Lipschitz positive decompositions of von Neumann preduals

We prove that the selfadjoint predual of a complex von Neumann algebra admits a Lipschitz positive decomposition if and only if the algebra is subhomogeneous. In particular, infinite-dimensional selfadjoint trace class gives a negative answer to the Lipschitz decomposition problem, confirming Messerschmidt's 2018 conjecture. For a nonzero subhomogeneous algebra of maximal matrix degree $d$, the least Lipschitz constant of a positive majorant $q(x)\geq\pm x$ is attained and equals the corresponding constant $c_d$ for $d\times d$ trace class. Among nonzero von Neumann algebras, the optimal constant equals one exactly for the abelian algebras. The finite-dimensional constants satisfy $$c\log\log N\leq c_N\leq C\log N$$ for all sufficiently large $N$, with absolute constants $c,C>0$. The lower bound applies to arbitrary positive majorants; the upper bound comes from the classical absolute-value estimate.

GitHub repository: https://github.com/eulogica/trace-class-lipschitz-decomposition