[Paper Review] Polynomially spectrum-preserving maps between commutative Banach algebras

Osamu Hatori, Takeshi Miura+1 more|ArXiv.org|Apr 15, 2009

Advanced Topics in Algebra12 references3 citations

TL;DR

This paper investigates polynomially spectrum-preserving maps between unital, semi-simple, commutative Banach algebras, showing that for certain two-variable polynomials $ p(z,w) $, a surjective map $ T $ satisfying $ \sigma(p(Tf,Tg)) = \sigma(p(f,g)) $ for all $ f,g \in A $ must be an algebra isomorphism. The key result establishes that such spectrum-preserving conditions force linearity and multiplicativity under mild assumptions, generalizing classical results on spectrum-preserving maps.

ABSTRACT

Let $A$ and $B$ be unital semi-simple commutative Banach algebras. In this paper we study two-variable polynomials $p$ which satisfy the following property: a map $T$ from $A$ onto $B$ such that the equality \[ σ(p(Tf,Tg))=σ(p(f,g)), \quad f,g \in A \] holds is an algebra isomorphism.

Motivation & Objective

• To investigate the conditions under which spectrum-preserving maps between commutative Banach algebras are forced to be algebra isomorphisms.
• To extend classical results on spectrum-preserving linear maps to nonlinear settings using polynomial conditions.
• To determine for which two-variable polynomials $ p(z,w) $ the equality $ \sigma(p(Tf,Tg)) = \sigma(p(f,g)) $ implies that $ T $ is an algebra isomorphism.
• To generalize the Gleason-Kahane-Żelazko theorem to nonlinear, spectrum-preserving maps via polynomial identities.

Proposed method

• Utilizes the Kowalski-Słodkowski theorem to show that spectrum-preserving maps satisfying $ \sigma(Tf - Tg) \subset \sigma(f - g) $ are linear and multiplicative when $ B $ is semi-simple.
• Constructs a transformed map $ S(f) = T(f - b) + b $ to reduce the problem to a standard form where spectral equality implies invertibility preservation.
• Applies Gelfand theory and uniform algebras to analyze the maximal ideal space and spectral behavior of elements under $ T $.
• Uses uniform convergence on the Gelfand transform space to extend $ T $ to invertible elements and prove continuity of the extended map.
• Employs the Choquet boundary and peak functions to analyze the structure of the underlying algebras and ensure injectivity of the Gelfand transform.
• Demonstrates that the spectral condition $ \sigma(p(Tf,Tg)) = \sigma(p(f,g)) $ forces $ T $ to be an algebra isomorphism for specific polynomials $ p(z,w) $, including linear forms and certain quadratic forms.

Experimental results

Research questions

• RQ1 Under what conditions on a two-variable polynomial $ p(z,w) $ does the spectral equality $ \sigma(p(Tf,Tg)) = \sigma(p(f,g)) $ imply that $ T $ is an algebra isomorphism?
• RQ2 Can nonlinear, spectrum-preserving maps between commutative Banach algebras be shown to be linear and multiplicative using polynomial spectral conditions?
• RQ3 How does the structure of the maximal ideal space and the Gelfand transform influence the linearity and multiplicativity of such maps?
• RQ4 To what extent can the Gleason-Kahane-Żelazko theorem be extended to nonlinear, spectrum-preserving maps via polynomial identities?
• RQ5 What role does semi-simplicity play in ensuring that spectrum-preserving maps are isomorphisms?

Key findings

• For linear polynomials $ p(z,w) = az + bw $ with $ ab \neq 0 $, a spectrum-preserving map $ T $ is linear and multiplicative if $ a + b \neq 0 $, and $ T - T(0) $ is linear and multiplicative if $ a + b = 0 $.
• The map $ T $ is an algebra isomorphism if it satisfies $ \sigma(p(Tf,Tg)) = \sigma(p(f,g)) $ for all $ f,g \in A $, provided $ p(z,w) $ is a suitable two-variable polynomial.
• The proof relies on transforming $ T $ into a map $ S $ that preserves spectral inclusions, enabling application of the Kowalski-Słodkowski theorem.
• The extended map $ \bar{S} $ from $ A \cup (\mathrm{cl}(A))^{-1} $ to $ B \cup (\mathrm{cl}(B))^{-1} $ is a bijection that preserves spectral ranges.
• The Gelfand transform of $ T $ satisfies $ \widehat{Tf}(y) = \eta(y)\hat{f}(\Phi(y)) + a(\eta(y) - 1) $, where $ \eta $ is a sign-valued function and $ \Phi $ is a homeomorphism.
• The result generalizes classical spectrum-preserving theorems to nonlinear maps by using polynomial spectral conditions, establishing that such maps are necessarily algebra isomorphisms under the given spectral equality.