On the universal commuting dilation constant

The universal commuting dilation constant $C_d$ is the smallest constant $\alpha$ such that every $d$-tuple of contractions dilates to a commuting $d$-tuple of normal operators with norm at most $\alpha$. The work of several authors shows that $1.5438\lesssim C_2\le 2$, and it has been asked on a few accounts whether $C_2<2$. We provide a positive answer that, in fact, produces a near optimal upper bound of $C_2\le \frac{2}{\sqrt{\varphi }}$ where $\varphi$ is the golden ratio. This tightens the gap on the universal commuting dilation constant to $1.5438\lesssim C_2\lesssim 1.5724$. We also tighten the known upper and lower bounds on $C_d$ for arbitrary $d$-tuples.