Characterizations of seminorms given by continuous linear functionals on normed linear spaces and applications to Gel'fand measures (Q2709912)

The main result is the following theorem. Let \(q\not=0\) be a seminorm on a normed linear space \(E\). Then the following assertions are equivalent: NEWLINENEWLINENEWLINE(1) There exists a closed linear subspace \(M\) of \(E\) such that NEWLINE\[NEWLINEq(x)=q(y)\Leftrightarrow x=\lambda y+z,\;\text{ with} |\lambda |=1,\;z\in M.NEWLINE\]NEWLINE NEWLINENEWLINENEWLINE(2) There exists a continuous linear form \(f\) on \(E\) such that \(q=|f|\). NEWLINENEWLINENEWLINEAs an application one gives a characterization of Gel'fand measures. A Gel'fand measure on a locally compact group \(G\) is a bounded measure \(\mu \) with \(\mu =\mu *\mu =\mu ^*\) such that \(L_1^{\mu }(G)=\mu *L_1(G)*\mu \) is commutative, or, equivalently, \(\pi (\mu)\) has rank \(\leq 1\) for all \(\pi \in \widehat G\). By using the preceding theorem this condition is equivalent to the following: for every \(\pi \in \widehat G\) on a Hilbert space \(H_{\pi }\), there exists a closed linear subspace \(M_{\pi }\) of \(H_{\pi }\) such that NEWLINE\[NEWLINE|\langle \pi (\mu)x|x\rangle |=|\langle \pi (\mu)y|y\rangle |\Leftrightarrow x=\lambda y+z\;\text{ with} |\lambda |=1,\;z\in M_{\pi }.NEWLINE\]