C* identity

Summary

The C* identity is the condition $\norm{a^* a} = \norm{a}^2$ on a Banach algebra with an involution. A Banach $*$-algebra whose norm satisfies it is a C*-algebra. This one equation ties the norm to the algebra so tightly that the algebraic structure determines the norm. Indeed, $\norm{a}$ is the square root of the spectral radius of $a^* a$.