Let A and B be Banach algebras. It is well-known that the Cartesian product space A × B equipped with the L1-norm and coordinatewise operations is a Banach algebra. In order to provide new examples of Banach algebras as well as a wealth of (counter) examples in different branches of functional analysis, the construction of an algebra product on the Cartesian product space A×B to make it a Banach algebra has been studied in a series of papers recently. The first important paper related to this construction is Lau’s paper [15] which he defined a new algebra product on the Cartesian product space A× B for the case where B is the predual of a van Neumann algebra M such that the identity of M is a multiplicative linear functional on B. Later on, Monfared [17] extended the Lau product to arbitrary Banach algebras.