In the context of $C^*$-algebras , why is the $C^*$-identity a "natural" one to choose ? ($||a^* a||=||a||^2$).
Some books try to motivate this by noting that bounded operators on a Hilbert space have the above property. Is this an optimal choice in some sense ? ($||a^*||=||a||$ could, for example, be another candidate for defining $C^*$-algebras).
From what I gather, historically $B^*$-algebras were defined first. ($||a^* a||=||a|| \cdot ||a^*||$). In that case, the same question may be directed towards the $B^*$-identity for $B^*$-algebras.
The fact that this identity is strongly used throughout and I haven't found a natural explanation for the choice, slightly dims the awesomeness of the GNS representation theorem for $C^*$-algebras for me.
Thanks.
Recall that in a unital $C^\ast$-algebra $A$, you have the spectral radius formula, i.e., for $a \in A$ normal, $$ \|a\| = \rho(a) := \sup\{|\lambda| \mid \text{$a-\lambda 1$ is not invertible in $A$}\}. $$ So, the $C^\ast$-identity $\|a\|^2 = \|a^\ast a\|$ gives you the norm of an arbitrary element $a$ in terms of the norm of self-adjoint element $a^\ast a$, which can be computed in terms of the spectral radius formula: $$ \|a\| = \sqrt{\|a^\ast a\|} = \sqrt{\rho(a^\ast a)} = \sqrt{\sup\{|\lambda| \mid \text{$a^\ast a-\lambda 1$ is not invertible in $A$}\}}. $$ Hence, phrasing things in terms of the $C^\ast$-identity at least emphasizes, if not makes manifest, the deep fact that the norm of a unital $C^\ast$-algebra $A$ is entirely determined by the (purely!) algebraic properties of the unital $\ast$-algebra $A$, or in other words, a unital $\ast$-algebra $A$ admits at most one norm making it into a pre-$C^\ast$-algebra.