1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

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 *-homomorphism 7r Q9 (} : A Q9 B -+ C Q9 D such that

7r Q9 (}(a Q9 b) = 7r(a) Q9 (J(b).

Exercise 3.3.6. Prove that JIB(.€^2 ) 0 lIB(.€^2 ) c JIB(.€^2 Q9 £^2 ) (see Lemma 3.3.9)

is not dense in norm.

3.4. Takesaki's Theorem

This section is devoted to the fact that 11 · llmin is really the smallest possible

C*-norm on A 0 B. The main ingredient in the proof is Proposition 3.4.7

• for any pair of functionals <p E A, 'I/; E B and any C* -norm 11 · 11 a on
  A 0 B, the natural linear functional cp 0 'I/; is automatically continuous with
  respect to II· Ila· There are two routes to this result. We first give Takesaki's
  original argument - a lovely display of soft analysis - then, at the end of the
  section, we sketch an alternate proof based on excision.
  The first lemma may already be familiar. If not, it is a fact worth
  remembering.

Lemma 3.4.1. Let S(A) denote the state space of a C*-algebra A. Assume

that S C S(A) is a set of states with the property that for each self-adjoint

a EA we have

llall = sup{lcp(a)I}.

cpES

Then the weak-* closed convex hull of S contains S(A) (hence is equal to

S(A) when A is unital).

Proof. Let S denote the closed convex hull of S and assume there is a state

'I/; which doesn't belong to S. By the Hahn-Banach Theorem we can find

a E A and a real number t such that Re(cp(a)) < t < Re('l/;(a)) for every

cp E S. (Recall that A is the dual of A* with respect to the weak-* topology.)

. Replacing a with its real part, we may assume a is self-adjoint. Note that
if a happened to be positive, then we would have our contradiction since
'l/;(a) ~ llall· So the trick is to replace a with a positive element. In the
unital case one considers a+ llalllA 2::: 0 and notes that

0 ~ <p(a + llalllA) = <p(a) + llall < t + liall < 'l/;(a + iialilA)·

Hence

ila+ilalllAll = sup{l<p(a+ilalilA)i} ~ t+llali < 'l/;(a+iialilA) ~ iia+iialllAll,

cpES

which gives the contradiction. In the nonunital case one passes to unitiza-

tions and applies the argument above. 0