Norms on Lau product of normed algebras and their comparison

Given regular normed algebra U with the norm & Vert;.& Vert;(U) and normed algebra A with the norm & Vert;.& Vert;(A), and given theta is an element of sigma(A), we define a new norm & Vert;.& Vert;(R) on U x( theta)A; the theta-Lau product of U and A, which is a generalization of the operator norm on the unitization of a regular normed algebra. We study this norm and compare it with the & ell;(1)-norm and other regular norms on Lau product of normed algebras (Banach algebras) under suitable conditions. Some of our results are generalizations of similar previous results on unitization. If U and A are C-& lowast;-algebras, we show that U x (theta)A with the pointwise-defined involution and & Vert;.& Vert;(R) is a C-& lowast;-algebra, and in this way we introduce the Lau product of C & lowast;-algebras.