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. 1. Introduction 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. In this duality, any compact space corresponds to , the C*-algebra of all continuous complex valued maps on , and any commutative unital C*-algebra corresponds to its maximal ideal space. Thus as the fundamental concept in noncommutative topology, a noncommutative unital C*-algebra is considered as the algebra of continuous functions on a symbolic compact noncommutative space . In this correspondence, *-homomorphisms interpret as symbolic continuous maps . Since the coordinates observable of a quantum mechanical systems are noncommutative, some-times noncommutative spaces are called quantum spaces. Woronowicz [1] and So？tan [2] have defined a quantum space of all maps from to and showed that exists under appropriate conditions on and . In [3], we considered the functorial properties of this notion. In this paper, we show that if is a compact finite dimensional (i.e., is unital and finitely generated) quantum semigroup, and if is a finite commutative quantum space (i.e., is a finite dimensional commutative C*-algebra), then has a canonical quantum semigroup structure. In the other words, we construct the noncommutative version of semigroup described as follows. Let be a finite space and be a compact semigroup. Then the space of all maps from to is a compact semigroup with compact-open topology and pointwise multiplication. In Section 2, we define quantum families of all maps and compact quantum semigroups. In Section 3, we state and prove our main result; also we consider a result about quantum semigroups with counits. At last, in Section 4, we consider some examples. 2. Quantum Families of Maps and Quantum Semigroups All C*-algebras in this paper have unit and all C*-algebra homomorphisms preserve the units. For any C*-algebra , and denote the identity homomorphism from to , and the unit of , respectively. For C*-algebras , denotes the spatial tensor product of and . If and are *-homomorphisms, then is the *-homomorphism defined by ( ). Let , and be three compact Hausdorff spaces and be the space of all continuous maps from to with compact open topology. Consider a continuous map . Then the pair is a continuous family of maps from to indexed by with parameters in . On the other hand, by topological