Some Formulae for Norms of Elementary Operators

We present a formula for the norm of an elementary operator on a C∗-algebra that seems to be new. The formula involves (matrix) numerical ranges and a kind of geometrical mean for positive matrices, the tracial geometric mean, which seems not to have been studied previously and has interesting properties. In addition, we characterise compactness of elementary operators. We consider an elementary operator Tx = ∑l j=1 ajxbj (with x ∈ A a C∗-algebra and aj , bj ∈ M(A), with M(A) the multiplier algebra of A). We denote the class of elementary operators T : A → A by El(A). Specifically, we address the question of finding a concrete formula for the operator norm ‖T‖. This problem has been considered (at least implicitly) over a long period by several authors and there are solutions known under various special circumstances (generalised derivations [25], antiliminal by abelian C∗-algebras [5], for example). See [19] for a recent survey of the problem, or see [3, §5.4] for a brief summary of its importance. One way to view the literature that relates to the problem is to separate two strands of problems. One strand concentrates on elementary operators of a rather special form (with l ≤ 2) and the other (for arbitrary l) has relied largely on dealing with the completely bounded norm ‖T‖cb and the Haagerup tensor norm estimate ‖T‖cb ≤ ∥∥ ∑l j=1 aj ⊗ bj ∥∥ h . For special forms where l ≤ 2, the case l = 1 is well understood (see [19]). There is a significant body of literature dealing with (inner) derivations δa(x) = ax−xa and the estimate ‖δa‖ ≤ 2 inf ‖a−z‖ with the infimum over z in the centre Z(M(A)) of M(A) (see references in [3, §4.1, §4.6] and [19]). In the case A = B(H) is the algebra of all bounded linear operators on a Hilbert space H (or A = K(H), the compacts) Z(M(A)) is just scalar multiples of the