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?


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 $$\|\sum b_j\otimes a_j\|_\max=\|\sum a_j\otimes b_j\|_\max.$$

  • $\begingroup$ Oh, yes. Thanks Martin. :P $\endgroup$ – Yan kai May 14 '14 at 1:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.