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The algebra of symmetric analytic functions on L∞

Part of: Measures, integration, derivative, holomorphy Commutative Banach algebras and commutative topological algebras Linear function spaces and their duals

Published online by Cambridge University Press:  31 May 2017

Pablo Galindo ,
Taras Vasylyshyn  and
Andriy Zagorodnyuk
Pablo Galindo
Affiliation:
Departamento de Análisis Matemático, Universidad de Valencia, Av. Doctor Moliner 50, Burjasot (Valencia) 46100, Spain (pablo.galindo@uv.es)
Taras Vasylyshyn
Affiliation:
Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Street, Ivano-Frankivsk 76000, Ukraine (taras.v.vasylyshyn@gmail.com; andriyzag@yahoo.com)
Andriy Zagorodnyuk
Affiliation:
Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Street, Ivano-Frankivsk 76000, Ukraine (taras.v.vasylyshyn@gmail.com; andriyzag@yahoo.com)
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Extract

We consider the algebra of holomorphic functions on L∞ that are symmetric, i.e. that are invariant under composition of the variable with any measure-preserving bijection of [0, 1]. Its spectrum is identified with the collection of scalar sequences such that is bounded and turns to be separable. All this follows from our main result that the subalgebra of symmetric polynomials on L∞ has a natural algebraic basis.

Keywords

polynomials and analytic functions on Banach spacesmeasure preservingsymmetric polynomialsspectra of algebras

MSC classification

Secondary: 46J15: Banach algebras of differentiable or analytic functions, $H^p$-spaces 46J20: Ideals, maximal ideals, boundaries 46E15: Banach spaces of continuous, differentiable or analytic functions 46E25: Rings and algebras of continuous, differentiable or analytic functions 46G20: Infinite-dimensional holomorphy
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 147 , Issue 4 , August 2017 , pp. 743 - 761
Copyright
Copyright © Royal Society of Edinburgh 2017 

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