The Larson-Sweedler theorem for weak multiplier Hopf algebras

TL;DR

This paper extends the Larson-Sweedler theorem to weak multiplier Hopf algebras, showing that under certain conditions, such structures with faithful integrals are indeed weak multiplier Hopf algebras, advancing quantum groupoid theory.

Contribution

It proves the Larson-Sweedler theorem for weak multiplier Hopf algebras, broadening the class of algebraic structures where the theorem applies.

Findings

Weak multiplier bialgebras with faithful integrals are weak multiplier Hopf algebras.

The result supports the development of locally compact quantum groupoids.

The theorem generalizes previous results to infinite-dimensional settings.

Abstract

The Larson-Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra. The result has been generalized to finite-dimensional weak Hopf algebras by Vecserny\'es. In this paper, we show that the result is still true for weak multiplier Hopf algebras. The notion of a weak multiplier bialgebra was introduced by B\"ohm, G\'omez-Torrecillas and L\'opez-Centella. In this note it is shown that a weak multiplier bialgebra with a regular and full coproduct is a regular weak multiplier Hopf algebra if there is a faithful set of integrals. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting. Our treatment of this material is motivated by the prospect of such a theory.