2024/01/01 by Andrzej Wiśnicki
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Fixed Point Theorems Analysis
paper · doi:10.1515/forum-2023-0193
openalex publication_date 2024/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>μ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> L1(μ) is an essential Banach <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> L1(G) -bimodule, then any continuous derivation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>δ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>→</m:mo> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>μ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> δ:L1(G)→ L∞(μ) is inner. This extends a theorem of B. E. Johnson (1991) asserting that the convolution algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> L1(G) is weakly amenable if G is a locally compact group.