Douglas-Rudin Approximation theorem for operator-valued functions on the unit ball of ${\mathbb{C}}^d$

Douglas and Rudin proved that any unimodular function on the unit circle $\text{T}$ can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of ${\mathbb{C}}^d$. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay’s result [Proc. Lond. Math. Soc. 2009] on the approximation of matrix-valued unimodular functions on $\text{T}$.