Affiliation:
1. Department of Applied Mathematics and Computer Science , University of Life Sciences in Lublin , 20-950 Lublin , Poland
Abstract
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
L
1
(
μ
)
{L_{1}(\mu)}
is an
essential Banach
L
1
(
G
)
{L_{1}(G)}
-bimodule, then any continuous derivation
δ
:
L
1
(
G
)
→
L
∞
(
μ
)
{\delta:L_{1}(G)\rightarrow L_{\infty}(\mu)}
is inner. This extends a theorem of
B. E. Johnson (1991) asserting that the convolution algebra
L
1
(
G
)
{L_{1}(G)}
is
weakly amenable if G is a locally compact group.
Subject
Applied Mathematics,General Mathematics
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