On Positivity and Roots in Operator Algebras

Abstract

In earlier papers the second author and Charles Read have introduced and studied a new notion of positivity for operator algebras, with an eye to extending certain C * -algebraic results and theories to more general algebras. The present paper consists of complements to some facts in the just mentioned papers, concerning this notion of positivity. For example we prove a result on the numerical range of products of the roots of commuting operators with numerical range in a sector.

Roots in Operator and Banach Algebras

“…Let a = e iθ x, then e −iθ a = x. By the case from [2] we have (e −iθ a) s = e −isθ a s , so that e isθ x s = (e iθ x) s as desired. Next, if W (x) contains numbers in the interior of the third quadrant and θ negative but very small, choose ρ > 0 with e i(θ+ρ) x accretive.…”

“…This shows that Corollary 5.5 is in some sense a noncommutative variant of the fact from [2] that a 1 2 b 1 2 is accretive for accretive commuting elements in a unital operator algebra. We noted in [8,Example 3.13] that the latter fact is false in a Banach algebra.…”

“…where in the second last equality we used the case from [2] again. The next case we consider is if x is accretive, and θ < 0.…”

“…Now the product of two commuting operators with numerical range in S π 4 is accretive [2]. Hence X k+1 is accretive, being the average of two accretives.…”

“…The numerical range W (c) is contained in S θ where θ = ρ/2 (see e.g. [26,2]). We will assume for clarity that a is accretive, and so we may take θ = π 4 , the case that π 4 < θ < π 2 will be discussed at the end.…”