Estimating Norms of Commutators

We find estimates on the norms commutators of the form [f(x), y] in terms of the norm of [x, y], assuming that x and y are contactions in a C*-algebra A, with x normal and with spectrum within the domain of f . In particular we discuss ‖[x, y]‖ and ‖[x, y]‖ for 0 ≤ x ≤ 1. For larger values of δ = ‖[x, y]‖ we can rigorous calculate the best possible upper bound ‖[f(x), y]‖ ≤ ηf (δ) for many f . In other cases we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the best upper bound. 1. Norms of Commutators and functional calculus In this paper we investigate upper and lower bounds on the norms of commutators in relation to the continuous functional calculus for normal elements of a C∗-algebra. For upper bounds we utilize simple nearby functions to the original function under study, building on a technique introduced by G. K. Pedersen in [11]. For lower bounds, we use Monte Carlo techniques to generate examples. The only norm we consider on matrices is the operator norm (largest singular value) as this is tied most closely to the norms in C∗-algebras. Definition 1.1. Suppose Ω is a compact subset of the complex plane and that f : Ω → C is continuous. Define ηf : [0,∞)→ [0,∞) by ηf (δ) = sup { ‖[f(X), A]‖ ∣∣X,A ∈ A, ‖A‖ ≤ 1, ‖[X,A]‖ = δ} and the supremum is taken over every possible C∗-algebra A and taking X and A in A with X normal with σ(X) ⊆ Ω. A convention is that if ‖[X,A]‖ = δ is not possible, we set ηf (δ) = 0. We shall see that so long as there is one example where ‖[X,A]‖ = δ occurs, whenever δ0 < δ1 we have ηf (δ0) ≤ ηf (δ). Therefore we have an equivalent definition of ηf (δ) if we allow in the supremum all pairs with X normal with spectrum in Ω, the norm of A at most one and ‖[X,A]‖ ≤ δ, so long as δ ≤ diam(Ω). We will focus on two special cases, where Ω is the unit interval or the unit circle. When Ω = [0, 1] we are studying the size of ‖[f(X), A]‖ where 0 ≤ X ≤ 1 and A has norm at most one, in any C∗-algebra. Following the arguments used in [10, Corollary 5.4] we can prove the following. Lemma 1.2. If f : [0, 1]→ C is continuous then ηf (δ) = sup { ‖[f(X), A]‖ ∣∣X,A ∈Mn(C), 0 ≤ X ≤ 1, ‖A‖ ≤ 1} the supremum taken over all n. 2010 Mathematics Subject Classification. 47B47, 47A56, 47A58.