- (
*x*+*y*)^{*}=*x*^{*}+*y*^{*}for all*x*,*y*in*A*

- (the involution of the sum of x and y is equal to the sum of the involution of x with the involution of y)

- (λ
*x*)^{*}= λ^{*}*x*^{*}for every λ in**C**and every*x*in*A*; here, λ^{*}stands for the complex conjugation of λ.

- (
*xy*)^{*}=*y*^{*}*x*^{*}for all*x*,*y*in*A*

- (the involution of the product of x and y is equal to the product of the involution of x with the involution of y)

- (
*x*^{*})^{*}=*x*for all*x*in*A*

- (the involution of the involution of x is equal to x)

If the following property is also true, the algebra is actually a '''C^{*}-algebra''':

- ||
*x x*|| = ||^{*}*x*||^{2}for all*x*in*A*.

- (the norm of the product of x and the involution of x is equal to the norm of x squared )