If I have a $Z_2$ (group with two elements) action on a $C^*$-algebra $A$, i.e. $A$ is graded by the definition of Ralf Meyer for example, then how may I decompose $A$ into a direct sum $A_0\oplus A_1$ such that I get the definition of a graded $C^*$-algebra of Blackadar for example.
The reason for this question is that with the definition of a $Z_2$ action I wonder how to declare the degree of an element, in particular how to define graded commutator.
If $\sigma$ is the automorphism corresponding to the nontrivial group element, then $$A_0=\{a\in A:\sigma(a)=a\}$$ and $$A_1=\{a\in A:\sigma(a)=-a\}.$$