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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 $$\mathbb{S}$$.

##### MSC:
 46H05 General theory of topological algebras
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