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# On certain decompositional properties of von Neumann algebras

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It is well known that if α and β are commuting *-automorphisms of a von Neumann algebra M satisfying the equation α + α-1 = β + β-1 then M can be decomposed into a direct sum of subalgebras Mp and M(l − p) by a central projection p in M such that α = β on Mp and α = β-1 on M(1 − p) (see, for instance, [6], [7], [2]). Originally this equation arose in the Tomita-Takesaki theory (see, for example, [11]) in the form of one-parameter modular automorphism groups and later on it has been studied for arbitrary automorphisms and one-parameter groups of automorphisms on von Neumann algebras [7], [8], [9]. In the case of automorphism groups satisfying the above equation, one has a similar decomposition but this time without assuming the commutativity condition (cf. [7], [8]). For another relevant work on one-parameter groups of automorphisms which is close to our papers [7] and [8], we refer to Ciorănescu and Zsidó [1]. Regarding applications, this equation has been used for arbitrary automorphisms in the geometric interpretation of the Tomita-Takesaki theory [2] and in the case of automorphism groups it has been a fundamental tool in the generalization of the Tomita-Takesaki theory to Jordan algebras [3]. We may remark that the decomposition in the commuting case [6], [7] is much simpler than in the case of automorphism groups in the non-commutative situation [8]. In some cases one can obtain the decomposition for an arbitrary pair of automorphisms without assuming their commutativity but the problem in the general case has been unresolved. Recently we have shown that if α and β are *-automorphisms of a von Neumann algebra M satisfying the equation α + α-1 = β + β-1 (without assuming the commutativity of α and β) then there exists a central projection p in M such that α2= β2 on Mp and α2 = β−2 on M(l − p) [10].

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## References

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1.Ciorănescu, I. and Zsidó, L., Analytic generators for one-parameter groups, Tôhoku Math. J. (2) 28 (1976), 327362.
2.Haagerup, U. and Skau, C. F., Geometric aspects of the Tomita-Takesaki theory II, Math. Scand. 48(1981), 241252.
3.Haagerup, U. and Hanche-Olsen, H., Tomita-Takesaki theory for Jordan algebras, J. Operator Theory 11 (1984), 343364.
4.Rudin, W., Functional analysis (Tata McGraw-Hill, 1974).
5.Thaheem, A. B., Decomposition of a von Neumann algebra relative to a *-automorphism, Proc. Edinburgh Math. Soc. (2) 22 (1979), 910.
6.Thaheem, A. B., Decomposition of a von Neumann algebra, Rend. Sem. Mat. Univ. Padova 65 (1981), 17.
7.Thaheem, A. B. and Awami, M., A short proof of a decomposition theorem of a von Neumann algebra, Proc. Amer. Math. Soc. 92 (1984), 8182.
8.Thaheem, A. B., Daele, A. van and Vanheeswijck, L., A result on two one-parameter automorphism groups, Math. Scand. 51 (1982), 261274.
9.Thaheem, A. B., A bounded map associated to a one-parameter group of *-automorphisms of a von Neumann algebra, Glasgow Math. J. 25 (1984), 135140.
10.Thaheem, A. B., On pairs of automorphisms of von Neumann algebras, preprint.
11.van Daele, A., A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras, J. Fund. Anal. 15 (1974), 387393.

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