1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

Cuntz-Pimsner algebras 145 r-^1 (v) = 0. Then the set coo(E) of all finitely supported functions on Eis naturally a co(V)- ...

146 4. Constructions Let ( 7r, T) be a representation of a C -correspondence 1{ over A. Then, for every n 2: 1, we define Tn: 1- ...

4.6. Cuntz-Pimsner algebras 147 B1. It follows that rn(μ)*xrn(v) E 7r(I(7r, 7 ))· Hence, for any "finite-rank operator" e = l:i ...

148 4. Constructions Here is the gauge-invariant uniqueness theorem for Toeplitz-Pimsner al- gebras. Theorem 4.6.18. Let 1i be a ...

4.6. Cuntz-Pimsner algebras 149 In particular, the covariant representation (Q1o1r:F(7i)' S) of Jt on O(Jt) is universal. Proof. ...

150 4. Constructions we observe first that Adu: Iffi(1i) 3 x 1--+ U xu-^1 E Iffi(1i) is a well-defined *-automorphism such that ...

Cuntz-Pimsner algebras 151 Let Ji= A0e L^2 (B, EE) be the Hilbert D-module obtained from A0B equipped with the D-valued se ...

152 4. Constructions (See Exercise 4.6.4.) Hence 'H&;;B is a C*-correspondence over A&;;B. This correspondence is called ...

Cuntz-Pimsner algebras 153 and 9.4.4). Unfortunately there is presently no direct proof of this fact, however. There is an ...

154 4. Constructions Exercise 4.6.5. Let Ji be a C*-correspondence over A and assume that the quotient map from A onto A/ I1i is ...

Reduced amalgamated free products 155 Definition 4.7.1 (Reduced amalgamated free product). The reduced amal- gamated free ...

156 4. Constructions algebraic amalgamated free product^12 of Ai's over D and let 7r: A ----+ A and 7r^1 : A----+ A' be the cano ...

4.8. Maps on reduced amalgamated free products 157 Exercises Exercise 4.7.1. Let (A, E) = D(Ai, Ei), where 1 ED C Ai are unital ...

158 4. Constructions Let p = 1-T^2 (T)^2 E C(Ai E9A 2 , T) be the orthogonal projection onto EB( 1-{i E9 (1-ii @n 1-ij) E9 ((1-i ...

4.8. Maps on reduced amalgamated free products 159 where the @ is just a marker to indicate when the first eiD in the middle app ...

160 4. Constructions and w i,3 ( >-.j (b)) ( (i ® (2 ® ... ® ei ® ... ) = Vi73 ( (bej) ® (i ® (2 ® · · · ® ei ® · · · ) = ei ...

4.8. Maps on reduced amalgamated free products 161 But since Ai 1 ( ai) · · · Ain ( an)~D C 1-lf, 1 @v · · · @v Hin and in -/= j ...

162 4. Constructions Since A is a quotient of a C* -subalgebra of pC* (Ai EBA2, T)p, the composition of these maps is the desire ...

4.8. Maps on reduced amalgamated free products 163 and such that V;an(a)Vn = PK7r(a)PK, Vn7r(,6n(x))V; = Vn V;xVn v; for all a E ...

164 4. Constructions described in Proposition 1.5.12, the c.p. map f3n corresponds to a positive element an E B(KnEB..Cn)@A**. W ...