Completely equivalent operator norms on $*$-Banach algebras. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:39:05Z http://mathoverflow.net/feeds/question/35110 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35110/completely-equivalent-operator-norms-on-banach-algebras Completely equivalent operator norms on $*$-Banach algebras. Kolya Ivankov 2010-08-10T12:40:27Z 2010-08-10T15:28:00Z <p>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. </p> <p>Should it also be completely bounded? If not, are there any criteria to know when they are.</p> <p>To this end, it could also be supposed that $A$ is unital and $\|1\|_i=1$.</p> http://mathoverflow.net/questions/35110/completely-equivalent-operator-norms-on-banach-algebras/35126#35126 Answer by Andreas Thom for Completely equivalent operator norms on $*$-Banach algebras. Andreas Thom 2010-08-10T15:28:00Z 2010-08-10T15:28:00Z <p>A priori, it does not make sense to talk about complete boundedness, since there are no specified operator space structure on $A_1$ and $A_2$.</p> <p>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 (<a href="http://books.google.de/books?id=0pKL-o7WUOAC" rel="nofollow">see here</a>). These two almost never the same.</p> <p>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 <em>length</em> of a $C^*$-algebra. This is also explained in his book.</p>