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.

Key Words:

• C-algebra
• Dual C-algebra
• C-algebra homomorphisms
• Dual morphisms
• C-subsets
• Quotient subsets

Acknowledgments

The author would like to thank Dr. Blau for the useful discussion while he wrote the notes for this research. The author would also like to thank the referee; his comments improved the exposition of the paper.

Notes

^#Communicated by A. Turull.