A notion of an amenable action of a Lau algebra on a von Neumann algebra is introduced. Amenable action of Lau algebras are characterized by a fixed point property. In particular, a fixed point characterization of H-amenable representation of groups is given.
Let $A$ and $B$ are Banach algebras such that $B$ is an algebraic Banach $A$-bimodule. We define a product $A\bowtie B$, which is a strongly splitting Banach algebra extension of $A$ by $B$. We obtain characterizations of bounded approximate identities, spectrum, topological center, and study the amenability and weak amenability of these products. This provides a unified approach for obtaining some known results of both module extensions and Lau product of Banach algebras.