2018
DOI: 10.4153/cjm-2018-010-3
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Uniqueness of the von Neumann Continuous Factor

Abstract: For a division ring D, denote by 𝓜D the D-ring obtained as the completion of the direct limit with respect to themetric induced by its unique rank function. We prove that, for any ultramatricial D-ring 𝓑 and any non-discrete extremal pseudo-rank function N on 𝓑, there is an isomorphism of D-rings , where stands for the completion of 𝓑 with respect to the pseudo-metric induced by N. This generalizes a result of von Neumann. We also show a corresponding uniqueness result for *-algebras over fields F with p… Show more

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Cited by 9 publications
(16 citation statements)
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“…Using this, we show that there exists a unique Sylvester matrix rank function rk A on A such that rk A (χ U ) = µ(U ) for all clopen subsets U of X . Moreover we show that rk A ∈ ∂ e P(A), the set of extreme points of the compact convex set P(A) of all the Sylvester matrix rank functions on A, and using a recent result by the authors [3], we identify the algebra R rk with the continuous von Neumann factor M K , in complete analogy with the analytic setting described above.…”
Section: Introductionmentioning
confidence: 82%
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“…Using this, we show that there exists a unique Sylvester matrix rank function rk A on A such that rk A (χ U ) = µ(U ) for all clopen subsets U of X . Moreover we show that rk A ∈ ∂ e P(A), the set of extreme points of the compact convex set P(A) of all the Sylvester matrix rank functions on A, and using a recent result by the authors [3], we identify the algebra R rk with the continuous von Neumann factor M K , in complete analogy with the analytic setting described above.…”
Section: Introductionmentioning
confidence: 82%
“…We also briefly study, in §3.2, a motivating example, the lamplighter group algebra K [Z 2 Z]. We use the above construction in §4, together with the main result in [3], to obtain an embedding of A = C K (X ) T Z into the well-known von Neumann continuous factor M K (Theorem 4.7, Proposition 4.8, Theorem 4.9). As a consequence, we get a faithful extremal Sylvester matrix rank function on A, and prove a uniqueness statement for such a rank function provided that a suitable compatibility condition with µ is satisfied.…”
Section: Introductionmentioning
confidence: 99%
“…If G is ICC, all the known examples of R K [G] are either isomorphic to Mat n (D) or to M D for some division * -ring D. Moreover, in [33] G. Elek has shown that if H is countable and amenable, then R C[C 2 H] is isomorphic to M C . It seems that his proof can be adapted to show that R K[C 2 H] is isomorphic to M K for any subfield K of C. Interesting related results have been proved in [7] by P. Ara and J. Claramunt. All this together suggests the following question.…”
Section: The Hanna Neumann Conjecturementioning
confidence: 95%
“…are isomorphic as epic * -regular K[F ]-rings. 7 The solution of the sofic Lück approximation conjecture for amenable groups over fields of arbitrary characteristic…”
Section: A Structural Reformulation Of the General Lück Approximationmentioning
confidence: 99%
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