scholarly journals On some groups of automorphisms of von Neumann algebras with cyclic and separating vector

1969 ◽  
Vol 13 (2) ◽  
pp. 142-153 ◽  
Author(s):  
A. Z. Jadczyk
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann ◽  
Groups Of Automorphisms

scholarly journals On certain decompositional properties of von Neumann algebras

1987 ◽  
Vol 29 (2) ◽  
pp. 177-179 ◽  
Author(s):  
A. B. Thaheem
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Automorphism Groups ◽  
Geometric Interpretation ◽  
Jordan Algebras ◽  
Central Projection ◽  
Von Neumann ◽  
Modular Automorphism ◽  
Groups Of Automorphisms ◽  
Relevant Work

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].

scholarly journals On certain pairs of automorphisms of C*-algebras

1989 ◽  
Vol 46 (2) ◽  
pp. 197-211 ◽  
Author(s):  
C. J. K. Batty
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann ◽  
C Algebras ◽  
Groups Of Automorphisms

AbstractLet α and β be *-automorphisms of a C*-algebra A such that α + α β + β-1. There exist invariant ideals I1I2 and I3 of A, with II ∩ I2 ∩ I3 = {O}, containing, respectively, the range ofβ − α, the range of β − α-1, and the union of the ranges of β2 − α2 and β2 − α-2. The induced actions on the quotient algebras give a decomposition of the system (A, α, β) into systems where β = α, β = α-2 and α2 = α-2.If α and β are one-parameter groups of *-automorphisms such that α + α-1 = β + β−1, then the corresponding result is valid, and may be strengthened to assert that I1 ∩ I2 = {0}.These results are analogues and extensions of similar results of A. B. Thaheem et al. for von Neumann algebras and commuting automorphisms.

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