Abstract
We prove a noncommutative analogue of the fact that every symmetric analytic function of (z,w) in the bidisc D2 can be expressed as an analytic function of the variables z+w and zw. We construct an analytic nc-map S from the biball to an infinite-dimensional nc-domain Ω with the property that, for every bounded symmetric function φ of two noncommuting variables that is analytic on the biball, there exists a bounded analytic nc-function Φ on Ω such that φ=Φ∘S. We also establish a realization formula for Φ, and hence for φ, in terms of operators on Hilbert space.
Highlights
Every symmetric polynomial in two commuting variables z and w can be written as a polynomial in the variables z + w and zw; every polynomial in z + w and zw determines a symmetric polynomial in z and w
Wolf showed in 1936 [11] that there is no finite basis for the ring of symmetric noncommuting polynomials over C. She gave noncommutative analogues of the elementary symmetric functions, but they are infinite in number
In this paper we extend Wolf’s results from polynomials to symmetric analytic functions in noncommuting variables within the framework of noncommutative analysis, as developed by J.L
Summary
Every symmetric polynomial in two commuting variables z and w can be written as a polynomial in the variables z + w and zw; every polynomial in z + w and zw determines a symmetric polynomial in z and w. In this paper we extend Wolf’s results from polynomials to symmetric analytic functions in noncommuting variables within the framework of noncommutative analysis, as developed by J.L. Taylor [9] and many other authors, for example [2,3,5,6,7,10]. If φ : D2 → C is analytic and symmetric in z and w there exists a unique analytic function Φ : π(D2) → C such that the following diagram commutes: D2 π π(D2) In this diagram the domain π(D2) is two-dimensional, in consequence of the fact that there is a basis of the ring of symmetric polynomials consisting of two elements, z + w and zw. An nc-domain in M∞ is a subset D of M∞ that is open in some union of Banach spaces contained in M∞ and satisfies conditions (1) and (2) of Definition 2.1. Where it is deemed helpful to indicate the space we shall use subscripts; 1n, 1 2 are the identity operators on Cn, 2 respectively
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