Determine the universal commuting dilation constant C2

Determine the exact value of the universal commuting dilation constant C_2, defined as c(u_u, u_0), where u_u denotes the universal 2-tuple of free unitaries (the canonical generators of C^*(F_2)) and u_0 denotes the universal 2-tuple of commuting unitaries (the canonical generators of C(T^2)); equivalently, C_2 is the smallest c such that every pair of contractions dilates to a pair of commuting normal operators whose norms are at most c.

Background

The constant C_2 is central in dilation theory and matrix convexity, capturing the minimal norm inflation needed to dilate any pair of contractions to commuting normal operators. Known general bounds satisfy sqrt(d) ≤ C_d ≤ sqrt(2d), and for d=2 the best upper bound currently equals the trivial bound 2. The paper provides numerical evidence suggesting that C_2 might be strictly less than 2, motivating a precise determination of C_2.

Establishing C_2 exactly would resolve the fundamental question for pairs of contractions and refine the broader understanding of commuting dilations, with implications in operator algebras, quantum information, and optimization.