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Владикавказский математический журнал Апрель-июнь, 2003, Том 5, Выпуск 2
УДК 517.98
ONE PROPERTY OF THE WEAK COVERGENCE OF OPERATORS ITERATIONS IN VON NEUMANN ALGEBRAS
A. A. Katz
Conditions are given for *-weak convergence of iterations for an ultraweak continuous fuctional in von Neumann algebra to imply norm convergence.
Let M be a von Neumann algebra [5], acting on a separable Hilbert space H. Let T be a contraction from Л/„, to Л/,„. so that TM*+ с M*+ . On the pre-conjugate to M space Л/„, there are two topologies selected: the weak, or the ct(M*,M) topology, and the strong topology of the convergence in the norm of the space M*.
Let now T = a*, where a be an automorphism of the algebra M. We will say that T in Л/„, is mixing, if for all x € M* and A £ M, the following condition is valid:
lim (Tnx, A) = 0,
га—» оо
where
M° = {у G М* : 2/(1) = 0}.
We will say that a positive contraction T in Л/„, is completely mixing, if for all x € M" the following condition is valid:
lim ||Т"ж|| = 0.
га—»оо
The following theorem is valid:
Theorem. Let T be a pre-conjugate operator to an automorphism a of a von Neumann algebra M for which there is no invariant normal state. Then, for x € A/„,. the weak convergence of Tnx implies the strong convergence of Tnx. In particular, if T is mixing, then T is completely mixing.
<1 Let us denote by \Tnx\ the sum
(Tnx)+ + (Tnx)-, where Tnx = (Tnx)+ - (Т"ж)_
is the Hahn decomposition of the functional Tnx [4]. The sequence {^"жЦ^^ is ст(М!И,М) pre-compact [4] and, therefore, the convex envelope of the set {\Tnx\}^=l is pre-compact as well. The sequence {An is also pre-compact because it belongs to the convex envelope
of the set {|Тиж|}~=1.
Because T is pre-conjugate to an automorphism, then \Tnx\ = Tn \x\. In fact, the support of T(Tnx)+ is orthogonal to the support of Т(Т"ж)_, T(Tnx)+ - T(Tnx)_ = T(Tnx) = Tn+lx, and from the uniqueness of the Hahn decomposition [4] it follows that \Tnx\ = Tn \x\.
© 2003 Katz A. A.
Weak Convergence Of Iterations In von Neumann Algebras 35
Let x be a(M*, M)-limit point of the set {An l®]}^^. Then the functional x will be T-invariant. In fact,
Tx = lim У2(ткх,у)
ТЬу—^oo \ /
= lim
k=1
11'у —1
n7 1 • Y1 (T*®> у) - n7 1 ' у) +n7 1 • (ТП~'Х, у}
к=О
= X.
It is easy to see that x ^ 0 and, therefore, from the conditions of the theorem it follows that x = 0. Now we know that the only weakly limit point of the set {An |a?|is the point x = 0. Therefore
0= lim P"b|||= lim (An |®|)(1) = lim (Tn b|)(l) = lim ||T"b|||,
n—>oo n—>oo n—>oo n—>oo
because (Tn |®|)(1) = (Tm |®|)(1) for all n, m € N. The theorem is proven. >
References
1. Bratteli O., Robinson D. Operator Algebras and Quantum Statistical Mechanics.—New York-Heidelberg-Berlin: Springer-Verlag, 1979,— 500 p.
2. Katz A. A. Ergodic Type Theorem in von Neumann Algebras.—Ph. D. Thesis.—Pretoria: University of South Africa, 2001.—84 p.
3. Pedersen G. K. C*-algebras and their automorphism groups.—London-New York-San Francisco: Academic Press, 1979.— 416 p.
4. Sakai S. C*-algebras and W*-algebras.—Berlin: Springer-Verlag, 1971.—256 p.
5. Takesaki M. Theory of Operator Algebras. I.—Berlin: Springer-Verlag, 1979,—vii+415 p.
Статья поступила 11 апреля, 2003 Alexander A. Katz, Ph.D.
Department of Mathematics & CS, St. John's University, 300 Howard Ave., Staten Island, NY 10301, USA. E-mail: [email protected]
