# Theory of Operator Algebras II by Masamichi Takesaki

By Masamichi Takesaki

to the Encyclopaedia Subseries on Operator Algebras and Non-Commutative Geometry the speculation of von Neumann algebras used to be initiated in a chain of papers via Murray and von Neumann within the 1930's and 1940's. A von Neumann algebra is a self-adjoint unital subalgebra M of the algebra of bounded operators of a Hilbert house that is closed within the vulnerable operator topology. in response to von Neumann's bicommutant theorem, M is closed within the vulnerable operator topology if and provided that it's equivalent to the commutant of its commutant. an element is a von Neumann algebra with trivial centre and the paintings of Murray and von Neumann contained a discount of all von Neumann algebras to elements and a class of things into kinds I, IT and III. C* -algebras are self-adjoint operator algebras on Hilbert area that are closed within the norm topology. Their examine used to be started within the paintings of Gelfand and Naimark who confirmed that such algebras might be characterised abstractly as involutive Banach algebras, enjoyable an algebraic relation connecting the norm and the involution. additionally they acquired the elemental end result commutative unital C* -algebra is isomorphic to the algebra of complicated valued non-stop services on a compact house - its spectrum. considering that then the topic of operator algebras has advanced right into a large mathematical endeavour interacting with nearly each department of arithmetic and several other parts of theoretical physics.

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**Example text**

2: (i) Suppose that 2t is a left Hilbert algebra and 2to is defined by (1). 17. Hence 2to c 2t n 2t'. Suppose ~ E 2to. 3. We want to show that L1 ia ~ E 2to, ex E C. Let ex = r + is and n = [s], the integral part of s. 19 and L1m~ E 2to, m E Z, by definition. For any ry E 2t', we have for each m E Z. sup tER Ilnr (ry) L1 it+m ~ I : : ; line (L1m~) 1111 ry II. By the Phragmen-Lindel6f theorem, we have so that L1ia~ is left bounded. Since L1ia~ E :D(L1 m ), m E Z, we conclude that L1 ia ~ E 2t, which means L1 ia ~ E 2to because ex is arbitrary.

Moreover, we have IIJre(~n)II2 = IIJre(~n)Jre(~n)*II = IIf(xnx;)xnx;f(xnx;) II ::::: supltf(t)21 = 1. {l IPn(t)-f(t)I:::: ;;rrun IIxnll' 1 I} ' lI~nll' II~£II Ilg~ -gnll:::: IIPn(XnX:) - f(XnX:)IIII~nll:::: lin; II(g~)U -g£11 :::: IIPn(X:Xn)IIII~£II:::: lin. Thus, IIg~ - gn lIu -+ 0 as n -+ Thus, setting g; = Now, if T/ E 00. Furthermore, we compute: (1 + ~rlg~, we obtain a sequence {g;} such that 2l', then we have, as n -+ 00, 1fl(g;)T/ = 1fr (T/)g; -+ 1fr (T/)g = 1fl(g)T/; 1ft;{g;)*T/ = 1fr (T/)(g;)U -+ 1fr (T/)gU = 1fl(g)*T/.

With this type of fundamental sequence, we can conclude the following result easily. 5. If Y f-+ ~(y) E 2(y) is measurable, then the field {Jrl(~(Y))} of operators in S)(y) is measurable. 6. If {2(y)} is a measurable field of left Hilbert algebras on a standard measure space {r (Y), /-L} then {2(y)'} is also a measurable field of right Hilbert algebras. PROOF: We know already that the field {L1(y)} of modular operators is measurable. } of {L1(y)} is measurable. In particular, {L1(y)-1/2} is measurable.