Reflexive Calkin algebras

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 \mathit{Cal}(X) = \mathcal{L}(X)/\mathcal{K}(X) the Calkin algebra of X . We prove that \mathit{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\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, \mathit{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 (\bigoplus_{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.