Twisted Algebras of Multiplier Hopf (${}^{\ast }$-)algebra

In this paper we study twisted algebras of multiplier Hopf (${}^{\ast }$-)algebras which generalize all kinds of smash products such as generalized smash products, twisted smash products, diagonal crossed products, L-R-smash products, two-sided crossed products and two-sided smash products for the ordinary Hopf algebras appeared in [P-O].