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

2010-08-10T12:40:27Z

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 $\|1\|_i=1$.

Answer by Andreas Thom for Completely equivalent operator norms on $*$-Banach algebras.

2010-08-10T15:28:00Z

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.

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

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

Answer by Andreas Thom for Completely equivalent operator norms on $*$-Banach algebras.