# Completely equivalent operator norms on $*$-Banach algebras.

Let $A$ be an algebra over $\mathbf{C}$ with an involution operator, and let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two $equivalent$ operator norms, making $A$ into a $*$-Banach algebra (we denote them as $A_1$ and $A_2$). The obvious morphism $A_1\to A_2$, $a\mapsto a$ is bounded by definition.

Should it also be completely bounded? If not, are there any criteria to know when they are.

To this end, it could also be supposed that $A$ is unital and $\|1\|_i=1$.

• What norm do you want to put on the matrix algebras $M_n(A)$ and $M_n(B)$? If A is a C*-algebra than $M_n(A)$ is also a C*-algebra, and so this question doesn't arise. But it seems that for a general Banach *-algebra, there are loads of options for the norm on $M_n(A)$. Aug 10 '10 at 15:27
• Thanks for the question! Well, presumably it is like this. The algebra $A$ is a dense subspace of a $C^*$-algebra. I have an $A$-valued inner product on $A^n$, defined in usual way, and this inner product defines a Banach norm on $A^n$ as $\|\xi\|_A=\|\langle \xi^*;\xi\rangle\|_A$. This norm, in turn, defines a norm on $M_n(A)$ as $\|(a_{ij})\|_A=\|sup_{\xi\in\mathcal{B}(A^n)}\|(a_{ij})\xi\|_A$ where $\mathcal{B}(A^n)$ is a unit ball. Hope I haven't written a complete nonsence. Aug 10 '10 at 16:16
• Could you please clarify what you mean by "an operator norm" and (as per the comments of Matthew Daws and the answer of Andreas Thom below) what operator space structure you are equipping $A_1$ and $A_2$ with? Aug 10 '10 at 17:55
• I think my question was not posed well enough. It seems I need to improve my background provided with the new references and to pose the original question, from which this one seemed to arrive. Thank you for the comment! Aug 11 '10 at 7:19

A priori, it does not make sense to talk about complete boundedness, since there are no specified operator space structure on $A_1$ and $A_2$.
There are criteria (also due to Pisier) which ensure that certain bounded maps between $C^*$-algebras are automatically completely bounded. This is related to the notion of length of a $C^*$-algebra. This is also explained in his book.