Weakly Compact Groups of Operators

It is shown that the weakly closed algebra generated by a weakly compact group of operators on a Banach space is reflexive and equals its second commutant. Also, an example is given to show that the generator of a monothetic weakly compact group of operators need not have a logarithm in the algebra of all bounded linear operators on the underlying space. Let X be a complex Banach space, B(X) the algebra of all bounded linear operators on X, and I the identity operator on X. By a group in B(X) we shall mean a multiplicative group with unit I. The weak operator topology on B(X) is denoted by the letter w. Given a nonempty subset & of B(X), &' and &' denote the first and second commutants of fe, and A(ë) is the 222-closed subalgebra of B(X) generated by ë and I. The lattice of all ©-invariant closed subspaces of X is denoted by Lat fe, and Alg Lat g = ÍT £ B(X): T(L) C L (L e Lat &)]. A subalgebra A of B(X) is reflexive if Alg Lat A = A. It is clear that reflexive algebras are 222-closed and contain I. Finally, C, R, Z and T are the complex numbers, the reals, the integers and the unit circle. We present several results concerning 2f-compact groups in B(X). Such groups come within the general framework discussed by de Leeuw in [1], where the underlying space is called a G-space. The monothetic (singly generated) case has been considered in [4], [5], where an operator in B(X) generating a i^-compact group (with unit I) is called a G-operator. It was shown in [4] that, if ^¡ is a monothetic nv-compact group, then A(§) is reflexive and (-/' = A(C¿). In fact the methods developed there and in [5] can be extended to prove Theorem 1. Let § be an abelian w-compact group in B(X) (with unit I). Then A(§) is reflexive and §" = A(§). Received by the editors February 4, 1974. AUS (UOS) subject classifications (1970). Primary 47D10, 47A15, 46L20; Secondary 43A22, 43A05. Copyright © 1975, American Mathematical Society 78 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use WEAKLY COMPACT GROUPS OF OPERATORS 79 Problem A. Does Theorem 1 remain valid if the hypothesis that § be abelian is omitted? G-operators have occurred in the work of Ljubic [8], where a study is made of the spectral properties of an operator S e B(X) satisfying ||exp (i r S)\\ < M (r £ R) and a certain almost-periodic condition. (The space X is taken to be weakly sequentially complete.) It is shown that each such S has a total set of eigenvectors corresponding to real eigenvalues, from which it is easily seen (via [4, Theorem 1.2]) that exp (iS) is a G-operator. Problem ß. Can every G-operator be written as exp (iS) fot some bounded S? Solution A. Theorem 1 does indeed extend to the nonabelian case and we sketch the main ideas of the proof. Let § be a i^-compact group in B(X). Lemma 2 [2, Theorem 8.1]. X is the closed linear span of finite dimen sional \yinvariant subspaces. An easy consequence of this is the following description of Lat y (cf. Corollary 1.4 of [4]). Lemma 3. Each subspace in Lat \¡ is spanned by finite dimensional y -irreducible subspaces. Write X(n) for the direct sum of n copies of X and T(n) e B(X(n>) for the 72th direct sum of T. Putting §'"^ = [T^: T e §!, it is easy to see that §("' is a »-compact group in B(X(n)) with unit I^n\ Lemma 4. Let S e Alg Lat §. Then S(n) e Alg Lat §(n) for n = 1, 2, • • •. This is the key result and we sketch its proof. A straightforward argument reduces the proof to the case n = 2. It is then sufficient, by Lemma 3, to show that each finite dimensional §^ '-irreducible subspace M of X^ is 5^ '-invariant. Using irreducibility, this is easily done in the case when M contains (0, x) for some x 4 0. Suppose therefore that M contains no elements of this form. Then there is a finite dimensional subspace N of X and a linear operator U: N — X such that M = \(x, Ux): x e N\. The hypotheses on M imply that N and UN belong to Lat § and that (UT TU)(N) = |0! (T e §). Also, N is §-irreducible and thus either (i) U(N) = N, or (ii) U(N)nN = io!. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 80 T. A. GILLESPIE AND T. T. WEST In case (i), U commutes with the irreducible set of operators §|/V on N. Hence U is a scalar, from which it follows that M e Lat .V '. In case (ii), the subspace L (I + U)N is )-¡and hence S-invariant. Therefore, given x e N, there exists y e N such that Sx y = SUx + Uy. The lefthand side of this equation is in N and the right in U(N), since N and U(N) ate 5-invariant. Therefore both sides are zero, giving SUx USx (x e N), and hence M e Lat S^h A standard argument (cf. [9, Lemma 1]) now gives Theorem 5. A(§) is reflexive. Given A = [a..] 6 m (C), the n x n complex matrices, and x = (x ,. . . , x ) e X^"\ let Ax denote the element y = (y .. . , y ) in X(n) defined by y . = 2" ,<2. x.. Let M be a finite dimensional 6-invariant subspace of X 'r; = l27 7 " r with basis \u,,---,u \. Given T e Q, let T22. = X" , a(T) = [a..(T)] is an anti-representation of §^în„(C). Define the operator P in B(X(n)) by Px= f a(T-1)T(")X22'7, where dT denotes Haar measure on ^. P is a projection, but this fact is not needed here. What is needed is the following result, which is easily verified using Lemma 6. Lemma 7. PT(n)x = Pa(T)x for T e§ and x £ X(n). Defining u in X'"* by u m. («.,..., u ), we have T^u = a(T)u for each T £ §. Therefore, from the definition of P, Pu = u. Since u 4 0, it follows that ker P is strictly smaller than X^nK Thus, if X* is the dual space of X, there exists f = (/ . .. , / ) e X*(n) with f 4 0 such that f annihilates ker P (making the obvious identification of the dual space of X<"> with X*(n)). Put 77 F= Y f. ®u.. *—i ' 1 1 2 = 1 Then F 4 0. Using the fact that T(n)x a(T)x belongs to ker P for every leSj and x £ X^"\ a routine calculation gives Lemma 8. F e §'. Lemma 9. Let M be ^-irreducible and let S e §". Then M e Lat 5. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use WEAKLY COMPACT GROUPS OF OPERATORS 81 To see this observe that F and S commute. Further \0\ 4 F(X) C M, and since F £ §', F(X) is ^-invariant. By irreducibility F(X) = M; but then S(M) = SF(X) = FS(X) C M. Using reflexivity, Lemmas 9 and 3 give )-¡" C Á((j). Since the reverse inclusion always holds, we have thus proved Theorem 10. §" = A(§). Solution B. We give an example of a G-operator on a weakly sequentially complete space which is not of the form exp(iS) with S bounded. This example depends on some general facts about logarithms of point measures. Let G be a LCAG and let M(G) be the commutative Banach algebra of bounded regular complex measures on G under convolution. Given x e G, we show that the point mass 8 has a logarithm in M(G) if, and only if, x is of finite order in G. The "if" proof follows from elementary spectral theory. For the converse, let MAG) be the Banach algebra of discrete measures on G. Lemma 11. If a discrete measure ß on G has a logarithm in M(G), then fi has a logarithm in MAG). This follows from the fact that if ß = exp v for some v £ M(G), then ß = exp vd where vd is the discrete part of v. Lemma 12. Let 8 = exp v in M(G). Then x is of finite order in G. By Lemma 11 we may (and do) assume that G is discrete. Then the maximal ideal space of M(G) is the compact group G dual to G. The proof can be completed by the following simple argument due to Gavin Brown. Taking Gelfand transforms in the equation 8 = exp v, we obtain x(y) = 8x(\) = exp ¿(y) (x e G), where, without loss of generality, viX\) = 0 f°r Xi tne unlt °^ G. Since x is a character on G, it follows that iXx ft e G) where N: G x G—>Z is continuous. Let H be the connected component containing Xl in G. Then 2mN(-^, ifj) =-j/(y,) = 0 on H. Hence v\yn) = nv(X) (y £ H, n £ Z). The boundedness of the continuous function v on the compact group H License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 82 T. A. GILLESPIE AND T. T. WEST gives ¿(y) = O (y e H). Hence x(y) =1 (-y e H) and so x is of finite order in G [6, 24.20]. We can now give the counterexample for Problem B. Let R be the translation operator on L (T) defined by (Rj)(t) = f(tco-1) if e lHT), t a.e.), where co £ T and arg co is an irrational multiple of 227. R is a G-operator on L (T) [4, Example 5.4] and L (T) is weakly sequentially complete [3, IV. 8.6]. Theorem 13. R does not have a logarithm in B(L (T)). For suppose R = exp S in B(L (T)). Since the powers of co are dense in T, it follows that S commutes with every translation R (t e T). Hence S is a multiplier on L (T) and there exists p. £ M(T) such that Sf = ¡i * f (f £ Ll(T)) [7, Theorem 0.1.1]. Therefore 8a*f~*J-b*PI* */ (/ei-HT)) and so 5 = exp p.. Lemma 12 gives the required contradiction.REFERENCES 1. K. de Leeuw, Linear spaces with a compact group of operators, Illinois J.Math. 2 (1958), 367-377. MR 21 #819-2. K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifi-cations, Acta Math. 105 (1961), 63-97. MR 24 #A1632.3. N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pureand Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302.4. T. A. Gillespie and T. T. West, Operators generating weakly compact groups,Indiana Univ. Math. J. 21 (1972), 671-688.5. -,Operators generating weakly compact groups. II, Proc. Royal IrishAcad. 73A (1973), 309-326.6. E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. 1: Structure oftopological groups. Integration theory, group representations, Die Grundlehren dermath. Wissenschaften, Band 115, Academic Press, New York; Springer-Verlag, Ber-lin, 1963. MR 28 #158.7. R. Larsen, An introduction to the theory of multipliers, Die Grundlehren dermath. Wissenschaften, Band 175, Springer-Verlag, Berlin, 1971.8. Ju. I. Ljubic, Almost periodic functions in the spectral analysis of operators,Dokl. Akad. Nauk SSSR 132 (1960), 518-520 = Soviet Math. Dokl. 1 (1960), 593-595.MR 22 #9863.9. H. Radjavi and P. Rosenthal, On invariant subspaces and reflexive algebras,Amer. J. Math. 91 (1969), 683-692. MR 40 #4796. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF EDINBURGH, EDINBURGH,SCOTLAND SCHOOL OF MATHEMATICS, TRINITY COLLEGE DUBLIN, IRELANDLicense or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use