Let $F\_{2}$ be the free group with two generators. 


Then $F\_{2}=\{\text{odd words}\}\sqcup\{\text{even words}\}$. This gives us a $Z\_{2}$ graded structure for $C^{\*}\_{red} (F\_{2})$, in a natural way.


My question:



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> Is there another $Z\_{2}$ graded structure for $C^{\*}\_{red} (F\_{2})$ which is not graded isomorphic to the above $Z\_{2}$- graded structure?
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> 
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Please consider the same question for $C^{\*}\_{red} (F\_{1}) \sim C(\mathbb{S}^{1})$.


Namely: is there a $Z\_{2}$ graded structure for $C(\mathbb{S}^{1})$ which is not graded isomorphism to the standard grading structure(decomposition to even and odd continuous functions)?



$Z_{2}$- graded structures for $C_{red} ^{*} (F_{2})$
It is well-known that $K\_1(C^\*\_{\rm red}(F\_2))={\mathbb Z^2}$ with generators given by $[u]$ and $[v]$, where $F\_2=\langle u,v\rangle$.


Now, the automorphism of order two associated with the even-odd grading is multiplying each generator by $-1$, which is homotopic to the multiplication by $1$. Hence it is trivial on $K\_1$.


On the other side, the automorphism of order two that comes mapping $u$ to $v$ and $v$ to $u$ gives rise to a non-trivial action on $K\_1$. Thus, the two automorphisms of order two cannot be conjugate - and hence the associated graded algebras cannot be isomorphic.


