Determine the precise value of Cd for all d

Determine the exact value of the universal commuting dilation constant C_d for each integer d ≥ 2, where C_d = c(u_u, u_0) for the corresponding d-tuples; equivalently, C_d is the smallest c such that every d-tuple of contractions dilates to a d-tuple of commuting normal operators with each coordinate having norm at most c.

Background

The universal commuting dilation constant C_d generalizes C_2 to d-tuples of contractions. Current bounds are sqrt(d) ≤ C_d ≤ sqrt(2d); for d=2 and d=3 it is known that C_d > sqrt(d), indicating the lower bound is not tight, while the upper bound may also be improvable. Precisely determining C_d for all d remains unresolved.

Progress on the exact values would sharpen several results in dilation theory and matrix convex sets, and link to estimates involving free and commuting unitaries used in operator system theory and quantum information.