Modulus support functionals, Rajchman measures and peak functions

TL;DR

This paper explores the properties of modulus support functionals and Rajchman measures, providing a counterexample related to the Bishop-Phelps theorem and demonstrating the non-triviality of certain functional sets.

Contribution

It introduces a $c_0$-analog of Lomonosov's counterexample, answering an open question about the set of sup-attaining functionals.

Findings

The set of sup-attaining functionals is non-trivial.

Counterexample to the complex Bishop-Phelps theorem on modulus support functionals.

Extension of Lomonosov's example to the $c_0$ setting.

Abstract

In 2000 V. Lomonosov suggested a counterexample to the complex version of the Bishop-Phelps theorem on modulus support functionals. We discuss the $c_0$-analog of that example and demonstrate that the set of sup-attaining functionals is non-trivial, thus answering an open question, asked in \cite{KLMW}.