A fixed point theorem in B(H,ℓ _∞ )

2021/12/30 by Wiśnicki, Andrzej
#Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Operator Algebras (math.OA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2112.15037

Abstract

We show that if X is a complete metric space with uniform relative normal structure and G is a subgroup of the isometry group of X with bounded orbits, then there is a point in X fixed by every isometry in G. As a corollary, we obtain a theorem of U. Lang (2013) concerning injective metric spaces. A few applications of this theorem are given to the problems of inner derivations. In particular, we show that if L₁(μ) is an essential Banach L₁(G)-bimodule, then any continuous derivation δ:L₁(G)→ L_∞ (μ) is inner. This extends a theorem of B. E. Johnson (1991) asserting that the convolution algebra L₁(G) is weakly amenable if G is a locally compact group.

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