# Holomorphic semi-groups and the geometry of Banach spaces

### Abstract

According to a fundamental theorem of Dvoretzky (cf. [24], [25]), any infinite dimensional Banach space $X$ contains for each integer $n$ and each $\varepsilon > 0$ a subspace $X_n$ which is $(1+\varepsilon)$-isomorphic to $l_2^{(n)}$. In the classical Banach spaces $L_p$, and $l_p$, with $1 < p < \infty$, it is well known that this result can be improved: These spaces contain "uniformly complemented $l_2^{(n)}$'s" (this means that we can find subspaces $X_n$ as above with the additional property that there exist projections $P_n\colon X\to X_n$ such that ${\rm sup}_n \Vert P_n\Vert < \infty$). Moreover, it is also well known that this property no longer holds for $l_1$ or $l_\infty$.

This motivated the question (cf. [13]) to determine which infinite dimensional Banach spaces contain “uniformly complemented $l_2^{(n)}$’s.” We will prove below that all uniformly complex spaces have this property. Moreover, we will show that a Banach space $X$ verifies this property “locally” (see definition 2.10 for more precision) if and only if $X$ does not contain $l_1^{(n)}$’s uniformly. Progressively, the work of previous authors (cf. [14], [12] and particularly [5]) has reduced such so-called “geometric” questions to a purely analytic problem consisting of verifying in $X$ a certain inequality referred to as “$K$-convexity”. In this paper, we connect this inequality with the holomorphy of a certain semi-group of operators on $L_p(X)$ and we show how this leads to the results already mentioned.

Gilles Pisier