Interpolation sets and the size of quotients of function spaces on a locally compact group (Q2826772)

The authors develop a general method for estimating the size of quotients between algebras of functions on a locally compact group. A main tool are interpolation sets.NEWLINENEWLINEFor a locally compact group \(G\) let \(l_\infty(G)\) be the \(C^\ast\)-algebra of bounded, scalar-valued functions on \(G\) with the supremum norm. In particular, the authors consider the following subalgebras of \(l_\infty(G)\): the algebra \(\mathcal{C}\mathcal{B}(G)\) of continuous bounded functions, the algebra \(\mathcal{L}\mathcal{U}\mathcal{C}(G)\) of bounded left uniformly continuous functions, the algebra \(\mathcal{W}\mathcal{A}\mathcal{P}(G)\) of weakly almost periodic functions, and the Fourier-Stieltjes algebra \(\mathcal{B}(G)\).\newline Let \(\kappa(G)\) be the minimal number of compact subsets of \(G\) which are necessary to cover \(G\).NEWLINENEWLINEThe comprehensive introduction is very informative and contains many references. After proving deep lemmas the following theorem is obtained: \newline Let \(G\) be a locally compact group with center \(Z(G)\), and let \(\kappa=\kappa(Z(G))\). For \(\kappa\neq 1\) there exists a linear isometry from \(l_\infty(\kappa)\) onto \(\mathcal{W}\mathcal{A}\mathcal{P}(G)/\mathcal{B}(G)\). \newline By the help of this theorem the following results are derived:\newline Let \(G\) be a non-compact, locally compact group with \(\kappa=\kappa(G)\).\newline (a) If \(G\) is an \(IN\)-group, then there exists a linear isometry from \(l_\infty(\kappa)\) onto \(\mathcal{W}\mathcal{A}\mathcal{P}(G)/\mathcal{B}(G)\). \newline (b) If \(G\) is nilpotent, then the quotient \(\mathcal{W}\mathcal{A}\mathcal{P}(G)/\mathcal{B}(G)\) contains a linear isometric copy of \(l_\infty(\kappa)\).NEWLINENEWLINEThe very interesting paper ends with the following theorem: Let \(G\) be a locally compact group. Then \(\mathcal{C}\mathcal{B}(G)/\mathcal{L}\mathcal{U}\mathcal{C}(G)\) contains a linear copy of \(l_\infty(\kappa(G))\) if and only if \(G\) is neither compact nor discrete.\newline This theorem implies a result of \textit{J. W. Baker} and \textit{R. J. Butcher} [Math. Proc. Cambridge Philos. Soc. 80, 103--107(1976; Zbl 0329.22003)]: Let \(G\) be a locally compact group. Then \(\mathcal{C}\mathcal{B}(G)\neq\mathcal{L}\mathcal{U}\mathcal{C}(G)\) if and only if \(G\) is neither compact nor discrete.