By Shavkat Ayupov, Abdugafur Rakhimov, Shukhrat Usmanov (auth.)
The thought of operator algebras performing on a Hilbert house was once initiated in thirties via papers of Murray and von Neumann. In those papers they've got studied the constitution of algebras which later have been referred to as von Neu mann algebras or W* -algebras. they're weakly closed advanced *-algebras of operators on a Hilbert area. at the present the speculation of von Neumann algebras is a deeply constructed idea with numerous purposes. within the framework of von Neumann algebras concept the examine of fac tors (i.e. W* -algebras with trivial centres) is essential, because they're relatively basic and research of common W* -algebras will be decreased to the case of things. hence the speculation of things is without doubt one of the major instruments within the constitution concept of von Neumann algebras. in the course of 60th Topping [To 1] and Stormer [S 2] have ini tiated the examine of Jordan (non associative and actual) analogues of von Neumann algebras - so referred to as JW-algebras, i.e. actual linear areas of self adjoint opera.tors on a posh Hilbert house, which include the identification operator 1. closed with admire to the Jordan (i.e. symmetrised) product advent 2 x zero y = ~(Xy + yx) and closed within the susceptible operator topology. The constitution of those algebras has occurred to be on the subject of the struc ture of von Neumann algebras and it used to be attainable to use principles and meth ods just like von Neumann algebras concept within the learn of JW-algebras.
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Extra resources for Jordan, Real and Lie Structures in Operator Algebras
Sample text
If we multiply this equality by e from the right, then we obtain eb = ebe. Hence eb = be, and thus e commutes with any skew-symmetric element from ~. Since any element x from ~ can be decomposed as a b, where a E ~s = A, b E~, b* = -b, we obtain that e commutes with any element from ~. Therefore, e belongs to the centre of the algebra U = ~ + i~. But U is a W* -factor, therefore e = 0, or 1. e. A is a purely real JW-factor. , not of type I). Suppose that contrary and let q be a minimal projection in A.
Consider the map al . BB from Ul into U2 . This map evidently, is a *-automorphism of Ul, which acts identically on AI. e. a2(B(x)) = B(x), and then B- 1 . a2 . B(x) = x, therefore al·rl·a2B(x)=al(x)=:r. e. alB-la2B = id, or a2B = Bal. sion to a *-isol1lorphism 1 . al . Proof of the "if" part. Let a2· B = B . e. a2 Al A2 = {:r E Ul = {y E U2 : y : x = B . al . B- 1 • Then = x* = al(x)}, = y* = B· al . rl(y)}. Hence :to E Al iff B( x) E A 2. Therefore the restriction of B on Al an isomorphism between Al and A 2 .
A trace on a JW-algebra A (respectively on a real W* -algebra~) is a function r on the set A + of positive elements of A (respectively on the set ~+ of positive elements of~) with values in [0, +00], satisfying the following conditions: = r(x) + 1) r(x + y) 2) r(Ax) = Ar(x), for all x E A+ (respectively x E ~+) and A E R, A ~ 0, where O· ( +(0) = 0; 3) r( sxs) = r( x), for an arbitrary symmetry s and x in A (respectively for each symmetry s and x in ~). r(y) , for all x,y E A+ (respectively X,y E ~+) The trace r is said to be faithful, if r( x) > 0 for all nonzero x E A +; if r(l) < +00; semifinite if given any x E A+ there is a nonzero yEA +, y ~ x with r(y) < +00.