Band-dominated Fredholm Operators on Discrete Groups

Transcription

1 Integr. equ. oper. theory 51 (2005), X/ , DOI /s c 2005 Birkhäuser Verlag Basel/Switzerland Integral Equations and Operator Theory Band-dominated Fredholm Operators on Discrete Groups John Roe 1. Introduction In [8] the authors introduce a class of operators on the Hilbert space l 2 (Z n )which they call band-dominated operators, and they construct a symbol calculus which provides a theoretical answer to the question whether or not a band-dominated operator is Fredholm. (The paper [8] also contains calculations for l p, p 2,which will not be discussed here.) The purpose of this note is to extend the main results of the symbol calculus of [8], in the Hilbert space case. We shall replace the group Z n by any (finitely generated) discrete group Γ which is exact in the sense of C -algebra theory, see [12]. The algebra of band-dominated operators is known to workers in coarse geometry as the translation C -algebra, and Skandalis, Tu and Yu showed that it is in fact the reduced C -algebra of a certain groupoid [11]. In the present context, this groupoid is amenable, and this gives a natural exact sequence associated to an open subgroupoid [5]. We conclude this note by relating the symbol calculus for band-dominated operators to this exact sequence. In a later paper we hope to discuss index theorems (along the lines of [9]) associated to the symbol calculus for band-dominated operators on Γ. 2. Basic definitions and properties Let Γ be a finitely generated discrete group. Equip it with a proper metric d which is left translation invariant, that is d(γ 1,γ 2 )=d(γγ 1,γγ 2 ). All such metrics are coarsely equivalent and explicit examples may be constructed by defining d(γ 1,γ 2 ) as the word-length of γ1 1 γ 2 with respect to a specific generating set. Let L γ and R γ be the unitary operators on l 2 (Γ) induced by left and right multiplication by γ Γ. The support of the National Science Foundation, grant DMS , is gratefully acknowledged.

2 412 Roe IEOT Definition 2.1. The uniform translation C -algebra or algebra of band-dominated operators associated to Γ is the C -algebra of operators on l 2 (Γ) generated by the unitaries R γ and the diagonal matrices l (Γ). In coarse geometry this algebra is denoted by C u( Γ ). In this paper we will denote it simply by A(Γ) or just A. For similar reasons we will usually denote l 2 (Γ) simply by H. Remark 2.2. One can define A using only the coarse-geometric structure of Γ. Indeed, consider a bi-infinite matrix parameterized by Γ Γ. One says that the matrix has finite propagation or finite bandwidth if there is a constant R>0such that its matrix entries a γγ vanish whenever d(γ,γ ) >R.(Thepropagation itself is the least R that has this property.) The bounded, finite propagation matrices form a -algebra of bounded operators on H and the norm closure of this algebra is A. Remark 2.3. It is a well-known fact, used in [9], that A is in fact the reduced crossed product C -algebra l (Γ) Γ. See [10] for the details of the proof. Notice that the unitaries L γ conjugate A to itself. Lemma 2.4. Given k, R > 0, lets k,r A denote the collection of operators T with norm T k and propagation Prop(T ) R. ThenS k,r is -strongly compact. Proof. Since the closed unit ball of B(H) is weakly compact, and the propagation is upper semicontinuous with respect to the weak topology, it is enough to show that the weak and the -strong topologies agree on S k,r. In fact, both topologies agree on this set with the topology of pointwise convergence of matrix entries. To see this, suppose that T α is a net in S k,r whose matrix entries converge pointwise to the matrix entries of T. We can write each T α as a sum γ F f αr γ γ, where the fα γ are l functions and the summation is over a fixed finite set F of group elements. The l functions fα γ converge pointwise and uniformly boundedly to f γ,wheret = γ F f γ R γ, so the corresponding operators converge -strongly. Since the sums appearing here are finite, we are done. Remark 2.5. This lemma remains true for any bounded geometry coarse space; one replaces the appeal to group structure by a standard decomposition argument in terms of partial translations (compare [10, Lemma 4.10]). Corollary 2.6. Let T be a band-dominated operator (that is, T A). Then the -strong closure of the set {L γ TL γ : γ Γ} B(H) is -strongly compact and is contained in A. Proof. It suffices to consider the case when T has finite propagation, R say. Let k = T. Then the set {L γ TL γ : γ Γ} B(H) is contained in S k,r.

3 Vol. 51 (2005) Band-dominated Fredholm Operators 413 Let βγ be the Stone- Cech compactification of Γ (the maximal ideal space of l (Γ))andlet Γ =βγ \ Γ. We identify points of Γ with free ultrafilters of subsets of Γ. By the universal property of the Stone- Cech compactification, any map from Γ to a compact Hausdorff space K extends uniquely to a continuous map from βγ to K. In particular, for each fixed T A the map γ L γ TL γ, Γ A has range in a -strongly compact set (by Corollary 2.6) and therefore extends uniquely to a -strongly continuous map Γ A. (2.1) Definition 2.7. Let T A. The -strongly continuous map defined by equation 2.1 above is called the symbol of T and denoted by σ(t ). Remark 2.8. Any -strongly continuous map from Γ (or any compact Hausdorff space) to B(H) is norm bounded. Indeed, if ω T ω is such a map, then for any v H the set {T ω v : ω Γ} is a compact, and hence bounded, subset of H. Thustheset{T ω } of maps H H is pointwise bounded, and therefore it is uniformly bounded by the Banach-Steinhaus theorem. It follows easily that the set of -strongly continuous maps from Γ tob(h), or to a C -subalgebra such as A B(H), is itself a C -algebra under the supremum norm. Definition 2.9. We will use the notation C s ( Γ; A) todenotethec -algebra of -strongly continuous maps Γ A. For ω Γ weuset ω to denote σ(t )(ω), and we call it the limit operator of T at ω (compare [8]). Using the facts that addition, multiplication and adjunction are -strongly continuous on the unit ball of B(H), one easily sees that T T ω is a -homomorphism A A. Thusthesymbolmapisa -homomorphism σ : A C s ( Γ; A). (2.2) What is the kernel of this homomorphism? To answer this we use the following definition, due to Yu. Definition An operator T A is a ghost if its matrix entries tend to zero at infinity, that is, for every ɛ>0there is a finite subset F Γ Γ with Te γ,e γ < ɛ whenever (γ,γ ) / F. Here e γ H denotes the characteristic function of the set {γ}. Proposition The kernel of the symbol homomorphism consists exactly of the ghosts in A. Proof. Suppose that σ(t ) = 0. Then the function γ L γ TL γ tends to zero ( strongly and therefore weakly) at infinity in Γ, which implies that for each fixed γ the matrix entries Te γ,e γγ (those along the diagonal defined by γ )tendto zero as γ.thus,givenɛ>0, T has only finitely many matrix entries greater

4 414 Roe IEOT than ɛ (in absolute value) on any diagonal; and, because of finite propagation, all but finitely many diagonals have all entries less than ɛ in absolute value. Thus T is a ghost. Conversely, suppose that T is a ghost. Then for any ɛ>0onecanwrite T = T 1 + T 2,whereT 1 hasnormlessthanɛand T 2 is of finite propagation and has all but finitely many matrix entries less than ɛ in absolute value. Therefore σ(t 1 )andσ(t 2 ) have all matrix entries less than ɛ in absolute value, so σ(t )has all matrix entries less than 2ɛ in absolute value. Since ɛ is arbitrary, σ(t )=0. 3. Exactness and Fredholm properties Recall that the reduced C -algebra of a discrete group Γ is the C -algebra Cr (Γ) generated by the translations R γ on l 2 (Γ). A group Γ is said to be exact if Cr (Γ) is an exact C -algebra, that is, if the minimal tensor product with Cr (Γ) is an exact functor. See [12] for general information on exact C -algebras. The following theorem is a compilation of results from [4, 3, 6, 7]. Theorem 3.1. The following properties of a finitely generated discrete group Γ are equivalent: (a) Γ is exact; (b) The action of Γ on Γ is topologically amenable [1]; (c) There exists a topologically amenable action of Γ on some compact Hausdorff space; (d) Thecoarsespace Γ has the property A of [13]. Amenable groups and hyperbolic groups, for example, are exact [1]. No explicit example of an inexact group is known, although on probabilistic grounds such groups must exist [2]. The following result is well-known. Lemma 3.2. If a bounded geometry coarse space X has property A, then an operator T A(X) is a ghost if and only if it is compact. Proof. The proof, which uses a characterization of property A in terms of positive definite kernels, is written up in [10, Proposition 11.43]. Thus from Proposition 2.11 we obtain Proposition 3.3. For an exact group Γ the kernel of the symbol homomorphism 2.2 is the C -algebra of compact operators. It follows of course that an operator T A(Γ) is Fredholm if and only if its symbol is invertible in the C -algebra C s ( Γ; A(Γ)). We obtain therefore the following result which generalizes the main theorem of [8]. Theorem 3.4. Let Γ be an exact group. For T A(Γ) the following are equivalent: (a) T is Fredholm;

5 Vol. 51 (2005) Band-dominated Fredholm Operators 415 (b) σ(t ) is invertible in C s ( Γ; A(Γ)); (c) Each limit operator T ω is invertible, and the norms Tω 1 are uniformly bounded. Proof. All we need to show is that (c) implies (b). The formula S 1 T 1 = S 1 (T S)T 1, together with the -strong continuity of multiplication on the unit ball, shows that if ω T ω is -strongly continuous and the norms Tω 1 are uniformly bounded, then ω Tω 1 is -strongly continuous also; this is what is required. 4. Relation to groupoid algebras The symbol map that we have defined is not surjective. In fact, symbols have a certain equivariance property. Note that Γ acts on Γ by left translation. Relative to this action, one has T gω = L g T ω L g, (4.1) as is easily checked from the definition. As a consequence of this equivariance property, one sees that σ(t ) is completely determined by the functions g γ : ω T ω e 0,e γ (4.2) defined on Γ. Now the theory of [11] associates to each coarse space a groupoid G, insucha way that the uniform C -algebra of the coarse space is just the reduced C -algebra of the groupoid. In the case at hand (of a discrete group), the groupoid G is the crossed product βγ Γ. There is an open subgroupoid G 0 corresponding to the Γ-invariant open subset Γ βγ, and there is a complementary closed subgroupoid G 1 = Γ Γ. If, as we are assuming, Γ is an exact group, then we obtain an exact sequence of groupoid C -algebras 0 C r (G 0 )=K C r (G) =A C r (G 1 ) 0. (See [5] for the relevance of the potential inexactness of this sequence for other groups to the failure of some versions of the Baum-Connes conjecture.) We therefore see that our symbol σ(t ) must be determined by the image of T in C r (G 1 )= C( Γ) Γ. In fact, one easily sees that if the image of T in C( Γ) Γ is written as a formal sum gγ [γ], then the g γ appearing here are the same as those appearing in equation 4.2 above.

6 416 Roe IEOT References [1] C. Anantharaman-Delaroche and J. Renault. Amenable groupoids, volume 36 of Monographies de L Enseignement Mathématique [Monographs of L Enseignement Mathématique]. L Enseignement Mathématique, Geneva, With a foreword by Georges Skandalis and Appendix B by E. Germain. [2] M. Gromov. Spaces and questions. Geom. Funct. Anal., Special Volume, Part I: , GAFA 2000 (Tel Aviv, 1999). [3] Erik Guentner and Jerome Kaminker. Addendum to: Exactness and the Novikov conjecture. Topology, 41(2): , [4] Erik Guentner and Jerome Kaminker. Exactness and the Novikov conjecture. Topology, 41(2): , [5] N. Higson, V. Lafforgue, and G. Skandalis. Counterexamples to the Baum-Connes conjecture. Geom. Funct. Anal., 12(2): , [6] N. Higson and J. Roe. Amenable group actions and the Novikov conjecture. Journal fur die reine und angewandte Mathematik, 519: , [7] Narutaka Ozawa. Amenable actions and exactness for discrete groups. C. R. Acad. Sci. Paris Sér. I Math., 330(8): , [8] V. Rabinovich, S. Roch, and B. Silbermann. Fredholm theory and finite section method for band-dominated operators. Integral Equations and Operator Theory, 30: , [9] Vladimir Rabinovich, Steffen Roch, and John Roe. Fredholm indices of banddominated operators. To appear in Integral Equations and Operator Theory, [10] J. Roe. Lectures on Coarse Geometry. American Mathematical Society, [11] G. Skandalis, J.-L. Tu, and G. Yu. Coarse Baum Connes conjecture and groupoids. Topology, 41: , [12] S. Wassermann. Exact C -algebras and related topics. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, [13] G. Yu. The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Inventiones Mathematicae, 139: , John Roe Department of Mathematics Penn State University University Park, PA USA Submitted: March 5, 2004