1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

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 since e is an extension of elA 0 elB· D

Lemma 3.4.6. Let B be unital, A be unital and abelian and <p be a pure

state on A. Then for every C*-norm II· Ila on A0B and every state 1/J E S(B)

the linear functional <p 01/J : A 0 B -+ (['. extends to a state on A ©a B.

Proof. Let S<p = {1/J E S(B): l<p07/J(x)I :'S llxlla, Vx E A0B}. Note that S<p
is convex and weak- closed in S(B). This lemma asserts that S<p = S(B)
and to show this, we invoke Lemma 3.4.1. So let h E B be any self-adjoint
element and let C = C(h, lB) be the unital C*-algebra generated by h.
Let A ©a C be the completion of A 0 C with respect to the restriction
of II · Ila· Let 1/J E P(C) be a pure state such that 17/J(h)I = llhll and let
<p 0 'ljJ : A ©a C-+ (['. be the extension given by the last lemma. Note that
<p 0 1/J is a pure state and hence extends to a state e on A ©a B. Since
elA = <p is pure, Corollary 3.4.3 tells us that e is an extension of <p 0 elB·
Hence elB E S<p. Moreover, llhll = lelB(h)I since elB(h) = 1/J(h). Thus
Lemma 3.4.1 implies that S<p = S(B). D

We finally come to the main ingredient in the proof of Takesaki's result.

Proposition 3.4. 7. Let A and B be unital C -algebras with states <p E S(A)
and 1/J E S(B). For any C -norm II · Ila on A 0 B, <p 01/J extends to a state
on A@a B.

Proof. We apply essentially the same trick as in the last lemma - but we
must do it twice and with a bit of care. So first fix a pure state <p E S(A)
and again let S<p = {1/J E S(B) : l'P 01/J(x)I :'S llxllm Vx EA 0 B}. The first
step is to show that for every self-adjoint h E B,

llhll = sup 17/J(h)I.

'1/;ES'P

So choose a pure state 'ljJ on C = C*(h, lB) such that 17/J(h)I = llhll· By the
previous lemma we may extend <p 01/J to a state on A ©a C c A ®a B. We
further extend to a state eon A@aB. Since elA =<pis pure, Corollary 3.4.3
implies e is an extension of <p 0 e1B and as in the last lemma we conclude
that S<p = S(B).

Now we reverse the roles a bit. Fix a state 'ljJ E S(B) and let S'l/; = { <p E
S(A) : l'P 01/J(x)I :'S llxlla, Vx EA 0 B}. The previous paragraph shows S'1/J
contains all the pure states of A. Thus, the Krein-Milman Theorem implies
S'1/J = S(A), as desired. D

Theorem 3.4.8 (Takesaki). For arbitrary C-algebras A and B, 11 · llmin is
the smallest C -norm on A 0 B.