The Larson–Sweedler theorem for weak multiplier Hopf algebras

Abstract

ABSTRACTThe Larson–Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [15]. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyés [44]. 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 Böhm et al. in [4]. 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. Weak multiplier Hopf algebras are introduced and studied in [40]. Integrals on (regular) weak multiplier Hopf algebras are treated in [43]. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting, see [13] and [14]. Our treatment of this material is motivated by the prospect of such a theory.