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 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., $I_sI_t = 0$, for $s\neq t$, 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 coordinate-wise multiplication. Banach spaces $U$ meeting these criteria include $\ell_p$ and $(\oplus_n\ell_\infty^n)_p$, $1\leq p<\infty$, 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.