1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

3.9. Exactness and tensor products 105 and mimic exactly what we did in the proof of Theorem 3.8.5. It is a good exercise to che ...

106 3. Tensor Products Then a perturbation argument (Lemma 3.9.7 plus Arveson's Extension The- orem) would imply that we can fin ...

3.9. Exactness and tensor products 107 The converse follows from the previous lemma since the top row of the commutative diagram ...

108 3. Tensor Products What one must check is that this isomorphism takes a® (bn)n E A ® (fJ Bn) C IB\ ( K ® ( E9 'Hn)) n n to t ...

3.9. Exactness and tensor products 109 Proposition 3.9.6. Let A C IIB(H) be a unital ®-exact C* -algebra, { Pn} be any increasin ...

110 3. Tensor Products and let X be the image of X. By definition of the spatial norm, we have an inclusion E ® (fin Mk(n) (C)) ...

3.9. Exactness and tensor products 111 The final step is an immediate consequence of a basic c.b.-perturbation fact. Indeed, the ...

112 3. Tensor Products Exercise 3.9"6. Prove that A is exact if and only if the sequence O, ( ~Mn(C)) Q9A, (I]Mn(C)) Q9A, (~:::~ ...

3.10. References 113 adapted from [113], where Lance introduced the weak expectation property inspired by Tomiyama's extensive ...

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Chapter 4 Constructions There are numerous ways of creating new C* -algebras out of old ones. Our goal in this chapter is not to ...

116 4. Constructions It turns out that nuclearity and exactness are reasonably well behaved under this construction and, in most ...

4.1. Crossed products 117 that Us1f(a)u; = 7r(a 8 (a)) for every s E r and a E A. rt· is not hard to see that every covariant re ...

118 4. Constructions Hence we get an induced covariant representation (10.) X7r, called a regular representation.^2 Definition 4 ...

4.1. Crossed products L a;^1 (a) 0 ep,s-lp EA 0 MF(<C). pEFnsF Now if x =I: asAs E Cc(r, A) c IIB(?i 0 .e^2 (r)), then we hav ...

120 4. Constructions Lemma 4.1.8. Let 'ljJ be a faithful state on B. Then idA ® 'ljJ: A® B -+ A is faithful. Proof. Observe that ...

4.2. Integer actions 121 When proving that a particular map into a crossed product is completely positive, it often suffices to ...

122 4. Constructions where and so on. Since Aj Ai = 0 whenever i # j, a straightforward calculation completes the proof. D Lemma ...

4.2. Integer actions 123 By Lemma 4.2.1, we only need to check that for every set {ap}pEF c A, 'I/;(~ a;aq ® ep,q) 2:: 0. But 'I ...

124 4. Constructions Corollary 4.2.5. The rotation algebras Ae are nuclear. At this point, the reader may feel lied to - we said ...