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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 $\|1\|_i=1$.

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    $\begingroup$ 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 $M_n(A)$. $\endgroup$
    – Matthew Daws
    Commented Aug 10, 2010 at 15:27
  • $\begingroup$ 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. $\endgroup$
    – Kolya Ivankov
    Commented Aug 10, 2010 at 16:16
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    $\begingroup$ 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? $\endgroup$
    – Yemon Choi
    Commented Aug 10, 2010 at 17:55
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    $\begingroup$ 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! $\endgroup$
    – Kolya Ivankov
    Commented Aug 11, 2010 at 7:19

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A priori, it does not make sense to talk about complete boundedness, since there are no specified operator space structure on $A_1$ and $A_2$.

In general, an infinite-dimensional Banach space can carry many incomparable operator space structure. Most prominently, there is the minimal and the maximal operator space structure (see Chapter 3 in the book of Gilles Pisier (see here). These two almost never the same.

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.

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    $\begingroup$ Thanks for references. I think I have to analyze what I actually want more precisely, and ask the original one. $\endgroup$
    – Kolya Ivankov
    Commented Aug 11, 2010 at 8:13

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