Is there a compact Reiemannian manifold $M$ for which the following complex $C^{\*}$ algebra does not have a nontrivial idempotent:


$A=Hom(E,E)$ where $E$ is the complexification of $TM$.


Of course any such $M$ can not have an almost complex structure. 



Almost complex structure and nontrivial idempotents
No: if $M^n$ is $n$-dimensional then $E = TM \otimes\_{\mathbb{R}} \mathbb{C}$ is $n$-complex dimensional so classified by a map $M \to BU(n)$. By obstruction theory this lifts to $BU(n-1)$ (the obstructions are in the groups $H^{i+1}(M^n; \pi\_i(S^{2n-1}))$, which all vanish). 


Hence there is an isomorphism of vector bundles $E \cong \mathbb{C} \oplus E'$, and projection to $E'$ gives a nontrivial idempotent.


