TAILIEUCHUNG - Báo cáo toán học: "Similarities of II_1 factors with property $\Gamma$ "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Điểm giống nhau của II_1 bao thanh toán với tài sản Gamma $ \ $. | Copyright by INCREST 1986 J- OPERATOR THEORY 15 1986 281-288 SIMILARITIES OF II FACTORS WITH PROPERTY r ERIK CHRISTENSEN Dedicated to Professor R. V. Kadisọn on his 60th birthday I. INTRODUCTION Shortly after Haagerup s proof of the Grothendieck-Ringrose-Pisier inequality 11 appeared J found the result stated in Proposition . The result tells that a representation of a IT factor can be extended to the bounded Hilbert-Schmidt operators in the semifinite properly infinite von Neumann algebra y X R T2 N . Unfortunately I have not been able to prove that is in general completely bounded but one can fairly easily see that 0 must be completely bounded when restricted to certain big subalgebras . the relative commutant of an abelian von Neumann subalgebra without atoms. A demonstration of this result is the main ingredient in the proof of Theorem . Recently Popa 15 has discussed derivations of a finite factor into the compacts and he discovered also that the set of subalgebras which behave nicely do have a remarkable structure. 1 will not discuss the similar question for similarities but state that analogous results for the set of nice subalgebras are obtainable. When the above result concerning relative commutants has been shown one considers an ultraproduct. of the given factor and shows that the similarity lifts to a similarity of the ultraproduct. Since the von Neumann algebra is assumed to have property r the algebra of central sequences in the ultraproduct commutes with the constant sequences and we find that the constant sequences is contained in the relative commutant of something big and hence the similarity is completely bounded. It is of course a main point that the algebra of central sequences do not have atoms. It is well known that the similarity question is connected to various other problems concerning derivations. In Section 3 we mention some consequences of our results. 2. SIMILARITIES In this section denotes a von Neumann algebra factor of .

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