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Hermitian powers: A Müntz theorem and extremal algebras. (English) Zbl 0999.46021
In [Math. Z. 66, 121-128 (1956; Zbl 0071.11503)] I. Vidav noticed that a bounded linear operator \(T\) on a Hilbert space is self-adjoint if \(\|e^{itT}\|= 1\) for all \(t\in \mathbb{R}\). Thus the algebraic property \(T^*= T\) is determined by the norm. This suggested that a “self-adjointness” concept could be introduced into an arbitrary unital Banach algebra \(A\): Call \(h\in A\) Hermitian if \(\|e^{ith}\|= 1\) for all \(t\in \mathbb{R}\). This has proved a fruitful idea, and Hermitian \(h\) share many properties with self-adjoint operators \(T\); for example, the spectrum is real and the norm coincides with the spectral radius. However, some properties are lost: \(h\) may be Hermitian and \(h^2\) not.
This paper investigates what powers of Hermitian elements are necessarily Hermitian, independently of the specific algebra \(A\). More precisely, for non-void \(\mathbb{S}\subset \mathbb{N}\) let \(\widehat{\mathbb{S}}\) denote the set of \(n\in\mathbb{N}\) such that \(h^n\) is Hermitian for every unital Banach algebra \(A\) and every \(h\in A\) with \(h^k\) Hermitian for all \(k\in\mathbb{S}\). Of special interest are sets \(\mathbb{S}\) with \(\widehat{\mathbb{S}}=\mathbb{N}\) or \(\widehat{\mathbb{S}}= \mathbb{S}\). In the first direction a beautiful Müntz-type theorem is proved: If \(1\in \mathbb{S}\) and \(\sum\{1/n: n\in \mathbb{S}\cap 2\mathbb{N}\}=\infty\), then \(\widehat{\mathbb{S}}\supset 2\mathbb{N}\), with an analogous result about \(2\mathbb{N}-1\). The proof is based on a \(C^1[-1,1]\)-functional calculus for Hermitian \(h\) in the unit ball of \(A\), whose proof in turn makes elegant use of the Paley-Wiener theorem. A consequence is that if \(h^n\) is Hermitian for all \(n\) in an arithmetic progression containing some evens and some odds, then \(h^n\) is Hermitian for all \(n\in\mathbb{N}\). On the other hand, if \(\mathbb{S}:= 2\mathbb{N}-1\) or \(\mathbb{S}= m\mathbb{N}\) with integer \(m\geq 2\), then \(\widehat{\mathbb{S}}= \mathbb{S}\). It is harder to prove that \(\mathbb{S}:= \{1\}\cup 2\mathbb{N}\) satisfies \(\widehat{\mathbb{S}}= \mathbb{S}\); in fact, two different examples of an \(h\) with \(h^n\) Hermitian iff \(n\in \{1\}\cup 2\mathbb{N}\) are constructed. For finite \(\mathbb{S}\) and \(\mathbb{N}:= 1+|\mathbb{S}|\) there is a norm \(\|\cdot\|\) on \(\mathbb{C}^{2N+1}\) and an operator \(T\) such that in the algebra of operators on \(\mathbb{C}^{2N+1}\) which \(T\) generates, with the operator norm determined by \(\|\cdot\|\), the Hermitian elements coincide with the real linear span of \(T^m\), \(m\in \{0\}\cup \mathbb{S}\), and only these powers of \(T\) are present. In particular, \(\widehat{\mathbb{S}}= \mathbb{S}\).
Given unital Banach algebra \(A\) and \(h\in A\), which powers of \(h\) are Hermitian depends on just two things: the subalgebra \(\mathbb{C}[h]\) of complex polynomials in \(h\), and the norm in \(A\); and we may assume that the norm of \(h\) is not greater than 1. For a given \(\mathbb{S}\subset \mathbb{N}\) we are therefore led to consider all possible algebra semi-norms in \(\mathbb{C}[h]\) that satisfy \(|h|\leq 1\) and make \(h^n\) Hermitian for every \(n\in\mathbb{S}\). The supremum of these is another one, a norm \(\|\cdot\|\). The \(\|\cdot\|\)-completion of \(\mathbb{C}[h]\) is denoted \(\text{Ea}(\mathbb{S})\). It is a unital Banach algebra in which \(h^n\) is Hermitian iff \(n\in\mathbb{S}\). In the second half of the paper for various \(\mathbb{S}\) the (extremal) algebra \(\text{Ea}(\mathbb{S})\) is described in ways that do not explicitly involve the generator \(h\). It is from this study that we learn that \(\widehat{\mathbb{S}}= \mathbb{S}\) whenever \(\mathbb{S}= m\mathbb{N}\), and get as well a second proof that the equality holds for every finite \(\mathbb{S}\).

46H05 General theory of topological algebras
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