The linear isometry group of the Gurarij space is universal

TL;DR

This paper constructs the Gurarij space using a Katetov-like method and proves that its linear isometry group is a universal Polish group, answering a longstanding open question.

Contribution

It introduces a novel construction of the Gurarij space based on a generalized Arens-Eells space, demonstrating the universality of its linear isometry group.

Findings

The linear isometry group of the Gurarij space is a universal Polish group.

The construction adapts Katetov's technique to the setting of normed spaces.

Provides a positive answer to Uspenskij's question about the Gurarij space.

Abstract

We give a construction of the Gurarij space, analogous to Katetov's construction of the Urysohn space. The adaptation of Katetov's technique uses a generalisation of the Arens-Eells enveloping space to metric space with a distinguished normed subspace. This allows us to give a positive answer to a question of Uspenskij, whether the linear isometry group of the Gurarij space is a universal Polish group.