1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

/ 3.1. Algebraic tensor products 65 We complete the *-algebra structure on A0B by defining multiplication. As with the involutio ...

66 Mn(C) 8 A~ Mn(A) given by L ei,j 0 ai,j 1---+ [ ai,j]. i,j Tensor Products Exercise 3.JL.4. For an arbitrary nonunital alge ...

3.2. Analytic preliminaries 67 form, it is not complete (unless either 1i or K is finite dimensional) in the corresponding norm. ...

68 3. Tensor Products is an orthonormal basis. Moreover, for each x E 1{ ® K there is a unique set of vectors { ki} C K such tha ...

3.2. Analytic preliminaries 69 if we take unit vectors hn E 1-l and kn E JC such that llSll = lim llShnll and llTll = lim llTknl ...

70 3. Tensor Products Proof. Since positive elements span, we may assume a 2: 0. In this case, for each positive b E B we have t ...

3.2. Analytic preliminaries 71 Of course 7rB is defined similarly and it is routine to check that we get a pair of *-homomorphis ...

72 3. Tensor Products is replaced by a C -algebra. Indeed, let A be a unital C -algebra and let SA= Co(R, A) = {f : R .,A: f is ...

3.3. The spatial and maximal C*-norms 73 It requires a little work, but it's a fact that C*-norms on algebraic tensor products a ...

74 3. Tensor Products The remainder of this section is devoted to resolving the following tech- nical issues. (1) Are // · I/max ...

3.3. The spatial and maximal C*-norms 75 Rearranging terms, we get ((LSi0Ti)v0e,w0'1J) = L:\si0Ti(v0e),w0'1J) i i = L:\Siv,w)(Ti ...

76 3. Tensor Products But since Pn'iPJ{H)Pn is naturally isomorphic to Mn(<C), we have II L(Pn7r(ai)Pn)@ o-(bi)ll =II L(Pn7r( ...

3.4. Takesaki 's Theorem 77 Exercise 3.3.5. If 7r: A -+ C and (}: B -+ D are *-homomorphisms, prove that there is a unique *-hom ...

78 3. Tensor Products The next two results will allow us to deduce that certain states on tensor products can always be decompos ...

3.4. Takesaki 's Theorem 79 = elA(a)elB(b). Hence e(x) = elA 0 elB(x) for all x EA 0 B. To prove the second assertion, we assume ...

80 3. Tensor Products that eu 0 g) > 0 and let e1A and elB be the restrictions. Corollary 3.4.3 gives the desired conclusion ...

3.4. Takesaki 's Theorem 81 Proof. Assume first that both A and B are unital and separable. Then we can find faithful states c.p ...

82 3. Tensor Products Thus, for a unit vector~ E eJH, we have ll(a@b)(~) -~II:::; 2c:. Since E was arbitrary, Ila Q9 bll = 1. 0 ...

3.5. Continuity of tensor product maps 83 Proof. It's well known, and easily verified, that the transpose map is a positive isom ...

84 3. Tensor Products -):iomomorphism 1fA ® 1fB: A® B ------ llll(H ® K). Hence we may define cp ® 'ljJ: A® B------ C ® D by the ...