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Spectrally additive maps on the positive cones of the Wiener algebra

Shiho Oi, Kaito Sato

TL;DR

The paper addresses rigidity of spectrum-preserving maps on the Wiener algebra's positive cone, specifically maps that preserve the spectrum of sums. It develops Kowalski-Słodkowski-type arguments to constrain the action of such maps on the monomial basis, leading to a dichotomy on the exponent map $\psi$. The main result is that any surjective spectrum-additive map $T$ on $A(T)_+$ extends to an isometric real-linear isomorphism of $A(T)$ and must act as either the identity or complex conjugation on $A(T)_+$. This work provides a sharp Kaplansky-type rigidity phenomenon for the Wiener algebra, clarifying the structure of spectral-preserving maps in this setting.

Abstract

We study surjective maps between the positive cones of the Wiener algebra that preserve the spectrum of the sum of every two elements. We show that such maps can be extended to isometric real-linear isomorphisms of the Wiener algebra.

Spectrally additive maps on the positive cones of the Wiener algebra

TL;DR

The paper addresses rigidity of spectrum-preserving maps on the Wiener algebra's positive cone, specifically maps that preserve the spectrum of sums. It develops Kowalski-Słodkowski-type arguments to constrain the action of such maps on the monomial basis, leading to a dichotomy on the exponent map . The main result is that any surjective spectrum-additive map on extends to an isometric real-linear isomorphism of and must act as either the identity or complex conjugation on . This work provides a sharp Kaplansky-type rigidity phenomenon for the Wiener algebra, clarifying the structure of spectral-preserving maps in this setting.

Abstract

We study surjective maps between the positive cones of the Wiener algebra that preserve the spectrum of the sum of every two elements. We show that such maps can be extended to isometric real-linear isomorphisms of the Wiener algebra.

Paper Structure

This paper contains 2 sections, 9 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Proofs

Key Result

Theorem 2

Let $T:A(\mathbb{T})_{+} \to A(\mathbb{T})_{+}$ be a surjective map which satisfies Then $T(f)=f$ for all $f \in A(\mathbb{T})_{+}$ or $T(f)=\overline{f}$ for all $f \in A(\mathbb{T})_{+}$ holds.

Theorems & Definitions (16)

  • Theorem 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5: LO
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 6 more