By Ola Bratteli, Derek William Robinson
During this publication we describe the trouble-free concept of operator algebras and elements of the complex idea that are of relevance, or very likely of relevance, to mathematical physics. as a result we describe quite a few functions to quantum statistical mechanics. on the outset of this venture we meant to hide this fabric in a single quantity yet during enhance ment it used to be discovered that this may entail the omission ofvarious fascinating issues or info. hence the e-book was once cut up into volumes, the 1st dedicated to the overall concept of operator algebras and the second one to the purposes. This splitting into concept and purposes is traditional yet a little arbitrary. within the final 15-20 years mathematical physicists have learned the significance of operator algebras and their states and automorphisms for difficulties of box thought and statistical mechanics. however the concept of twenty years aga was once principally constructed for the research of crew representations and it was once insufficient for lots of actual purposes. therefore after a brief honey moon interval during which the recent came across instruments of the extant idea have been utilized to the main amenable difficulties an extended and extra attention-grabbing interval ensued within which mathematical physicists have been pressured to redevelop the idea in suitable instructions. New strategies have been brought, e. g. asymptotic abelian ness and KMS states, new concepts utilized, e. g. the Choquet idea of barycentric decomposition for states, and new structural effects received, e. g. the life of a continuum of nonisomorphic type-three components.
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Extra info for Operator Algebras and Quantum Statistical Mechanics: C*- and W*-Algebras Symmetry Groups Decomposition of States
E/):S; lim. ) :s; Ilwll. Thus Ilwll = lim. ) and the last statement of the proposition is established. (2) ~ (1) We may assume Ilwll = 1. II and we have lim. = ~. Hence w(~) = I. If mdoes not have an identity we adjoin one and extend w to a functional (u on ill: = IC~ + mby E. 2 W(A~ + A) = A + meA). Because A - AE. ) + (A - AEJE. we have lim. AE. 5, we then have IW(A~ + A) I = IA + w(A) I = lim IAw(E,z) :s; lim supllAE/ + AE,z11 + w(AE/) I :s; IIA~ + All. = A. Using Representations and States Thus in any case we may assume that wO) ~ has an identity and = I 51 Ilwll.
I. This follows because A. E. 2 converges uniformly, and monotonically, to some E2 E m+ and hence, by linearity and positivity, I A,w(E. 2) :s; W(E2). An immediate consequence of this remark is that M < + 00 because M = + 00 gives a contradiction. 10 to obtain Iw(AE,) 12 :s; w(A *A)w(E. 2). Taking the limit over C/. one finds Iw(AW :s; MW(A*A). A*AE. ) :s; IIAI1 2 w(E/). Again taking a limit over rt. one has w(A*A):s; MIIAI12. Combination of these two bounds gives Iw(A)1 :s; MIIAII, which establishes that w is continuous and that Ilwll :s; M.
W over a C*-algebra m with Notice that we have not demanded that the positive forms be continuous. 11. Note also that every positive element of a C*algebra is of the form A *A and hence positivity of W is equivalent to w being positive on positive elements. The origin and relevance of the notion of state is best illustrated by first assuming that one has a representation (~, 'Tt) of the C*-algebra m. Now let n E ~ be any nonzero vector and define Wg by wg(A) = (n, 'Tt(A)n) for all A E m. It follows that Wg is a linear function over m but it is also positive because Wg(A*A) = 11'Tt(A)nI1 2 ;;::: O.