Twisted tensor product of multiplier Hopf (*-)algebras

We combine two (not necessarily commutative or co-commutative) multiplier Hopf (*-)algebras to a new multiplier Hopf (*-)algebra. We use the construction of a twisted tensor product for algebras, as introduced by A. Van Daele. We then proceed to find sufficient conditions on the twist map for this twisted tensor product to be a multiplier Hopf (*-)algebra with the natural comultiplication. For usual Hopf algebras, we find that Majidʹs double crossed product by a matched pair of Hopf algebras is exactly the twisted tensor product Hopf algebra according to an appropriate twist map. Starting from two dually paired multiplier Hopf (*-)algebras we construct the Drinfelʹd double multiplier Hopf algebra in the framework of twisted tensor products.