Local units and multiplier algebras of AF corona ideals

Let A be a simple unital separable infinite-dimensional AF algebra, fix a tracial state τ, and put D = A ⊗ K. Let I be the closed trace ideal of M(D) associated with τ, and set J = I/D and Q = M(D)/D. We prove that, under the Continuum Hypothesis, the canonical inclusion Q ⊂ M(J) is proper. More precisely, there is a positive contraction m ∈ M(J) such that dist(m,Q) = 1/2 and |M(J)| = dens‖·‖ M(J) = 2^c, where c = 2^ℵ₀. The construction produces globally compatible multiplier actions on an exhaustive family of projection corners. Countable Riesz separation suffices for the limit-stage interpolation. In ZFC, every countable subset of J has a projection unit, with a lift of arbitrarily small finite trace, and every multiplier agrees on that subset on both sides with an element of J.

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