1、A vector space with a multiplicative structure is called an algebra, and if the vector space is also a Banach space, and if the multiplication has the proverty the $\left \| xy \right \|\leq \left \| x \right \|\left \| y \right \|$ for any two elements x and y, then it is called a Banach Algebra.

2、A C*-algebra is a Banach algebra with an involution, which means a function that associates with each element x another element x* in such a way that x**=x, $\left \| x* \right \|=\left \| x \right \|$ , (x+y)*=x*+y*, and (xy)*=y*x* for any elements x and y; this involution is required to satisfy the C*-identity $\left \| xx*\right \|=\left \| x \right \|^{2}$ .