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# Compact Groups of Operators on Subproportional Quotients of $l^m_1$

Published:2000-10-01
Printed: Oct 2000
• Piotr Mankiewicz
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## 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]