The commutative tensor product norm

Question

Definition 3.3.3 (Maximal norm) Given $A$ and $B$, we define the maximal C*-norm on $A\odot B$ to be $$||x||_{max}=sup\{||\pi(x)||:\pi:A\odot B\rightarrow B(H) a *-homomorphism\}.$$ for $x\in A\odot B$. We let $A\otimes_{max} B$ denote the completion of $A\odot B$ with respect to $||.||_{max}$. (The $\odot$ denotes the algebraic tensor product.)

Exercise 3.3.1. Show that $||.||_{max}$ is a commutative tensor product norm - i.e., there are canonical isomorphism $A\otimes_{max} B\cong B\otimes_{max}A.$

Well, could someone give me some hints of this exercise?

Answer 1

Use Theorem 3.2.6 to, given any $*$-homomorphism $\pi:A\odot B\to B(H)$, construct a $*$-homomorphism $\pi':B\odot A\to B(H)$ with $$\|\pi'(\sum b_j\otimes a_j)\|=\|\pi(\sum a_j\otimes b_j)\|.$$ Then deduce that $$\|\sum b_j\otimes a_j\|_\max=\|\sum a_j\otimes b_j\|_\max.$$