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10. Testing membership in a subalgebra of the free algebra


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      Problem 10.1.

      [org.aimpl.user:maugat@wustl.edu] If $\mathcal{P}$ is a finite collection of noncommutative polynomials and $q$ is a noncommutative polynomial, then is $q$ in the unital subalgebra generated by $\mathcal{P}$ if and only if $q(X) = q(Y)$ for all matrices $X$ and $Y$ such that $p(X) = p(Y)$ for all $p\in \mathcal{P}$.
          The answer is no. The the coordinates of $P = (x, xy, yx, y + yxy)$ do not generate the free algebra, yet $P$ is injective on matrices.
        1. Remark. There may be reasonable extensions of this question to the free skew field.

              Cite this as: AimPL: Noncommutative inequalities, available at http://aimpl.org/noncommineqV.