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Continuous Selection of Comparison Unitaries in II1 Factors
In a II1 von Neumann factor, one projection is Murray-von Neumann subequivalent to another exactly when its trace is no larger; this Idea covers the strengthening that the implementing partial isometries can be chosen to vary continuously over a compact parameter space. The construction feeds the tracial comparison data into Michael's continuous selection theorem for lower-semicontinuous set-valued maps to produce a continuous field of witnessing unitaries. The payoff is a resolution of the trace problem for trivial W*-bundles C_sigma(X, M) whose base space X has covering dimension at most one, showing every tracial state on such a bundle is realized. Learners see how comparison theory, the Cuntz semigroup, and topological selection combine.