If X is a separable reflexive Banach space, there are several natural Polish topologies on \({\cal{B}}_{1}(X)\) , the set of contraction operators on X (none of which being clearly “more natural” than the others), and hence several a priori different notions of genericity—in the Baire category sense—for properties of contraction operators. So it makes sense to investigate to which extent the generic properties, i.e., the comeager sets, really depend on the chosen topology on \({\cal{B}}_{1}(X)\) . In this paper, we focus on ℓp-spaces, 1 < p ≠ 2 < ∞. We show that for some pairs of natural Polish topologies on \({\cal{B}}_{1}(\ell_{p})\) , the comeager sets are in fact the same; and our main result asserts that for p = 3 or 3/2 and in the real case, all topologies on \({\cal{B}}_{1}(\ell_{p})\) lying between the Weak Operator Topology and the Strong* Operator Topology share the same comeager sets. Our study relies on the consideration of continuity points of the identity map for two different topologies on \({\cal{B}}_{1}(\ell_{p})\) . The other essential ingredient in the proof of our main result is a careful examination of norming vector for finite-dimensional contractions of a special type.
