Analytic functionals for the non-commutative disc algebra

A non-commutative F&M Riesz Theorem

“…We remark that the NC F&M Riesz theorem obtained here is related, but not equivalent to, one previously developed in [3, Theorem A] using different techniques. Both Theorem 5.5 of the present paper and Theorem A of [3] conclude that an analytic NC measure need not be absolutely continuous, however the results of this paper and those of [3] describe the obstruction in different ways. We discuss the relationship between these two results in Remark 5.10.…”

“…As each φ j,ℓ is of type t, it follows that µ (k) has a Wittstock decomposition ψ such that | ψ| is of type t, and therefore µ (k) is of type t. To compare the two sets of results, fix λ ∈ A † d and suppose λ(A d ) = {0}, as this is the definition of analyticity used in [3]. There, the free disk system is viewed as embedded, completely isometrically, inside the Cuntz algebra, O d , via the quotient map q : E d → O d whose kernel is the compact operators.…”

“…, π(L d )) to the norm closure of π(A d )h 1 is unitarily equivalent to L. The triple (π, h 1 , h 2 ) is referred to as a "Riesz representation" of the functional Λ. The particular form of the representation implies that π(L) has no von Neumann-type summand, as is noted in[3]. We note thatλ(b) = h 2 , π(q(b))h 1 , b ∈ A d .The representation π and the vector h 1 are such that | Λ|(f ) = h 1 , π(f )h 1 , where Λ and π denote their weak− * continuous extensions to the second dual of O d and | Λ| is the 'radial' part of the polar decomposition a normal linear functional.…”