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Uniqueness of uniform norm and \(C^*\)-norm in \(L^p(G,\omega)\). (English) Zbl 1349.43002
A uniform norm on a Banach algebra \(A\) is a norm satisfying \(\| x^2\| =\| x\| ^2\) for all \(x\in A\). \(A\) is said to have the unique uniform norm property (UUNP) if it admits exactly one uniform norm, and the unique \(C^*\)-norm property (UC*NP) if it admits exactly one \(C^*\)-norm. The authors consider these properties for the class of weighted \(L_p\)-algebras. Given a locally compact abelian group \(G\) and a positive measurable function \(w\), the algebra \(L_p^w(G)\) is defined as \(\bigl \{f\: fw\in L_p(G)\bigr \}\), with the norm \(\| f\|_{p,w} = \| fw\|_p\).
The main result (Theorem 3.2) is that for a translation invariant algebra \(L_p^w(G)\) with \(p>1\), (b) UUNP is equivalent to (a) the existence of a minimal uniform norm and to (c) the Shilov regularity. There is however a problem in the proof of the implication (b)\(\Rightarrow \)(c): it is assumed implicitly that the spectrum of \(L_p^w(G)\) is equal to the dual group \(\widehat G\); this is always true for regular algebras, and might be true for algebras with UUNP, but the paper does not contain a proof of the latter fact.
43A10 Measure algebras on groups, semigroups, etc.
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