Spectra of Invariant Uniform and Transform Algebras

For G a locally compact abelian group, any closed invariant proper subalgebra of CQ(G) has analytic discs in its spectrum. Related results are given for A(G) and B(G). For a compact group G it is an easy consequence of Bishop's generalized Stone-Weierstrass theorem and some well-known facts on representations that a proper (doubly) invariant uniform algebra on G cannot have G as its spectrum. (Indeed, Haar measure of a nontrivial subgroup must be multiplicative.) The corresponding result for closed subalgebras of C0(C7), G locally compact abelian, seemingly required more harmonic analysis (and, originally, the argument yielding Bishop's result); but more emerged, namely, that the spectra of (nonzero) proper closed invariant subalgebras of C0(G) contain analytic discs. This follows easily from the following more specific result. Theorem 1. Let A be a separating translation invariant closed proper subalgebra of C0(G), G I.ca. Then: (1) there is a compact subgroup H of G whose normalized Haar measure m is multiplicative on A, so a -» m * a is a multiplicative projection of A onto its subalgebra A n C0(G/H), while (2) there is a nontrivial homomorphism a: R -» G/H for which [A n C0(G/H)] ° a consists of boundary value functions (onRor R/Z) of continuous functions analytic on the closed half-plane or disc. Alternatively, A D C0(G/H) consists of functions analytic with respect to the induced flow on G/H; in particular we have analyticity relative to a flow on G when G has no nontrivial compact subgroups. Of course one can exploit such analyticity; for example, if / G C0(R") has f~l(c) of positive Lebesgue measure for some c =£ 0 then f and its translates generate C0(R") as an algebra. Such corollaries are given in §2, with the first section devoted to the proof of our theorem. After a revised version of this paper was essentially complete I found a result of J. L. Taylor [Tl, Lemma 2] (extending, and using, an earlier result of Gleason and Rieffel [Ri, Theorem 6.4]), which is closely related to Theorem 2 below; at least two thirds of the proof of Theorem 1 is devoted to covering essentially the same ground Received by the editors February 11, 1981 and, in revised form, May 10, 1982. 1980 Mathematics Subject Classification. Primary 46J10, 43A25. 1 Work supported in part by the NSF. ©1983 American Mathematical Society 0002-9947/82/0000-1545/S05.O0 381 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use