Modular Paulsen Problem and Modular Projection Problem

Based on the solution of $\mathbf{\text{Paulsen}}\mathbf{\text{Problem}}$ by Kwok, Lau, Lee, and Ramachandran [$\mathit{\text{STOC’18-Proceedings}}\mathit{\text{of}}\mathit{\text{the}}\mathit{\text{50th}}\mathit{\text{Annual}}\mathit{\text{ACM}}\mathit{\text{SIGACT}}\mathit{\text{Symposium}}\mathit{\text{on}}\mathit{\text{Theory}}\mathit{\text{of}}\mathit{\text{Computing,}}\mathit{\text{2018}}$] and independently by Hamilton, and Moitra [$\mathit{\text{Isr.}}\mathit{\text{J.}}\mathit{\text{Math.,}}\mathit{\text{2021}}$] we study Paulsen Problem and Projection Problem in the context of Hilbert C*-modules. We show that for commutative C*-algebras, if Modular Paulsen Problem has a solution, then Modular Projection Problem also has a solution. We formulate the problem of operator scaling for matrices over C*-algebras.