Here, diag ci , ci , Then, by repeatedly applying Lemma 3. We may assume that A is unital. Definition 3. With notation as in Definition 3. Compare with Proposition 2.
For the convenience of the reader, we recall some notation introduced in the introduction. Suppose that either A is nuclear or M is semidiscrete. By , we may assume that M is a continuous properly infinite factor. We may assume that the elements of F have norm less than or equal to one. Therefore, by Corollary 3.
Hence, by Proposition 2. From 3.
In fact, in these papers, the convergence for the approximate unitary equivalence is stronger in the norm topology. The next result seems standard, but we did not find an exact reference. Sketch of Proof. We may assume that the elements of F all have norm less than or equal to one. In Corollary 3. Also, A is the norm- closed linear span its unitaries. NIU Corollary 3. That 1 is equivalent to 2 follows from Corollary 3. That 2 is equivalent to 3 follows from Lemma 3. Uniqueness for general von Neumann algebra codomains In this section, we will generalize the results of section 3 to a general von Neumann algebra M with separable predual by using its direct integral decomposition along its centre.
Without loss of generality, we can assume that M acts on a separable Hilbert space H. We refer the reader to  Chapter 14 for notation and preliminary results. For a positive element a in a von Neumann algebra N , let supp a denote the support projection of a. The first lemma is a standard computation. Lemma R4. Proposition 4. To prove the sufficiency of the condition i. By  Lemma Recall see  Theorem By Lemma 4.
Hence, by Corollary 3. Therefore, by  Theorem NIU The proof of the necessity of the condition i. Let us only prove the statement in the case that A is nuclear the proof for the case that M is semidiscrete is similar. Therefore, we may assume that A is separable. Then M is a direct product of von Neumann algebras with separable predual. Since any von Neumann algebra is a direct sum of a finite von Neumann algebra and a properly infinite von Neumann algebra, the proof is reduced to the case that M is finite or properly infinite, and it follows from Proposition 2.
The following exposition follows . We let T A denote the collection of lower semicontinuous tracial weights on A.
Lemma 5. Then the converse of the above statements hold. The above are  Corollary 5. Definition 5.
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With notation as in Definition 5. Note that by Lemma 5. Consider the following statements: 1 A is nuclear. That 1 implies 2 follows from Proposition 4. We now prove that 2 implies 3. Hence, by Lemma 5. In section 7, we will show that all the statements in Lemma 5. We also note that in Lemma 5. In other words, if we modify the definition of the weak uniqueness property i.
Recall also that k. Lemma 6. We now prove that 2 implies 1. NIU By Lemma 6. Hence, by Lemma 3. From this and 6. Then the following statements are equivalent: 1 M is injective. That 1 implies 2 follows from Proposition 2. As M is type II1 and A is k. Nuclearity and uniqueness Theorem 7.
We now prove that 3 implies 1. Let H be a separable infinite dimensional Hilbert space.
Uniqueness of the Injective III-1 Factor
Note that 7. From this and 7. Hence, by , A is nuclear. In Theorem 7.
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Indeed, if we modify Definition 5. From this and the remarks after Lemma 5. References  C. Akemann, J. Anderson and G. Anantharaman—Delaroche and J. Havet, On approximate factorizations of com- pletely positive maps, Journal of Functional Analysis, 90 , — Blackadar, Operator Algebras. Springer—Verlag, Berlin, Choi and E.
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Connes, A classification of injective factors. Effros and C. Lance, Tensor products of operator algebras, Adv. Gauthier—Villars Editeur, Paris, Elliott, On approximately finite-dimensional von Neumann algebras. II, Canad. Elliott and D. Asked 3 months ago.
Active 2 months ago. Viewed times. Then, again as they are fibrant-cofibrant, any good path-object has all the necessary homotopies, and choosing the correct path-object shows that you have actual homotopies. The complication is in proving the lifting properties and the factorisation. Unfortunately, I don't know much about model categories.
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