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By Vadim G. Korneev, Ulrich Langer

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Pk are minimal projections in C and C is generated as a linear space by them. (ii) Since U' is finite dimensional, U' is strong-operator closed. Thus Q' is a von Neumann algebra and Ql'P is a von Neumann algebra with center C P . From (i), CP consists of scalar multiples of P , whence U'P is a factor. Since U'P is finite dimensional, U'P is a factor of type I, with n finite. If the linear dimension of U P is less than n2,then U P is a factor of type I, with m < n. Now Pso is a generating vector for UP.

Let 7-l be a non-separable Hilbert space and let { E , : A} be an orthogonal family of projections with sum I such that E , ( X ) is separable for each a in A. Let R' be the commutant of {Ea : a E A} and let R be R" (so that R is the (abelian) von Neumann algebra generated by {EQ : a E A}). Note that T' E R' if EaT'E, = T' for some a in A. Let 3 be the family of finite subsets of A partially ordered by inclusion. Then Ep E R. Suppose a vector pa is given for each a in F. Choose Ti (effectively in B ( E , ( X ) ) ) such that and Ti E B ( H ) .

Iv) Let (Q,}be a maximal orthogonal family of central projections in R each with the property of Q in (iii). If 0 # I - C,Qc (= Q O ) , then RQo is a von Neumann algebra of type I, and AQo is a maximal abelian subalgebra of it with the property that QO is the union of projections in AQo finite in RQo. Thus there is a non-zero central projection Q1 in RQo that is the sum of projections in AQo abelian with central carriers Q1 in RQo (hence, in R). Adjoining Q1 to {Q,} produces a family that contradicts the maximality of {Q,}.

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Approximate solution of plastic flow theory problems by Vadim G. Korneev, Ulrich Langer

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