On representations of isometric isomorphisms between some monoid of functions

TL;DR

This paper characterizes isometric isomorphisms between monoids of nonnegative 1-Lipschitz functions on invariant metric groups, showing they are derived from isometric isomorphisms of the groups themselves.

Contribution

It establishes a canonical correspondence between isometric isomorphisms of function monoids and their underlying groups of units, revealing structural insights.

Findings

Isometric isomorphisms correspond to group isometries.

The monoid structure is preserved under these isomorphisms.

The result applies to nonnegative 1-Lipschitz functions on invariant metric groups.

Abstract

We prove that each isometric isomorphism, between the monoids of all nonegative 1-Lipschitz maps defined on invariant metric groups and equiped with the inf-convolution law, is given canonically from an isometric isomorphism between their groups of units.