Properties of the sequence of closed powers of a maximal ideal in a sup-norm algebra

Abstract

1. A sup-norm algebra is a closed point-separating subalgebra A of C(X) containing the constants, where X is a compact Hausdorff space. MA will denote the (compact Hausdorff) space of multiplicative linear functionals p on A in the Gelfand topology, and to each such p we associate its kernel A,; the A, are precisely the maximal ideals of A. X can be viewed as a closed subset of MA, and there is an isomorphism f -> of A into C(MA) such that f' X=f. The isomorphism is given by I(p) =(pf). We will frequently write f for f! SA will denote the Silov boundary of A, that is, the smallest closed subset of X on which each function in A attains its maximum modulus. Eachfs A has the same supremum over SA, X, and MA; this common supremum is the norm off, written 11f 1. We intend to study the ideals (A')-. [Note: If I is an ideal in A, In is the ideal generated by products fi... fn with f, E I if n > 1, and I? = A; also I denotes the closure of L] Evidently (An) D (A n+ 1)and if the inclusion is an equality for some n, then it is an equality for all subsequent n. A problem of the sort that will interest us-indeed, in great part the stimulus for this research-is to determine whether one can preassign the cutoff point in this chain of inclusions: given n, is it possible to have (A n) =A (A n + 1) -but (An + 1) -= (A n + 2) -? More generally, how much control do we have over the sequence of numbers {dim ((An) /(An + 1)-): n= 1 2,.. . }? While we will not give an explicit arithmetic answer, we will reduce the problem to a purely algebraic one. In particular, it is possible to obtain any nonincreasing sequence of nonnegative integers, and hence to preassign the cutoff point. We now will begin to formulate the problem we will actually solve. First, consider a graded algebra (all algebras in this paper are commutative algebras over the complex field C) Q = ? (1) Qn and suppose that Qo consists of scalar multiples of an identity for Q, and that Qo and Q, generate Q. If {F,: y E L'} is a basis for Q, and R = C[{XY : y E ]}] is the polynomial ring in P variables with its usual graded algebra structure, the linear extension of the map X Y n Yn * . is a graded algebra homomorphism of R onto Q. Thus Q is isomorphic as a