The universal C*-algebra of a contraction

We say that a contractive Hilbert space operator is universal if there is a natural surjection from its generated C*-algebra to the C*-algebra generated by any other contraction. A universal contraction may be irreducible or a direct sum of (even nilpotent) matrices; we sharpen the latter fact and its proof in several ways, including von Neumann-type inequalities for noncommutative *-polynomials. We also record properties of the unique C*-algebra generated by a universal contraction, and we show that it can be used similarly to C*(F_2) in various Kirchberg-like reformulations of Connes' Embedding Problem (some known, some new). Finally we prove some analogous results for universal C*-algebras of noncommuting row contractions and universal Pythagorean C*-algebras.