Example 6

Let A be an involutive normed algebra and let $\widetilde{A}$ be the normed algebra deduced from A by adjunction of a unit element. Endowed with the involution defined in No.$^o2$, the algebra $\widetilde{A}$ is an involutive normed algebra. If A is an involutive Banach algebra, then $\widetilde{A}$ is likewise an involutive Banach algebra.

If A is an involutive normed algebra, the closure of an involutive subalgebra is an involutive subalgebra. If $M\subset A$, the smallest closed involutive subalgebra containing M is called the closed involutive subalgebra generated by M; it is the closure of the subalgebra generated by $M\cup M^*$. If M reduces to a normal element, the closed involutive algebra generated by M is commutative, and all its elements are normal.

Analogously, if A is a unital involutive normed algebra and M a subset of A, the smallest closed involutive unital subalgebra containing M is called the closed involutive unital subalgebra generated by M; it is the closure of the unital subalgebra generated by $M\cup M^*$. If M reduces to a normal element, the closed involutive unital algebra generated by M is commutative, and all its elements are normal.

The quotient of an involutive normed algebra by a closed self-adjoint two-sided ideal, the completion, and the opposite of an involutive normed algebra are naturally involutive normed algebras.

If A is an involutive normed algebra, the set $A_h$ of hermitian elements of A is a normed real vector space.