1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

1.4. Double duals 5 x 0 e E IIB(X, M)*, where x EX, e E M* and x 0 e is defined by x 0 e(T) = e(T(x)). Then IIB(X, M) is isometr ...

6 1. Fundamental Facts Theorem 1.4.1. The enveloping van Neumann algebra of A is isometrically isomorphic to the double dual A** ...

1.4. Double duals 7 Proposition 1.4.5. The representations 7r and p are quasi-equivalent if. and only if c( 7r) = c(p). Since ev ...

8 1. Fundamental Facts and 7r(a^112 )Q = 7r(bi)Q, for all i EI. Since a and Q commute, one has 7r(br)Q = 7r(bi)(7r(a^1 l^2 )Q) = ...

1.5. Completely positive maps g such that llek(a - 'lj!k(a))ekll < e for every a E ~. Let P;t 0 = 1 - PJC 0 and notice that P ...

10 1. Fundamental Facts 7r and an operator V is c.p. (One should verify this. Don't forget that a 2 0 <=? a = x*x.) A positiv ...

1.5. Completely positive maps 11 Remarks 1.5.5. We call the triplet (1f, it, V) in Theorem 1.5.3 a Stinespring dilation of rp. W ...

12 1. Fundamental Facts Multiplicative domains. Proposition 1.5.7. Let A and B be C-algebras and <p: A-+ B be a c.c.p. map. ( ...

1.5. Completely positive maps 13 (2) E is c.c.p.; (3) E is contractive. Proof. We only have to prove that the last condition imp ...

14 1. Fundamental Facts Indeed, the first inequality is due to Cauchy-Schwarz, while the second fol- lows from the general fact ...

1.5. Completely positive maps 15 Proof. Since [ei,j] E Mn(Mn(C)) is positive (it's a multiple of a rank-one projection), the "on ...

16 1. Fundamental Facts There is a similar characterization of c.p. maps from a C*-algebra A into Mn(q. For a linear map cp: A-+ ...

1.6. Arveson's Extension Theorem 17 Corollary 1.5.16. Let EC A be an operator subsystem and cp: E ----t Mn(C) be a c.p. map. The ...

18 1. Fundamental Facts easily seen from explicit form of the correspondence). Unfortunately the cp .A's need not be contractive ...

Voiculescu's Theorem 19 Definition 1. 7.4. A representation Ti: A ---+ IIB(1i) is called essential if ri(A) contains no no ...

20 1. Fundamental Facts p1-= 1 - P. It follows from the identity (P1-p(a)P)*(P1-p(a)P) = V(rp(a*a) - rp(a*)rp(a))V* that llP1-p( ...

Voiculescu's Theorem 21 be the unitary V EIH. Note that V(cp(a) EB p'^00 (a))V* = V cp(a)V* EB p'^00 (a) = p( a )11 EB p'^ ...

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Part 1 · Basic Theory ...

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