Real Algebraic and Analytic Geometry
Submission: 2010, May 8.
We prove that every positive trace on a countably generated $*$-algebra can be approximated by positive traces on algebras of generic matrices. This implies that every countably generated tracial $*$-algebra can be embedded into a metric ultraproduct of generic matrix algebras. As a particular consequence, every finite von Neumann algebra with separable pre-dual can be embedded into an ultraproduct of tracial $*$-algebras, which as $*$-algebras embed into a matrix-ring over a commutative algebra.
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