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A SPHERICAL VERSION OF THE KOWALSKI–SŁODKOWSKI THEOREM AND ITS APPLICATIONS

Part of: Commutative Banach algebras and commutative topological algebras Spaces and algebras of analytic functions Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  03 December 2020

SHIHO OI Open the ORCID record for SHIHO OI [Opens in a new window]
SHIHO OI*
Affiliation:
Niigata Prefectural Hakkai High School, Minamiuonuma949-6681, Japan Current address: Department of Mathematics, Faculty of Science, Niigata University, Niigata 950-2181, Japan e-mail: shiho-oi@math.sc.niigata-u.ac.jp
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Abstract

Li et al. [‘Weak 2-local isometries on uniform algebras and Lipschitz algebras’, Publ. Mat.63 (2019), 241–264] generalized the Kowalski–Słodkowski theorem by establishing the following spherical variant: let A be a unital complex Banach algebra and let $\Delta : A \to \mathbb {C}$ be a mapping satisfying the following properties:

  1. (a) $\Delta $ is 1-homogeneous (that is, $\Delta (\lambda x)=\lambda \Delta (x)$ for all $x \in A$ , $\lambda \in \mathbb C$ );

  2. (b) $\Delta (x)-\Delta (y) \in \mathbb {T}\sigma (x-y), \quad x,y \in A$ .

Then $\Delta $ is linear and there exists $\lambda _{0} \in \mathbb {T}$ such that $\lambda _{0}\Delta $ is multiplicative. In this note we prove that if (a) is relaxed to $\Delta (0)=0$ , then $\Delta $ is complex-linear or conjugate-linear and $\overline {\Delta (\mathbf {1})}\Delta $ is multiplicative. We extend the Kowalski–Słodkowski theorem as a conclusion. As a corollary, we prove that every 2-local map in the set of all surjective isometries (without assuming linearity) on a certain function space is in fact a surjective isometry. This gives an affirmative answer to a problem on 2-local isometries posed by Molnár [‘On 2-local *-automorphisms and 2-local isometries of B(H)', J. Math. Anal. Appl.479(1) (2019), 569–580] and also in a private communication between Molnár and O. Hatori, 2018.

Keywords

2-local mapssurjective isometriesthe Kowalski–Słodkowski theorem

MSC classification

Primary: 46B04: Isometric theory of Banach spaces
Secondary: 46B20: Geometry and structure of normed linear spaces 46J10: Banach algebras of continuous functions, function algebras 46J15: Banach algebras of differentiable or analytic functions, $H^p$-spaces 30H05: Bounded analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 3 , December 2021 , pp. 386 - 411
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Lisa Orloff Clark

Dedicated to Professor Osamu Hatori on the occasion of his retirement from Niigata University

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