Banach spaces generated by the coefficients of the conjugation representation (Q2367636)

Following the achievements of Eymard and Herz, \textit{G. Arsac} developed the general theory of the Banach space generated by the coefficients of an arbitrary unitary representation \(\pi\) of a locally compact group \(G\) [Publ. Dépt. Math., Lyon 13, No. 2, 1-101 (1976; Zbl 0365.43005)]. Let \(\Delta\) be the modular function on \(G\) and let \(*\) denote the convolution product; if \(1\leq p\leq\infty\), define \(\pi_ p f(y)= f(x^{-1} yx) \Delta(x)^{1/p}\) for \(f\in L^ p(G)\), \(x,y\in G\), with the convention \(1/\infty =0\). For \(1<p<\infty\), \(1/p+ 1/p'=1\), the author performs a remarkable presentation of the theory of the Banach space \(A_{\pi_ p}(G)\) generated by the functions \(u\) on \(G\) of the form \[ u(x)= \sum_{n=1}^ \infty \langle \pi_ p(x) h_ n,g_ n\rangle, \] where \(g_ n\in L^ p(G)\), \(h_ n\in L^{p'}(G)\), \(\sum_{n=1}^ \infty \| h_ n\|_ p \| g_ n\|_{p'}< +\infty\); \(\| u\|_{A_{\pi_ p}}\) is defined as the infimum of the set of the latter sums. The dual space of \(A_{\pi_ p}(G)\) is characterized. The Banach space \(A_{\pi_ p}(G)\) is basic for the study of inner amenability on \(G\) [\textit{C. K. Yuan}, J. Math. Anal. Appl. 130, 514-524 (1988; Zbl 0639.43002)]: \(G\) is inner amenable, i.e., there exists a mean \(m\) on \(L^ \infty(G)\) such that \(m(\pi_ \infty(x) f)= m(f)\) whenever \(f\in L^ \infty(G)\), \(a\in G\), if and only if anyone of the following conditions holds: (i) There exists a bounded net in \(A_{\pi_ p} (G)\) converging uniformly to \(1_ G\) on compact subsets of \(G\). (ii) For every \(f\in L_ +^ 1(G)\), \(\| \lambda_ p(f)\|= \| f\|_ 1\), where the linear operator \(\lambda_ p\) is defined on \(A_{\pi_ p}(G)\) by \(\lambda_ p(f)u =f*u\), \(u\in A_{\pi_ p} (G)\).