A Kind of Compact Quantum Semigroups

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. Let X be a finite space and Y be a compact space. Then the space F (X,Y ) of all maps from X to Y , is a compact space with compact-open topology. The notion of quantum family of all maps, defined by Woronowicz [8] and Soltan [5], is a noncommutative version of F (X,Y ), where X and Y are noncommutative spaces recognized by the C*-algebras of functions on them. Categorical aspects and noncommutative metric structures of quantum families of maps are studied in [3] and [4]. In this short note, we describe a special kind of quantum semigroup structures on quantum families of all maps. More precisely, we construct a noncommutative analogue of the semigroup F (X,S) constructed as follows: Let S be a compact finite dimensional semigroup and X be a finite space. Then F (X,S) is a compact semigroup with the multiplication ff (x) = f(x)f (x) (f, f ′ ∈ F (X,S), x ∈ X). By ”finite dimensional” in above, we mean that the underlying space of S is a finite dimensional (topological or smooth) manifold. Note that if a space X is compact and finite dimensional, then its corresponding C*-algebra C(X), the algebra of all continuous complex valued maps on X , is finitely generated. For this reason, in the literature of Non Commutative Geometry, a noncommutative space that its C*-algebra is finitely generated, is called finite dimensional. For any C*-algebra A, iA denotes the identity homomorphism from A to A. If A is unital then 1A denotes the unit of A. For C*-algebras A,B, A ⊗ B denotes the spatial tensor product of A and B. If Φ : A −→ B and Φ : A −→ B are *-homomorphisms, then Φ⊗Φ : A ⊗ A −→ B ⊗ B is a *-homomorphism defined by Φ⊗Φ(a⊗a) = Φ(a)⊗Φ(a) (a ∈ A, a ∈ A). We use the dual terminology ”quantum family of morphisms” instead of ”quantum family of maps”: 2000 Mathematics Subject Classification. 46L05, 16W30.