The isometry degree of a computable copy of ℓ p 1

Suppose p is a computable real so that \(p \ge 1\). We define the isometry degree of a computable presentation of \(\ell ^p\) to be the least powerful Turing degree \(\mathbf {d}\) by which it is \(\mathbf {d}\)-computably isometrically isomorphic to the standard presentation of \(\ell ^p\). We show that this degree always exists and that when \(p \ne 2\) these degrees are precisely the c.e. degrees.