PERMANENT WEAK AMENABILITY OF GROUP ALGEBRAS OF FREE GROUPS

Transcription

1 PERMANENT WEAK AMENABILITY OF GROUP ALGEBRAS OF FREE GROUPS B. E. JOHNSON ABSTRACT We show that all derivations fro the group algebra (G) of a free group into its nth dual, where n is a positive even integer, are inner. Cobined with the known result for odd values of n, this shows that these algebras are peranently weakly aenable. 1. Introduction and notation Dales, Ghahraani and Grønbæk in [1] have defined a Banach algebra A to be peranently weakly aenable if every derivation fro A into its nth dual An is inner for n 1, 2,. They raised the question of whether the group algebra A ( )of the free group on two generators has this property, and the purpose of this paper is to show that it does (Theore 3.1). In [1] they covered the case in which n is odd, so we need consider only even values of n. Note that by [3, Theore 4], the result which we are trying to prove holds for n 0. We solve the proble by transposing it to another setting. If X is a topological space and G is a (discrete) group, then we say that X is a G space if the product gx is defined for all g in G and x in X in such a way that (gh)x g(hx)(g, h G, x X) and x gx is a hoeoorphis of X onto X for each g in G. A ap α fro one G space X to another G space Y is a G orphis if it is continuous and α(gx) gα(x)(gg, x X). If X is locally copact, then putting ( fg)(x) f(gx) (f C (X), g G, x X) gives a right action of G on C (X), and the adjoint action fdgµ f(gx) dµ(x) akes C (X)M(X) a left G odule. A crossed hooorphis φ is a ap fro G to M(X) with φ(gh) φ(g)gφ(h) (g, h G). It is principal if there is µ M(X) with φ(g) gµµ(g G). We denote the space of bounded crossed hooorphiss (that is, those with φ sup φ(g): g G ) odulo the principal ones by H(G, M(X)), so that saying H(G, M(X)) 0 is a quick way of saying that every bounded crossed hooorphis is principal. If is a Banach (G) biodule, then any continuous derivation D into is deterined by the eleents D(g) of, where we write g for the probability easure supported at g. Putting φ(g) g D(g), the derivation law for D shows that φ(gh) φ(g)gφ(h)g ; that is, φ is a crossed hooorphis if we consider G as acting on by g ξ gξg (g G, ξ ). It is easy to check that D is inner if and only if φ is principal. In ters of G spaces, for (G), we are considering X G and G acting on X by conjugation; the question of whether every continuous derivation is inner is equivalent to asking whether H(G, (G)) 0. For convenience, we write A for (G). Because the second dual of a coutative unital C*-algebra is a coutative unital C*-algebra, for each non-negative integer Received 7 May Matheatics Subject Classification 46M20, 43A20. Bull. London Math. Soc. 31 (1999)

2 570 B. E. JOHNSON there is a copact space X() with X() βg and C(X()) A(+). This gives A(+) M(X()). Thus if we apply the above with M(X()), where G acts on G by conjugation, and lift this action to M(X()), then the question of whether all derivations fro (G) to((g))(+) are inner is equivalent to asking whether H(G, M(X())) 0. In Section 2 we shall prove that in certain circustances we always have H(G, M(X)) 0. In Section 3 we see how when G, the result in Section 2 applies when we take X as the space of words of infinite length in the generators, and show how this gives the ain result of the paper. Finally, in Section 4 we show that the result applies to all free groups. 2. Nowhere totally recurrent actions DEFINITION 2.1. Let X be a G space. We say that the action of G on X is nowhere totally recurrent (NTR) if there is a partition X B B r of X into a finite nuber of disjoint G-invariant Borel sets, and eleents a,, a r of G such that for j 1,, r and x B j, the sequence an j x n= has no point of accuulation in B j ; that is, there is no y B j with an j x N for an infinite nuber of values of n, for each neighbourhood N of y. The usefulness of this definition coes fro the following two results. LEMMA 2.2. Let X and Y be G spaces, and let α be a G space orphis fro Y to X. If the action of G on X is NTR, then the sae is true for Y. Proof. Take the sae a,, a r as for X, and the Borel sets α (B ),, α (B r ). THEOREM 2.3. Let X be a locally copact G space. If the action of G on X is NTR, then H(G, M(X)) 0. Proof. Let φ be a bounded crossed hooorphis fro G into M(X). We shall say that a easure ν is concentrated on a Borel subset B of X if ν(xb) 0. Suppose that the values of φ are concentrated on B. By the aenability of, there is µ M(X) with φ(an) anµµ (n ). Multiplying µ by the characteristic function of B, we can assue that, in addition, µ is concentrated on B, so putting ψ(g) φ(g)(gµµ), ψ is a bounded crossed hooorphis fro G to M(X) with values concentrated on B, and ψ(an) 0(n). We shall prove that ψ 0. Suppose not. Then Q supψ(g): g G is positive, so there is g G with ψ(g) 0.9Q, and a copact subset K of B with ψ(g)(k) 0.9Q and g ψ(g)(k) 0.9Q (because g ψ(g) ψ(g) 0.9Q). If k Ka n K, then k K and a nk K. Ifk li sup Ka nk, then k K and there are infinitely any positive integers n with ank K.AsKis copact, these points would have a point of accuulation in K. By hypothesis, this does not happen, so li sup Ka n K. Hence ψ(g)(ka nk) 0asn, and we can choose n with ψ(g)(ka nk) 0.9Q. Now consider ψ(g)a n g ψ(g) a n g (ψ(g)ganψ(g)) ψ(gang) Q.

3 PERMANENT WEAK AMENABILITY OF GROUP ALGEBRAS OF FREE GROUPS 571 We have ψ(g)(ka n K) 0.9Q and a n g ψ(g)(a n K) g ψ(g)(k) 0.9Q, so ψ(g)(a n K) 0.1Q and a n g ψ(g)(ka nk) 0.1Q. Thus ψ(g)a n g ψ(g)(ka nk) 0.8Q, ψ(g)a n g ψ(g)(a nk) 0.8Q, which shows that ψ(g)a n g ψ(g) 1.6Q, a contradiction. To coplete the proof, note that any easure on X is the su of r easures, one concentrated on each of the B j. This decoposes any bounded crossed hooorphis into the su of r crossed hooorphiss to which we can apply the above arguent. It would be interesting to know if this result still holds if we consider decopositions of X into countably any G-invariant Borel sets. 3. Peranent weak aenability of ( ) Before we consider the action of on β etc., we consider a sipler space. Let a and b be the two generators of the free group. Let be the set of infinite strings, where each i a, b, a, b, and for all i, i+. Thus consists of i the reduced infinite words in a, b, a and b. It is a closed subset of the copact space a, b, a, b, and so is a copact topological space. The action of on is given by taking g (g, ) to be the eleent of obtained by juxtaposing g and and reducing the resulting infinite word. It is straightforward to check that acts on by hoeoorphiss. The eleent a has two fixed points in its action on, naely a aaa and a a a a. If and a, then an a as n. We take B ga : g ga : g, a countable set, and hence a Borel set, and B B. With a b and a a, we see that the action is NTR because for B the only possible points of accuulation of bn n= are b and b, which are not in B. THEOREM 3.1. The algebra ( ) is peranently weakly aenable. As explained in Section 1, to prove this we need only show that H(, M(X())) 0 for 0, 1, 2,. The canonical injection fro A( ) to A(+) is a unital injection J of C(X( )) into C(X()). For x X() and f C(X( )), f (J f )(x) is a ultiplicative linear functional on C(X( )) which takes the value 1 at the identity, so there is an eleent κ (x)ofx( ) with f(κ (x)) (J f )(x). If we identify eleents of X() with ultiplicative linear functionals, then κ is just the restriction of J to this subset of A(+). This shows that κ is continuous, and it is surjective because all ultiplicative linear functionals on a Banach algebra extend by weak* continuity to ultiplicative linear functionals on the second dual. We need to show that if x is an isolated point in X( ), then κ (x) consists of a single isolated point in X (). Let e be the idepotent in C(X( )) corresponding to x, and let φ be the linear functional given by evaluation at x. Thus for a C(X( )), we have ae φ(a)e. Taking weak* liits, this identity holds for all a in C(X()), so J e is a inial non-zero idepotent there too. Because κ is surjective, κ (x) is not epty, and if x, x κ (x), then (J e)(x ) J (e)(x ). Thus, by iniality, x x,soκ (x) is a copact open set consisting of only one point.

4 572 B. E. JOHNSON Put λ κ κ κ. Then λ is a continuous ap of X() onto X(). For any isolated point x of X(), λ (x) is a single isolated point of X (). Put Y(), Z() β, Y() λ Y (), Z() λ Z (). Then each Y() is open, each Z() is copact, X() is the disjoint union of Y() and Z(), the spaces Y() and Z() are closed under the left and right actions of, and hence under the conjugation action, and λ is a bijection of Y() onto Y(). To show that H(, M(X())) 0, we need only show that H(, M(Y())) H(, M( )) 0, which holds by [3, Theore 4], and that H(, M(Z())) 0, which will follow using Theore 2.3 if we show that the action of on Z() is NTR. Using Lea 2.2 and the aps λ,we need show this only for 0, and using Lea 2.2 again, this will follow if we construct a G ap α fro Z() β to, because the action of G on is NTR. Define a relation for k and or by k if k k k p p where, in reduced for, k k k p and or q with q p. For and p a positive integer, put N(, p) h:h, p h. We define N(, p) in the sae way if and q in reduced for with q p. Let w β. By the definition of an ultrafilter and the fact that for fixed p the sets N(, p) are disjoint, there is exactly one word of length p with N(, p) w. If is the word of length p1 with N(, p1) w, then, because otherwise the sets N(, p) and N(, p1) would be disjoint. Taking these choices for all p and putting the together, we find that there is exactly one in with N(, p) w for all p. The ap α is defined by α(w). It is straightforward to see that α is continuous. It is a G ap because N(a, p1) an(, p) ( N(a, p1) if a, if a, N(, p) a N(, p) p, so α(aw) aα(w) and α(wa) α(w), where acts on and hence β by left and right ultiplication. This gives α(awa ) aα(w), and a siilar result for b,soα is an ap when acts on β by conjugation. We have actually proved a little ore than peranent weak aenability. For each positive integer n, the ap δ which sends an eleent of A(n) to the inner derivation it generates is surjective, so there is a constant K n such that for each derivation D fro A to A(n), there is an eleent f of A(n) with δf D and f K n. We have shown that K n is bounded as n. 4. Peranent weak aenability for other free groups It is easy to see how the proof we have given extends to any finitely generated free group. With a little ore work, it can be extended to any free group. THEOREM 4.1. Let G be a free group. Then (G) is peranently weakly aenable. Proof. We need to consider only the case in which G has an infinite set of generators. Let H be the subgroup generated by a finite subset F of the generators. As we have seen, we consider G acting on itself by conjugation. Writing G H(GH), we see that the only eleent of H which coutes with any eleent of GH is e. SoH acts freely on GH and, choosing a transversal T for this action, that is, choosing one eleent fro each of the H orbits in GH, we see that, as an H space, GH is isoorphic to HT where H acts on the first factor by left ultiplication

5 PERMANENT WEAK AMENABILITY OF GROUP ALGEBRAS OF FREE GROUPS 573 and trivially on the second. Projection onto the first factor is thus an H space hoeoorphis fro GH with H acting by conjugation into H with H acting by left ultiplication, which extends to an H space hoeoorphis λ fro β(gh) to βh. The action of H on βh is NTR because, as we have seen, this is true for the action on βhh, and it is obviously true for the action on the discrete space H because there are no finite orbits. Thus if φ is a crossed hooorphis fro H into (GH) M(β(GH)), then φ is principal, by Lea 2.2 and Theore 2.3. The sae holds for any of the even duals (GH)() C(X ), where the copact space X is defined by this equation, because the canonical inclusions C(X ) C(X ) induces H space hoeoorphiss + fro X + to X, showing that the X are NTR just as in Theore 3.1. Thus (G)() (H)() (GH)(), and any bounded crossed hooorphis fro H into (G)() is principal because this is true for the suands. Suppose now that φ is a bounded crossed hooorphis fro G into (G)(). Restricting φ to H, we have shown that there is an eleent µ F of (G)() with φ(h) hµ F µ (h H). We need soe control over µ F. Writing φ supφ(g):g G, we can see that when we apply Theore 2.3, each of the µ i has µ i φ, so the resulting easure µ has µ rφ. In the proof that is NTR, we had r 2 (note that this does not increase if we have soe other finite nuber of generators), so for β(gh), r 3. For βhh, r 2, and for H, where we used [3, Theore 4] to obtain µ, we have µ φ. Thus µ F 6φ. The µ F,asF ranges over finite subsets of the set of generators of G, for a bounded net which has a weak* accuulation point µ (using the topology on the dual of C(X )). The identity φ(h) hµ F µ F, which holds if h is in the group generated by F, shows that φ(h) hµµ for all h in G. References 1. H. G. DALES,F. GHAHRAMANI and N. GRØNBÆK, Derivations into iterated duals of Banach algebras, Studia Math. 128 (1998) B. E. JOHNSON, Cohoology in Banach algebras, Me. Aer. Math. Soc. 127 (1970). 3. B. E. JOHNSON and J. R. RINGROSE, Derivations of operator algebras and discrete group algebras. Bull. London Math. Soc. 1 (1969) Departent of Matheatics University of Newcastle Newcastle upon Tyne NE1 7RU