Example 5

Let $(A_i)$ be a family of involutive normed algebras. Let A be the product normed algebra of the $A_i$ (No.$^o1$ of I, p. 15). The algebra A, endowed with the involution $(x_i)^*= (x^*_i)$, is an involutive normed algebra. If each of the algebras $A_i$ is an involutive Banach algebra, then A is an involutive Banach algebra. One says that A is the product involutive normed algebra (resp. the product involutive Banach algebra) of the $A_i$.