Spaces of operators, the ψ-Daugavet Property, and numerical indices

Suppose ψ : [0, ∞) → [1, ∞) is a strictly increasing function. A Banach space X is said to have the ψ-Daugavet Property if the inequality ∥I_X+T∥≥(∥T∥) holds for every compact operator T : X → X. We show that, if 1 < p < ∞ and K(ℓ_p)→ X → B(ℓ_p), then X has the ψ-Daugavet Property with ψ(t)=(1+c_p t^q) ^1/q (here q = max{2,p} and c _p is an absolute constant). We also prove that a C *-algebra A is commutative if and only if 1 + ∥T∥ = sup{∥I_A+ω T∥ ∥ω| = 1} for any T: A → A. Together, these results allow us to distinguish between some types of von Neumann algebras by considering spaces of operators on them.