Hermitian powers:A Müntz theorem and extremal algebras
Given S ⊂ N, let Ŝ be the set of all positive integers m for which h is hermitian whenever h is an element of a complex unital Banach algebra A with h hermitian for each n ∈ S. We attempt to characterize when (i) Ŝ = N, or (ii) Ŝ = S. A key tool is a Müntz-type theorem which gives remarkable conclusions when 1 ∈ S and ∑{1/n : n ∈ S} diverges. The set Ŝ is determined by a single extremal Banach algebra Ea(S). We describe this extremal algebra for various S. 0. Introduction. Let A be a complex unital Banach algebra generated by an element h and let S be a subset of the positive integers N. We aim to make a general study of what can be said about A given that h is hermitian (that is, ‖exp(iαhn)‖ = 1 for all real α) for n ∈ S. When S = N, the Vidav–Palmer Theorem gives immediately that A is a C∗-algebra with Gelfand space Sp(h), the spectrum of h. We show that the same conclusion holds under much weaker assumptions for S. We prove in fact the following Müntz-type theorem. If h is hermitian and if h is hermitian for a large enough subset of odds (resp. evens) to give divergence for the corresponding series ∑ 1/n, then h is hermitian for all odds (resp. evens). In particular, if S = 3N− 2, then it follows that h is hermitian for all n ∈ N. Given that h is hermitian for n ∈ S in a unital Banach algebra A generated by h, it is natural to ask if there is a largest subset Ŝ such that h is hermitian for n ∈ Ŝ for every such unital Banach algebra A. This question is readily answered in the context of extremal Banach algebras. We discuss these in §3, but it is enough to note here that one definition of the extremal norm for polynomials in h is given by the pointwise supremum of all algebra seminorms | · | such that |h| ≤ 1 and h is hermitian for all n ∈ S. If a polynomial is hermitian for the extremal norm, then it is hermitian for any of these algebra seminorms. Thus we may define Ŝ to be the set of 2000 Mathematics Subject Classification: Primary 46H05.