Banach Symmetric Spaces

Abstract

A Banach symmetric space in the sense of O. Loos is a smooth Banach manifold $M$ endowed with a multiplication map $\mu\colon M \times M \to M$ such that each left multiplication map $\mu_x := \mu(x,\cdot)$ (with $x \in M$) is an involutive automorphism of $(M,\mu)$ with the isolated fixed point $x$. We show that morphisms of Lie triple systems of symmetric spaces can be uniquely integrated provided the first manifold is 1-connected. The problem is attacked by showing that a continuous linear map between tangent spaces of affine Banach manifolds with parallel torsion and curvature is integrable to an affine map if it intertwines the torsion and curvature tensors provided the first manifold is 1-connected and the second one is geodesically complete. Further, we show that the automorphism group of a connected Banach symmetric space $M$ can be turned into a Banach-Lie group acting smoothly and transitively on $M$. In particular, $M$ is a Banach homogeneous space.