On the homotopical significance of the type of von Neumann algebra factors
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The set of all projections and the set of all unitaries in a von Neumann algebra factorA are studied from the homotopical point of view relative to the operator norm topology.
Two projectionsE andF can be deformed continuously to each other if and only ifE∼F and 1−E∼1−F where ∼ denotes the equivalence of projections inA in the sense of von Neumann. In other words, the relative dimension and co-dimension are a complete homotopical invariants of projections inA and label pathwise connected components of the set of projections.
The first homotopy group π1(U(A)) of unitaries inA is shown to be 0 forA of infinite type. For typeII1 and typeI n factors, π1(U(A)) are isomorphic to additive groups of realsR and integersZ, respectively, in which the first homotopy group π1(F U(A)) of the center ofU(A) is imbedded asZ andnZ, respectively.
KeywordsNeural Network Statistical Physic Complex System Nonlinear Dynamics Operator Norm
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