Commutative matrix multiplication algorithms
In answer to the "Update": yes, for any $c$, the existence of an $O(n^c)$ non-commutative algorithm for matrix multiplication is equivalent to the existence of an $O(n^c)$ commutative algorithm for matrix multiplication (I am assuming, here, that all algorithms are algebraic, in the sense of algebraic circuits). This is because any algebraic circuit computing a bilinear map can be converted into a bilinear circuit at the expense of at most a factor of 2 in the size.