# Journal of Operator Theory

Volume 84, Issue 1, Summer 2020 pp. 153-184.

The universal $C^*$-algebra of a contraction**Authors**: Kristin Courtney (1), David Sherman (2)

**Author institution:**(1) Department of Mathematics, University of Virginia, Charlottesville, 22904, U.S.A. \textit{and} Mathematical Institute, WWU M\"{u}nster, M\"{u}nster, 48149, Germany

(2) Department of Mathematics, University of Virginia, Charlottesville, 22904, U.S.A.

**Summary:**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 $C^*(\mathbb{F}_2)$ 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.

**DOI:**http://dx.doi.org/10.7900/jot.2019mar11.2229

**Keywords:**universal contraction, universal $C^*$-algebra, von Neumann's inequality, Connes embedding problem

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