Projective operator spaces, almost periodicity and completely complemented ideals in the Fourier algebra
The author considers projective operator spaces in the sense of [\textit{D.\,P.\thinspace Blecher}, ``The standard dual of an operator space'', Pac.\ J.\ Math.\ 153, No.\,1, 15--30 (1992; Zbl 0726.47030)]: an operator space space \(E\) is called \textit{projective} if, for every operator space \(X\), each closed subspace \(Y\) of \(X\), each complete contraction \(\Gamma : E \to X/Y\), and each \(\varepsilon > 0\), there is a lifting \(\tilde{\Gamma} : E \to X\) of \(\Gamma\) with \(\| \tilde{\Gamma} \|_{cb} \leq 1 + \varepsilon\). The paper focuses on projective operator spaces that arise naturally in abstract harmonic analysis, in particular on subspaces of the Fourier algebra \(A(G)\) or of the Fourier-Stieltjes algebra \(B(G)\) of a locally compact group \(G\). Among other things, the author shows that \(A(G)\) is projective if and only if \(B(G)\) is projective if and only if \(G\) is compact. Applications to (complete) invariant complementation of ideals of \(A(G)\) are also given.
scientific article; zbMATH DE number 5858883
scientific article