-Banach algebra C = Ind G G/ B). The space of sections of C vanishing at infinity is denoted by Γ 0 X, C). It is a C 0 X)-Banach algebra and comes with a compatible G-action. Remark 2.1. Note that there are two viewpoints on G X-Banach algebras: We can consider them as upper semi-continuous fields of Banach algebras over X with an action of the groupoid G X) or as C 0 X)-Banach algebras which carry a compatible G-action). The two viewpoints are very close, though not equivalent: The upper semi-continuous fields of Banach algebras over X correspond via the map C Γ 0 X, C) to the C 0 X)-Banach algebras which satisfy a regularity property, namely local C 0 X)-convexity see [Par07c], for example). In accordance to [Laf06], we consider G X- Banach algebras as upper semi-continuous fields. Remark 2.2. In [CEOO03] it is shown, if C is a G X-C -algebra, that there is an isomorphism C r G X) = Γ 0 X, C) r G and that the following diagram commutes 3) K top G X; C) K top = K C r G X)) G; Γ 0 X, B)) K Γ 0 X, C) r G) where the horizontal arrows are the respective Baum-Connes assembly maps. Note that it is a nontrivial fact that the vertical arrow on the left-hand side is an isomorphism actually, this is the main result of [CEOO03]). We now consider the right half of diagram 3) after replacing the reduced crossed products with the corresponding L 1 -algebras, i.e., we compare the K-theories of the two algebras L 1 G, Γ 0 X, C)) and = 3

4 L 1 G X, C). Both algebras can be defined as completions of Γ c G X, r C); the first algebra is the completion for the norm ξ L 1 G) = sup ξg, x) dg, ξ Γ c G X, r C), x X the second algebra is the completion for the norm ξ L 1 G X) = sup ξg, x) dg, x X ξ Γ c G X, r C). We hence have a norm-decreasing homomorphism ψ from L 1 G, Γ 0 X, C)) to L 1 G X, C). In general, it will not be an isomorphism but, as above, it makes sense to ask: Question 2.3. Is K L 1 G, Γ 0 X, C))) K ψ) K L 1 G X, C)) an isomorphism? Note that Question 2.3 reduces to Question 1.3 if X = G/ and C = Ind G G/ B. 3 A solution for discrete X The method that we use to show that ψ is an isomorphism in K-theory is to produce a dense hereditary subalgebra of L 1 G, Γ 0 X, C)) which is also dense and) hereditary in L 1 G X, C). A first attempt would be Γ c G X, r C), but this seems to work only if G X is a proper groupoid; in the case that X = G/ this means that is compact, and this is a rather uninteresting case in the framework of [Par07b]. If X is discrete, we can easily construct an algebra which is larger than Γ c G X, r C) and hence dense in both completions), but which is also hereditary. It is not clear whether a similar idea works for general X. Assume from now on that X is discrete. For each subset M of X, define Note that A M is a right ideal of Γ c G X, r C). A M := {ξ Γ c G X, r C) : rsupp ξ) M}. Lemma 3.1. If M is a finite set, then the norm on A M which is inherited from L 1 G X, C) is equivalent to the norm on A M which is inherited from L 1 G, Γ 0 X, C)). Proof. Let ξ A M. Then ξ L 1 G X) ξ L 1 G) = sup ξg, x) dg x X = sup ξg, m) dg ξg, m) dg m M m M = ξg, m) dg M sup m M m M = M ξ L 1 G X). ξg, m) dg 4

5 It follows from this lemma that ψ maps the closure A M of A M in L 1 G, Γ 0 X, C)) bijectively and bicontinuously onto the closure of A M in L 1 G X, C) which we also call A M ). Note that A M is a right ideal of both, L 1 G, Γ 0 X, C)) and L 1 G X, C), because A M is a right ideal of Γ c G X, r C). If M and N are finite subsets of X with M N, then A M A N and hence A M A N. Define A := A M L 1 G, Γ 0 X, C)). M X finite Note that ψ is injective on A, so we can think of A as a subspace also of L 1 G X, C). Indeed, A is a linear subspace of both algebras, and because all the A M are right ideals, also A is a right ideal in both completions). Moreover, the union of all A M contains Γ c G X, r C), so A is dense in L 1 G, Γ 0 X, C)) and L 1 G X, C). In particular, A is a dense hereditary subalgebra of both Banach algebras. By Lemme in [Laf02], this means that ψ is an isomorphism in K-theory. So we have shown: Proposition 3.2. If X is discrete, then ψ is an isomorphism in K-theory. Corollary 3.3. If is an open subgroup of G, then G/ is discrete and K L 1 G, Ind G B)) is hence isomorphic to K L 1, B)). And using the results of [Par07b], we can conclude: Corollary 3.4. If G is a locally compact ausdorff group for which the Bost conjecture with C - algebra coefficients is true, then the Bost conjecture with C -algebra coefficients is true for every open subgroup of G. 4 A short look at the general case If X is not discrete, then the above argument is no longer valid. The crucial point is that, for the argument to work, the L 1 -norm and the L -norm for the continuous functions on any given compact subset of X have to be equivalent; this is a very restrictive condition on X. An instructive counterexample is G = R, = Z and X = G/ = S 1. It is easy to construct a continuous function f on G X = R S 1 which vanishes at infinity such that sup x S 1 t R ft, x) dt < and t R sup x S1 ft, x) dt =. owever, there might be a different way to construct a dense hereditary subalgebra of L 1 G X) which is also contained in L 1 G, C 0 X)). For example, the algebra A := C c G X) L 1 G X)C c G X) is certainly dense and hereditary in L 1 G X); the question is whether A is also contained in L 1 G, C 0 X)). More precisely, it is easy to see that the elements of A are indeed continuous functions on G X which vanish at infinity), so it makes sense to ask whether, for all f A, the norm t R sup x S1 ft, x) dt is finite. The intuitive idea behind this is that the continuous counterexamples f that come to mind in the case G = R and X = S 1 are constructed by the use of an infinite series of little bumps that become steeper and steeper, and maybe such an f cannot lie in A because the convolution product prevents bumps from becoming too steep. But this vague idea still has to be put work. 5

6 References [CEOO03] Jérôme Chabert, Siegfried Echterhoff, and ervé Oyono-Oyono. Shapiro s lemma for topological K-theory of groups. Comment. Math. elv., 781): , [Laf02] Vincent Lafforgue. K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes. Invent. Math., 149:1 95, [Laf06] Vincent Lafforgue. K-théorie bivariante pour les algèbres de Banach, groupoïdes et conjecture de Baum-Connes. Avec un appendice d ervé Oyono-Oyono. J. Inst. Math. Jussieu, Published online by Cambridge University Press 28 Nov [Par07a] Walther Paravicini. The Bost Conjecture and Closed Subgroups. Preprintreihe SFB Geometrische Strukturen in der Mathematik, 469, [Par07b] [Par07c] Walther Paravicini. Induction for Banach algebras, groupoids and KK ban. Preprintreihe SFB Geometrische Strukturen in der Mathematik, 478, Walther Paravicini. KK-Theory for Banach Algebras And Proper Groupoids. PhD thesis, Universität Münster,