Stone’s theorem and spectral subspaces of automorphisms

It is shown that spectral subspaces of automorphisms of a von Neumann algebra can be defined by use of Stone's theorem on unitary representations. It is well known that the theory of spectral subspaces of automorphisms as developed by Arveson [1] generalizes Stone's theorem for unitary representations. In this note we shall show a converse result, thus indicating how the theory of spectral subspaces of abelian groups of automorphisms of von Neumann algebras, see [1], [2], [3], can be developed from Stone's theorem. The idea is that a *-automorphism of the bounded operators B (H) on a Hilbert space H is implemented by a unitary operator, and so restricts to an isometry of the Hilbert-Schmidt operators H2 on H. Thus Stone's theorem can be used on unitary representations on H2 and then "lifted" to B (H). Throughout this note we let G be a locally compact abelian group with Haar measure dt and dual group r; t --*U, is a strongly continuous unitary representation of G on the Hilbert space H, and a, = Ad(u,). Then t-> a, is a continuous representation of G into the automorphism group Aut B (H) of B(H), i.e., t -> p(a,(x)) is a continuous function for all x E B(H), p E B(H)*. Recall from [1], [2], [3] that if f E L1(G) and x E B(H) then Tfa(f)(X) =fx f (t)a,(x) dt, Z(f) = {y E F:f (Y) = 01, Spa(X) = n( {Z(f):f E L1(G), 7Ta(f)(X) = 0). We assume M is a von Neumann algebra acting on H such that a,(M) = M, t E G. Then if E is a closed subset of r, its spectral subspace is Ma (E) = {x E M: Spa(X) C E. Moreover, the subspaces Ma(E) determine a [1]. LEMMA. Denote by &, the restriction of at to the Hilbert-Schmidt operators H2 on H. Then t -> &, is a weakly, hence strongly, continuous unitary representation of G on H2. PROOF. Let x, y E H2 and E > 0. Lety = y1 + Y2 with y1 of finite rank and Received by the editors January 12, 1977. AMS (MOS) subject classifications (1970). Primary 46L10; Secondary 46L05, 43A25. () American Mathematical Society 1978 18 This content downloaded from 157.55. 39.151 on Fri, 29 Jul 2016 05:54:13 UTC All use subject to http://about.jstor. org/terms SPECTRAL SUBSPACES OF AUTOMORPHISMS 19 IIY2112 1/2, and is the inner product on H2. Since a, (x) -> x ultraweakly as t -> e, the identity in G, there is a neighborhood N of e in G such that I J+JKd,(x) X,Y2>I dPx. If E and F are closed subsets of r we denote by E + F the closure of the set fy + X: y E E, X E F). We denote by P(F) the closed subspace of H2 obtained as the range of fFdPx, and by P (F)its ultraweak closure in B (H). THEOREM. With the above assumptions and notation, if E is a closed subset of r then Ma(E) = n N{M n P(E + N)-), where the intersection is taken over all compact neighborhoods of the identity in r. PROOF. Let F be a closed subset of r. Let x E M n P (F)and (x,) be a net in P (F) which converges ultraweakly to x. Let f E L1(G) have Fourier transform vanishing in a neighborhood of F. By [1, Proposition 1.6] 7lwa(P) is an ultraweakly continuous linear map on B(H). Thus by [1, Remark, ? 2] O = ra (j)(x,,) -> sa(j)(x), where we have identified the operator 7lw (f) defined by a on H2 and its extension Twa (f) to B (H). Thus Ta (j)(x) = 0 for all such f, so again by [1], x E Ma(F). Thus nN{M n P(E + N)-) C n NM (E + N) = Ma (E) [1, Proposition 2.2]. Conversely, let x E M' (E), so, in particular, x E B (H)a (E). If F is a closed subset of r let R2 (F) (resp. Ra(F)) denote the closed (resp. ultraweakly closed) subspace of H2 (resp. B (H)) generated by range 7T,a (f) in H2 (resp. in B (H)) for all f E L1(G) with supp f compact and contained in F. By [1, Proposition 2.2], B (H)' (E) = nN Ra (E + N), where the intersection is taken over all compact neighborhoods N of the identity in r. Let x E Ra(E + N) and assume there are f E L1(G) such that supp f is compact and contained in E + N, andy E B (H), such that x = Ta (f)(Y). Since H2 is ultraweakly dense in B (H) there is a net (y,) in H2 which converges ultraweakly toy. Since 7'Ta (f) is ultraweakly continuous, x = '7Ta(f)(y) = lim r7a(f)(Y) C R2 (E + N) But from the theory of spectral subspaces applied to unitary representations [1], R'(F) = P(F) for all closed sets F c r. Thus x E P(E + N)for all N, and since such x are dense in Ra(E + N), the proof is complete. REMARK 1. We cannot sharpen the theorem to a statement like "Ma(E) = M n P (E)-". Indeed, let H be a separable Hilbert space and u a unitary operator on H such that the von Neumann algebra A generated by u is a maximal abelian subalgebra of B (H) without minimal projections. Let a be This content downloaded from 157.55. 39.151 on Fri, 29 Jul 2016 05:54:13 UTC All use subject to http://about.jstor. org/terms