# Faithfulness of bi-free product states

## TL;DR

This paper investigates conditions under which bi-free product states are faithful, showing that faithfulness implies the pairs of faces are minimal tensor products, with some partial converse results.

## Contribution

It establishes a link between faithfulness of bi-free product states and the minimal tensor product structure of pairs of faces in unital C*-algebras.

## Key findings

- Faithfulness of bi-free product states implies minimal tensor product structure.
- Partial converse results connecting faithfulness and tensor products.
- Conditions for faithfulness in bi-free product states.

## Abstract

Given a non-trivial family of pairs of faces of unital C*-algebras where each pair has a faithful state, it is proved that if the bi-free product state is faithful on the reduced bi-free product of this family of pairs of faces then each pair of faces arises as a minimal tensor product. A partial converse is also obtained.