If A is a unital complex Banach algebra, and if σ(a) denotes the spectrum of an element a∈A, then the famous Gleason-Kahane-Żelazko Theorem says that any linear functional ϕ:A→C satisfying ϕ(a)∈σ(a) ...for each a∈A, is multiplicative and continuous. In this paper we establish a multiplicative Gleason-Kahane-Żelazko theorem for the case where A is a C⋆-algebra. Specifically, if A is a C⋆-algebra, then any continuous multiplicative functional ϕ:A→C satisfying ϕ(a)∈σ(a) for each a∈A, is linear and hence a character of A.
Let 𝑥 be a Banach algebra. In this article, on the one hand, we proved some results concerning the continuous projection from 𝑥 to its center. On the other hand, we investigate the commutativity of ...𝑥 under specific conditions. Finally, we included some examples and applications to prove that various restrictions in the hypotheses of our theorems are necessary.
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.
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