1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

3.6. Inclusions and The Trick 85 Corollary 3.5.6. Assume e: A--+ C and O": B--+ D are c.c.p. maps and en: A--+ C are c.c.p. maps ...

86 3. Tensor Products Applying this fact again on the right hand side implies that a pair of inclusions Ac Band Cc D induces a n ...

3.6. Inclusions and The 'Irick 87 We will soon introduce The Trick and provide the converse to the pre- vious result, but first ...

88 3. Tensor Products and u.c.p. maps are bimodule maps over their multiplicative domains, a simple calculation completes the pr ...

3.6. Inclusions and The Trick 89 Definition 3.6.7. A C*-algebra A C JE(H) is said to have Lance's weak expectation property (WEP ...

90 3. Tensor Products the extension <.p: JB(.€^2 (r)) --+ JB(.€^2 (r)) given by The Trick takes values in L(I'). D It turns o ...

3.6. Inclusions and The Trick 91 for all a E A, c E 0. Evidently \If is a *-homomorphism and it is injective since this is the c ...

92 3. Tensor Products Show that there is a natural isomorphism Co(X) ©max A~ Co(X) 129 A~ Co(X, A) such that h@a maps to the fun ...

Exact sequences 93 Did you catch the subtlety? Since J ©max B is much bigger than J 8 B, is it not possible that (J ©max B ...

94 3. Tensor Products In the previous section we saw that for certain inclusions A c B, we have inclusions A @max C C B @max C f ...

Exact sequences 95 Lemma 3. 7. 7. If J <l A is an ideal and B C C is a C*-subalgebra, then ( J 0 C) n (A 0 B) = J @ B. ...

96 3. Tensor Products Now let's give an example where a tensor product sequence fails to be exact.^22 Though elementary, the pro ...

Exact sequences 97 Proposition 3.7.10. If r is residually finite, then the product map ,\ x p: C(r) 0 C(r)-+ JB(£^2 (r)) i ...

98 3. Tensor Products However, it is evident that J 0 C*(r) belongs to the kernel of,\ Xmin p and hence this *-homomorphism fact ...

3.8. Nuclearity and tensor products gg Exercise 3. 7.3. Let A be nonunital and assume that 0------* J@A-----*B@A-----*B/J@A----- ...

100 3. Tensor Products Define new c.p. maps by and note that 11'¢~11 :S 1 + 8. Since 8 > 0 was arbitrary, it suffices to show ...

3.8. Nuclearity and tensor products 101 is continuous with respect to the spatial tensor product norm and let 7r: M -t lffi(JC) ...

102 3. Tensor Products Proof. ( =?-) This is very similar to the proof of Lemma 3.6.10. If 'Pn: A---+ Mk(n)(C) and '!/Jn: Mk(n)( ...

3.8. Nuclearity and tensor products 103 Now let P E IIB(JC) be the orthogonal projection onto the span of the vectors {v1, ... , ...

104 3. Tensor Products Theorem 3.8.7 (Choi and Effros, Kirchberg). For a C*-algebra A, the following statements are equivalent: ...