Positive cone of operators: if two selfadjoints $a$, $b$ obey $a^2 + b^2 =1$, must they commute? The question relates to the structure of the positive cone of operators, in C*-algebra. 


*

*If $a$ and $b$ are selfadjoints such that  $a^2 + b^2 = 1$ can one prove $a$ and $b$ commute? What one derives is $ ba = ba^3 + b^3a $ and $ ab = a^3b + ab^3 $ but I am stuck here, only higher powers seem to commute.

*Also, if for selfadjoints $a$ and $e$, $a^2 + e^2 = e$, can one prove that $e$ is actually a square of a selfadjoint? 

*Finally, is $1-e$ actually a square of a selfadjoint?
We know that $e$ is the square of a selfadjoint so that, by the textbooks, it is positive but I do not find a complete algebraic proof of this fact: a sum of squares is a square.
 A: OP wrote (v2):

1) If $a$ and $b$ are selfadjoints such that  $a^2 + b^2 = 1$ can one prove that $a$ and $b$ commute? 

No, it is easy to give two-dimensional counterexamples, e.g.
$$ a~=~\frac{1}{\sqrt{3}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\quad\text{and}\quad b~=~\frac{1}{\sqrt{3}}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}.$$ 
Or even simpler: Take $a$ and $b$ to be $\frac{1}{\sqrt{2}}$ times any two of the three Pauli matrices, as Trimok suggests in a comment.

2) Also, if for selfadjoints $a$ and $e$, $a^2 + e^2 = e$, can one prove that $e$ is actually a square of a selfadjoint? 

Yes, the LHS is a semipositive element, so the RHS $e$ is a semipositive element. Here we are using various characterizations of semipositive elements: 
$$a \text{ semipositive element} $$ 
$$\Updownarrow$$
$$ a \text{ selfadjoint with non-negative eigenvalues}$$
$$\Updownarrow$$ 
$$\exists b: a=b^{\dagger}b$$ 
$$\Updownarrow$$ 
$$\exists b\text{ selfadjoint}: a=b^2$$ 

3) And is $1-e$ actually a square of a selfadjoint?

Yes, rewrite the equation as $a^2 = e-e^2$. The LHS is a semipositive element, so the RHS $e-e^2$ is also a semipositive element. Hence the eigenvalues $\lambda$ of $e$ must obey $\lambda-\lambda^2\geq 0$. In other words, $0\leq \lambda\leq 1$. Hence $1-e$ is a semipositive element as well.
