Banach algebras with scattered structure spaces

Abstract

1. For a commutative semisimple Banach algebra B, let 9lZB denote its space of nonzero multiplicative linear functionals, and x->)x its Gelfand representation. A well-known theorem of Silov [1; 15 ] shows that for any compactopen subset U of 9)fB there is an x in B with xe the characteristic function 4u of U. An immediate consequence is the fact that B is regular if 9)lB iS totally disconnected. The present note is devoted to a similar application of gilov's result which has apparently escaped notice. Normally when A is a subalgebra of B (closed or not), 9MA may contain functionals other than those provided by the restrictions of the elements of 9)B. But at least when A is closed the Silov boundary dA of A is produced by the elements of dB. In any case, if we assume aA is produced by aB, and that dB is scattered (i.e., contains no nonvoid perfect subset), then Silov's theorem shows not only that B is regular but indeed that A is regular as well so that all of 9J1A = 9A arises from 9B = 9B. When d9B is discrete the same is true of dA, and we can trivially identify the smallest hull-less ideal jA( ?? ) of A; it is precisely the span of the idempotents in A. Consequently A is tauberian if and only if it is the closed span of its idempotents. As an application we can easily determine all closed tauberian subalgebras of L1(G) and L2(G) when G is a compact abelian group. In the L2 case every closed subalgebra A is tauberian, and is determined by just the sets of constancy of the Fourier transforms A'; the same prescription applies to the tauberian closed subalgebras A of L1(G), but whether nontauberian subalgebras exist seems to be a difficult problem. Borrowing from the L2 case we can easily see that A is tauberian if (and of course only if) A nL2 is dense in A. (Some application can also be made to closed commutative semisimple subalgebras of L1(G) and L2(G) when G is compact nonabelian, cf. ?4.) The notation used below is essentially standard, as in [9], with the exception of our use of scattered, as defined above. We denote the hull of an ideal I by hI, and by kF the kernel of a subset F of 9)A, while jA( CO) is the set of a in A for which a has compact support. A is called tauberian when every hull-less ideal is dense in A; when A is regular this amounts to the density of jA ( oo) [9]. All algebras will be assumed commutative semisimple. Although ^ may be used for any Gelfand representation which arises, it will always be clear from the context which is intended. Finally, Aa will be used to