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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.
##### MSC:
 43A10 Measure algebras on groups, semigroups, etc.
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