Abstract
We show that the quantum family of all maps from a finite space to a finite-dimensional compact quantum semigroup has a canonical quantum semigroup structure.
Highlights
According to the Gelfand duality, the category of compact Hausdorff spaces and continuous maps and the category of commutative unital C∗-algebras and unital ∗-homomorphisms are dual
We show that the quantum family of all maps from a finite space to a finite-dimensional compact quantum semigroup has a canonical quantum semigroup structure
We show that if QA is a compact finite dimensional i.e., A is unital and finitely generated quantum semigroup, and if QB is a finite commutative quantum space i.e., B is a finite dimensional commutative C∗-algebra, QC has a canonical quantum semigroup structure
Summary
According to the Gelfand duality, the category of compact Hausdorff spaces and continuous maps and the category of commutative unital C∗-algebras and unital ∗-homomorphisms are dual. As the fundamental concept in noncommutative topology, a noncommutative unital C∗-algebra A is considered as the algebra of continuous functions on a symbolic compact noncommutative space QA. In this correspondence, ∗-homomorphisms Φ : A → B interpret as symbolic continuous maps QΦ : QB → QA.
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