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A Spherical Version of the Kowalski-Słodkowski Theorem and Its Applications

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 Δ:A→C

be a mapping satisfying the following properties:

(a) Δ

is 1-homogeneous (that is, Δ(λx)=λΔ(x) for all x∈A , λ∈C

);

(b) Δ(x)−Δ(y)∈Tσ(x−y),x,y∈A

.

Then Δ is linear and there exists λ0∈T such that λ0Δ is multiplicative. In this note we prove that if (a) is relaxed to Δ(0)=0 , then Δ is complex-linear or conjugate-linear and Δ(1)¯¯¯¯¯¯¯¯¯¯¯Δ
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.

Ketersediaan

Informasi Detail

Judul Seri

JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY; Vol.111 Issue 3 December 2021

No. Panggil

-

Penerbit

: .,

Deskripsi Fisik

p. 386-411

Bahasa

English

ISBN/ISSN

-

Klasifikasi

NONE

Tipe Isi

-

Tipe Media

-

Tipe Pembawa

-

Edisi

-

Subjek

2-LOCAL MAPS
SURJECTIVE ISOMETRIES
THE KOWALSKI-SŁODKOWSKI THEOREM

Info Detail Spesifik

https://doi.org/10.1017/S1446788720000452

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