 |
|
Compact Groups of Operators on Subproportional Quotients of $l^m_1$ |
Canad. J. Math. 52(2000), 999-1017
Published:2000-10-01
Printed: Oct 2000
Printed: Oct 2000
- Piotr Mankiewicz
Abstract
It is proved that a ``typical'' $n$-dimensional quotient $X_n$ of
$l^m_1$ with $n = m^{\sigma}$, $0 < \sigma < 1$, has the property
$$
\Average \int_G \|Tx\|_{X_n} \,dh_G(T) \geq
\frac{c}{\sqrt{n\log^3 n}} \biggl( n - \int_G |\tr T| \,dh_G (T)
\biggr),
$$
for every compact group $G$ of operators acting on $X_n$, where
$d_G(T)$ stands for the normalized Haar measure on $G$ and the average
is taken over all extreme points of the unit ball of $X_n$. Several
consequences of this estimate are presented.
| MSC Classifications: |
46B20 - Geometry and structure of normed linear spaces 46B09 - Probabilistic methods in Banach space theory [See also 60Bxx] |