Edit(A confession): I just realized that the question is trivial: Since one can easily prove that the convex hull of the spectrum of every nontrivial homogeneous element of a $\mathbb{Z}_{n}$-graded $C^{*}$ algebra must contatin the origin. But is this statement true if we replace $\mathbb{Z}_{n}$ by an arbitrary finite group $G$?
So I confess that the last statement of this note was a trivial statement about Kaplanski-Kadison conjecture. http://arxiv.org/abs/1110.0091
But I am still interested to know:"Is there a $\mathbb{Z}_{2}$ graded structure for $C^{*}_{red} F_{2}$ such that a non trivial homogeneous element is invertible and can be connected to the identity?(A question which I explained in question 2 in the above paper)
Edit: According to the comment of Qiaochu Yuan I realize that $\mathbb{C}^{2}$ is a counter example. So I add the assumption "simplicity" to this edited version
Note: In this post, the cyclic group of order $n$ is denoted by $\mathbb{Z}_{n}$.
For a unital simple $C^{*}$ algebra $A$, are the following two statements equivalent?
A has no nontrivial projection.
For all natural number $n$ and every $\mathbb{Z}_{n}$-graded structure for $A$, the convex hull of the spectrum of every nontrivial homogeneous element $a\in A$, must contain the origin.
Note that 1. implies 2.(for this implication we do not need "simplicity")
By a non trivial homogeneous element of a $\mathbb{Z}_{n}$-graded algebra $A=\oplus_{i=0}^{n-1} A_{i}$, we mean an element $a\in A_{i}$ for some $i\neq 0$.
Can one consider the matrix algebras as a possible counter example? Is it easy to classify all $\mathbb{Z}_{m}$ graded structures for matrix algebra $M_{n}(\mathbb{C})$?
As another related question: What reference classified the $\mathbb{Z}_{n}$ graded structures for $C^{*}_{red} \Gamma$ where $\Gamma$ is a torsion free group?
