Banach space decomposition

Question

A graded $C^*$ algebra is a $C^*$ algbebra $A$ equipped with an order two $*$ automorhism $\phi_A$. $A$ can be decomposed into two eigenspaces for $\phi_A$,$A=A_0\oplus A_1$,where $A_0=\{a\in A:\phi_A(a)=a\}$,$A_1=\{a\in A:\phi_A(a)=-a\}$.

Why is the composition a Banach space decomposition,not a $C^*$-algebra decomposition?

Answer 1

Because $A_1$ is not an algebra: if $a\in A_1$, then $a^2\in A_0$.