A hierarchy of Banach spaces with \(C(K)\) Calkin algebras (Q2805999)

Let \(X\) be a Banach space. Denote by \(\mathscr B(X)\) the Banach algebra of all bounded linear operators on \(X\) and by \(\mathscr K(X)\) the ideal of compact operators on \(X\). The Calkin algebra of \(X\) is defined to be the quotient algebra \(\mathrm{Cal}\,X = \mathscr B(X) / \mathscr K(X)\). When \(X\) is an infinite-dimensional Hilbert space, \(\mathrm{Cal}\,X\) is humongous as it does not even embed into \(\ell_\infty\) as a Banach space (neither in \(\mathscr B(\ell_2)\) as a \(C^*\)-algebra). Similar phenomena occur for other classical Banach spaces. On the other end of the spectrum, there is the famous Argyros-Haydon space [\textit{S. A. Argyros} and \textit{R. G. Haydon}, Acta Math. 206, No. 1, 1--54 (2011; Zbl 1223.46007)] with the Calkin algebra being one-dimensional and its variants where the Calkin algebras are finite-dimensional [\textit{M. Tarbard}, J. Lond. Math. Soc., II. Ser. 85, No. 3, 737--764 (2012; Zbl 1257.46013)], [\textit{N. J. Laustsen} and the reviewer, ``Ideal structure of the algebra of bounded operators acting on a Banach space'', Preprint, \url{arXiv:1507.01213}]. The problem of describing those unital Banach algebras which are isomorphic to Calkin algebras of certain Banach spaces does not seem to be feasible at the moment, therefore every new concrete example of a Calkin algebra is particularly valuable.NEWLINENEWLINEThe paper under review builds upon fine techniques of Argyros and Haydon pushing them to the limit. The main result of the paper asserts that for every countable, compact metric space \(K\) there exists an \(\mathscr{L}_\infty\)-space \(X_K\) with \(X_K^*\) isomorphic to \(\ell_1\) such that \(\mathrm{Cal}X_K\) is isomorphic to \(C(K)\), the algebra of all continuous, scalar-valued functions on \(K\).NEWLINENEWLINEThe proof is based on recursive constructions on trees that involve the so-called AH-sums of Banach spaces, which generalise the original Argyros-Haydon construction in a highly non-trivial way. Given the inevitably quite technical nature of the proofs, it is perhaps better to send the reader directly to the paper itself, rather than to elaborate here further on proof techniques.NEWLINENEWLINEThe paper is well written and may be read also by those who have not read the Argyros-Haydon paper. It should perhaps be noted that the spaces \(X_K\) provide new examples of Banach spaces for which the lattice of all closed ideals of \(\mathscr B(X)\) is fully understood. Indeed, algebraically oriented readers will immediately recognise that \(\mathscr B(X_K)\) is the Banach-algebra semi-direct product of \(\mathscr K(X_K)\) with \(C(K)\) and so the closed ideals of \(\mathscr B(X_K)\) correspond to those of \(\mathscr K(X_K)\) (which are \(\{0\}\) and \(\mathscr K(X_K)\) itself, as \(X_K\) has the approximation property) and those of \(C(K)\). However, every closed ideal of \(C(K)\) consists of functions which vanish on some fixed closed set, so these ideals correspond precisely to closed subsets of \(K\).NEWLINENEWLINELet me finish the review by quoting two interesting questions left open in the paper:NEWLINENEWLINEDoes there exist a Banach space \(X\) for whichNEWLINENEWLINE(a) \(\mathrm{Cal}X\) is infinite-dimensional and reflexive?NEWLINENEWLINE(b) \(\mathrm{Cal}X\) is isomorphic to \(C(K)\) for some uncountable compact metric space?