Fixed product preserving mappings on Banach algebras
Hayden Julius
备注：
9 pages - this is a second version. There is a major flaw in Thm 3.3 from previous version when I approached the boundary of the open set of invertible elements. I removed that argument from the second version
In this paper, we describe linear maps between complex Banach algebras that preserve products equal to fixed elements. This generalizes some important special cases where the fixed elements are the zero or identity element. First we show that if such map preserves products equal to a finite-rank operator, then it must also preserve the zero product. In several instances, this is enough to show that a product preserving map must be a scalar multiple of an algebra homomorphism. Second, we explore a more general problem concerning the existence of product preserving maps and the relationship between the fixed elements. Lastly, motivated by Kaplansky's problem on invertibility preservers, we show that maps preserving products equal to fixed invertible elements are either homomorphisms or antihomomorphisms multiplied on the left by a fixed element.
Commutativity preserving mappings in Banach algebras
Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Faculty of Natural Sciences and Mathematics, University of Maribor（马里博尔大学自然科学与数学学院）；IMFM, Ljubljana（斯洛文尼亚科学院数学物理研究所）；Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada（格拉纳达大学理学院数学分析系）
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University of Agriculture in Krakow（克拉科夫农业大学）；Institute of Mathematics of NAS of Ukraine（乌克兰国家科学院数学研究所）；Lviv Polytechnic National University（利沃夫国立理工大学）；Comenius University in Bratislava（布拉迪斯拉发康斯坦丁尼查尔大学）
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