The Dual Morphisms in C-Algebras#

Abstract

Let (A, B, f) and (U, V, g) be C-algebras and ψ : (A, B, f) → (U, V, g) a C-algebra homomorphism. Let (A, Bˆ f , f) and (U, Vˆ g , g) be the dual C-algebras of (A, B, f) and (U, V, g), respectively. In this article we define a dual morphism ψˆ : (U, Vˆ g , g) → (A, Bˆ f , f) for ψ, and use this dual morphism to establish the bijection between the quotient subsets of B and the C-subsets of its dual Bˆ f obtained in [Blau, H. I. (1995a). Quotient Structures in C-algebras. J. Algebra 177:297–337]. This approach provides a conceptual proof of this very important bijection.