Hermitian powers: A Muntz theorem and extremal algebras

Given S subset of N, let (S) over cap be the set of all positive integers m for which h(m) is hermitian whenever h is an element of a complex unital B anach algebra A with h(n) hermitian for each n is an element of S. We attem pt to characterize when (i) (S) over cap = N, or (ii) (S) over cap = S. A k ey tool is a Muntz-type theorem which gives remarkable conclusions when 1 i s an element of S and Sigma {1/n : n is an element of S} diverges. The set (S) over cap is determined by a single extremal Banach algebra Ea(S). We de scribe this extremal algebra for various S.