论文标题
固定产品保存在Banach代数上的映射
Fixed product preserving mappings on Banach algebras
论文作者
论文摘要
在本文中，我们描述了复杂的Banach代数之间的线性图，该代数保留了等于固定元素的产品。这概括了固定元素为零或身份元素的一些重要特殊情况。首先，我们表明，如果此类地图保留了等于有限级运算符的产品，那么它也必须保留零产品。在某些情况下，这足以表明保留产品必须是代数同态的标量倍数。其次，我们探讨了一个关于保留地图的存在以及固定元素之间关系的更普遍的问题。最后，是由卡普兰斯基（Kaplansky）在可逆性保留者上的问题的动机，我们表明保留等于固定可逆元素的产品的地图是同构或抗Xi型的抗原态性，左侧乘以固定元素。
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.
