The isometry degree of a computable copy of ${\ell }^p$

Abstract

When $p$ is a computable real so that $p\ge 1$, the isometry degree of a computable copy $\mathcal{B}$ of ${\ell }^p$ is defined to be the least powerful Turing degree that computes a linear isometry of ${\ell }^p$ onto $\mathcal{B}$. We show that this degree always exists and that when $p\ne 2$ these degrees are precisely the c.e. degrees.