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