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10. Testing membership in a subalgebra of the free algebra


    1. Add remark

      Problem 10.1.

      [Meric Augat] 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.