Idempotent states on locally compact quantum groups

TL;DR

This paper characterizes idempotent states on unimodular coamenable locally compact quantum groups, linking them to invariant C*-subalgebras, and explores their properties and invariance under quantum group actions.

Contribution

It establishes a one-to-one correspondence between idempotent states and invariant C*-subalgebras, including Haar idempotents, extending the theory beyond unimodularity.

Findings

Idempotent states correspond to right invariant expected C*-subalgebras.

Haar idempotents are characterized and shown to be invariant under modular actions.

Coproducts restrict to continuous coactions on these subalgebras.

Abstract

Idempotent states on a unimodular coamenable locally compact quantum group A are shown to be in one-to-one correspondence with right invariant expected C*-subalgebras of A. Haar idempotents, that is, idempotent states arising as Haar states on compact quantum subgroups of A, are characterised and shown to be invariant under the natural action of the modular element. This leads to the one-to-one correspondence between Haar idempotents on A and right invariant symmetric expected C*-subalgebras of A without the unimodularity assumption. Finally the tools developed in the first part of the paper are applied to show that the coproduct of a coamenable locally compact quantum group restricts to a continuous coaction on each right invariant expected C*-subalgebra.