Maximal ideal spaces of $U$-algebras
Jonathan M. Kane · Illinois Journal of Mathematics · 1983
IntroductionFor n > 1 let Bn be the open unit ball in C and $2-1 be its boundary.The group of n x n unitary matrices, U(n), acts on C by multiplication on the right and this action takes B. onto itself and $2.-1 onto itself.A subspace, X, of C($2-1) is called a U-space if for each f X and V U(n), f V X.A U-space which is closed under multiplication is called a U-algebra.In this paper the maximal ideal spaces are found for every closed U-algebra which contains the constant functions.Let Z represent the natural numbers {0, 1, 2, ...} and Z/, the positive natural numbers.For p, q Z, le_t Hp, be the set of restrictions to S2._ of the harmonic polynomials in z and z which are homogeneous of degree p in z and qin z.A. Nagel and W. Rudin show in I-2] that if X is a closed subspace of C($2._ 1) and Y {(p, q) lX c H,, =p 0}, then X is a U-space if and only if X is th closure in C($2_ 1) of the direct sum X* ,) rH,.Thus, each closed U-space is associated with a set of lattice points, Y_ Z2, and a dense subspace, X*, equal to a direct sum of H, spaces.If X is a closed U-algebra, its associated set of lattice points is called an alaebra pattern.Note that _H,, is a U-space and is spanned by the unitary translates of the function zlz2.Define H,,q.H,,s to be the subspace of C(S2n_I) spanned by {f.gift H,,q, g H,,s} and (Hp,q) (Hp.q)H,,q for m > 1. Nagel and Rudin prove in l-2] and I-3-1 the results" cH, wherej=O, 1, min PROPOSITION 1.1.(a) H,,q H,,,_ +,_,q+,_j (p + q, r + s, p +,r, q + s).(b) If n > 3,( H,q) Hm,-j,mq-wherej 0, 1, min (mp, mq).(c) Ifn 2, (n,,q) 2 n2"_2,2q_2 wherej 0, 1, min (p, q).(d) If n 2 and m > 2, (Hp,c) E nml,-j,mq-j where j 0, 2, 3, 4, min (rap, mq).