The universal C∗-algebra of a contraction

Abstract

We call a contractive Hilbert space operator 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 in several ways, including von Neumann-type inequalities for ∗-polynomials. We also record properties of the unique C∗-algebra generated by a universal contraction and show that it can replace C∗(F2) in various Kirchberg-like reformulations of Connes' embedding problem (some known, some new). Finally we prove some analogous results for universal row contraction and universal Pythagorean C∗-algebras.