# Uniqueness of the Injective III1 Factor by Steve Wright

By Steve Wright

Based on lectures brought to the Seminar on Operator Algebras at Oakland collage through the wintry weather semesters of 1985 and 1986, those notes are a close exposition of modern paintings of A. Connes and U. Haagerup which jointly represent an explanation that every one injective elements of kind III1 which act on a separable Hilbert house are isomorphic. This consequence disposes of the ultimate open case within the category of the separably performing injective components, and is among the striking fresh achievements within the concept of operator algebras. The notes may be of substantial curiosity to experts in operator algebras, operator concept and staff in allied parts comparable to quantum statistical mechanics and the idea of workforce representations.

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**Example text**

11~ < 20~ ~ ~(~o(~)%(x)). 1 ! Since u = u,~(x) is a partial isometry, we have the conclusion of the temma except for u 2 = 0 . To get this, set y = u(1 - uu*). (,*y) _. ). 34) with u~(x) replaced by an appropriate us(y). Since u,~(y) 2 = 0 for each a, this gives us what we want. 36). 37) _< 11:(1 - i)a(au*J + ~,~ I l u ( a ( ~ - f ) J -< tl(zx2, - x ~ ) ~ l l a~(~ - f)),'il I - a~,)el II - (1-f))~,ll + ~'itl(af J- f)~,ll, since u {1 - f, ~il = u If, (i]. 38). 36), set p = u * . , ,~2 = E ~,(p).

Claim 2. If every element of the center Z of L is of the form ( ( \ Proof. ) a~ 0 0 )))EZ. O(a~ Thenz~l=(( a~ 0 o) O(a~) )' then a0n 00) ) ' )' and so T(zYla) = limn_~ ( qOOO 0(~'o0) ) ( a,~o 0)0 ( 00) lim ~o (an) = lira ~po(O-l(O(a,,)) = Iim ( ~--~ = ~0 0 0 )) 0(p0 o O(an) r(z~22). Since v is a faithful trace on L and z is arbitrary, ~11 "~ e22. We must hence prove that every element of Z has the indicated form. To do this, we need the asymptotic centralizer M,~ of M + ([6], Section II). +. is the yon Neumann algebra of w-centralizing sequences in M modulo the ideal of all w-centralizing sequences (x,~) such that lim Xn = 0, *-strongly (i.

Thus r ' E T~. We now prove that r' > r. We clearly have r I # r. (1) is clear by construction. (2) follows from the facts that a~ - oq e e l ( H ) and e £ el. 23), I]c~ -c~j]l 2 _< 6 ~ q o j ( e l ) <_ 6 ~ q o j ( e ' - e ) . Thus Claim 2 holds, and the proof of Theorem 2 will be finished modulo the lemma by producing a maximal element of "R. 36 We prove that T~, < is a partially ordered set to which Zorn's lemma can be applied. Reflexivity of <_. Obvious A n t i s y m m e t r y of <. Suppose r, r ' ~ 7~, r _< r', r' _< r.