Reflexive Calkin algebras

Abstract

For a Banach space $X$ denote by $\mathcal{L}(X)$ the algebra of bounded linear operators on $X$, by $\mathcal{K}(X)$ the compact operator ideal on $X$, and by $Cal(X)=\mathcal{L}(X)/\mathcal{K}(X)$ the Calkin algebra of $X$. We prove that $Cal(X)$ can be an infinite-dimensional reflexive Banach space, even isomorphic to a Hilbert space. More precisely, for every Banach space $U$ with a normalized unconditional basis $(u_s{)}_{s=1}^{\infty }$ not having a $c_0$ asymptotic version we construct a Banach space ${\mathfrak{X}}_U$ and a sequence of mutually annihilating projections $(I_s{)}_{s=1}^{\infty }$ on ${\mathfrak{X}}_U$, i.e., $I_s{I}_t=0$ for $s\ne t$, such that $\mathcal{L}({\mathfrak{X}}_U)=\mathcal{K}({\mathfrak{X}}_U)\oplus [(I_s{)}_{s=1}^{\infty }]\oplus \mathbb{C}I$ and $(I_s{)}_{s=1}^{\infty }$ is equivalent to $(u_s{)}_{s=1}^{\infty }$. In particular, $Cal({\mathfrak{X}}_U)$ is isomorphic, as a Banach algebra, to the unitization of $U$ with coordinatewise multiplication. Banach spaces $U$ meeting these criteria include ${\ell }_p$ and $({⨁}_n{\ell }_{\infty }^n{)}_p$, $1\le p<\infty$, with their unit vector bases, $L_p$, $1<p<\infty$, with the Haar system, the asymptotic-${\ell }_1$ Tsirelson space and Schlumprecht space with their usual bases, and many others.