A $C^{*}$ algebra $A$ is graded by $\mathbb{Z}_{n}$ iff it can be acted by $\mathbb{Z}_{n}$. So we associate the $C^{*}$ algebra $A\rtimes \mathbb{Z}_{n}$ to a $\mathbb{Z}_{n}$-graded $C^{*}$ algebra.

Now what about if the grading group is an arbitrary finite group $G$? Is a $G$-graded structure on $A$ equivalent to existence of a $G$-action on $A$? And is there a natural $C^{*}$ algebra associated to a $G$-graded $C^{*}$ algebra?

A $C^{*}$ algebra $A$ is graded by $\mathbb{Z}_{n}$ iff it can be acted by $\mathbb{Z}_{n}$. So we associate the $C^{*}$ algebra $A\rtimes \mathbb{Z}_{n}$ to a $\mathbb{Z}_{n}$-graded $C^{*}$ algebra.

Now what about if the grading group is an arbitrary finite group $G$? Is it true to say that: existence of a $G$-graded structure on $A$ is equivalent to existence of a $G$-action on $A$? And is there a natural $C^{*}$ algebra associated to a $G$-graded $C^{*}$ algebra?

A $C^{*}$ algebra $A$ is graded by $\mathbb{Z}_{n}$ iff it can be acted by $\mathbb{Z}_{n}$. So we associate the $C^{*}$ algebra $A\rtimes \mathbb{Z}_{n}$ to a $\mathbb{Z}_{n}$-graded $C^{*}$ algebra.

Now what about if the grading group is an arbitrary finite group $G$? Is it equivalent to existence of a $G$-action on $A$? And is there a natural $C^{*}$ algebra associated to a $G$-graded $C^{*}$ algebra?

A $C^{*}$ algebra $A$ is graded by $\mathbb{Z}_{n}$ iff it can be acted by $\mathbb{Z}_{n}$. So we associate the $C^{*}$ algebra $A\rtimes \mathbb{Z}_{n}$ to a $\mathbb{Z}_{n}$-graded $C^{*}$ algebra.

Now what about if the grading group is an arbitrary finite group $G$? Is a $G$-graded structure on $A$ equivalent to existence of a $G$-action on $A$? And is there a natural $C^{*}$ algebra associated to a $G$-graded $C^{*}$ algebra?