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.
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  and Sołtan  have defined a quantum space of all maps from to and showed that exists under appropriate conditions on and . In , 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 exponential law we know that is characterized by a continuous map defined by . Thus can be considered as a family of maps from to . Now, by Gelfand's duality we can simply translate this system to noncommutative language.
Definition 2.1 (see [1, 2]). Let and be unital C*-algebras. By a quantum family of maps from to , we mean a pair , containing a unital C*-algebra and a unital *-homomorphism .
Now, suppose instead of parameter space we use (note that in general this space is not locally compact). Then the family of all maps from to has the following universal property.
For every family of maps from to , there is a unique map such that the following diagram is commutative: (2.2)
Thus, we can make the following definition in noncommutative setting.
Definition 2.2 (see [1, 2]). With the assumptions of Definition 2.1, is called a quantum family of all maps from to if for every unital C*-algebra and any unital *-homomorphism , there is a unique unital *-homomorphism such that the following diagram is commutative: (2.3)
By the universal property of Definition 2.2, it is clear that if and are two quantum families of all maps from to , then there is a *-isometric isomorphism between and .
Proposition 2.3. Let be a unital finitely generated C*-algebra and be a finite dimensional C*-algebra. Then the quantum family of all maps from to exists.
Definition 2.4 (see [2, 4–6]). A pair consisting of a unital C*-algebra and a unital *-homomorphism is called a compact quantum semigroup if is a coassociative comultiplication: .
A *-homomorphism induces a binary operation on the dual space defined by for . Now, suppose that is a compact Hausdorff topological semigroup. Using the canonical identity , we define a *-homomorphism by for and . Then is a coassociative comultiplication on and thus is a compact quantum semigroup. Conversely, if is a compact quantum semigroup such that is abelian, then the character space of , with the binary operation induced by , is a compact Hausdorff topological semigroup . It is well known that a compact semigroup with cancellation property is a compact group [8, Proposition 3.2]. Analogous cancellation properties for quantum semigroups are defined as follows.
Definition 2.5. Let be a compact quantum semigroup. (i)(see ) has left (resp., right) cancellation property if the linear span of (resp., ) is dense in . (ii)(see ) has weak left cancellation property if, whenever are such that for all , we must have or . Similarly, has weak right cancellation property if, whenever for all , we must have or . (iii) (see ) A left (resp., right) counit for , is a character on (a unital *-homomorphism ), satisfying (resp., ). A left and right counit is called (two-sided) counit.
In the above definition the functionals and are defined by and .
Remark 2.6. In , counits are characters on special dense subalgebras of compact quantum groups. These subalgebras are constructed from finite dimensional unitary representations of compact quantum groups. In this paper we mainly deal with quantum semigroups and since it is not natural to define unitary representations for (quantum) semigroups, we use the above notion for counits.
It is clear that the left (resp., right) cancellation property implies weak left (resp., weak right) cancellation property. The converse is partially satisfied [5, Theorem 3.2]:
Theorem 2.7. Let be a compact quantum semigroup. Then has both left and right cancellation properties if and only if it has both weak left and weak right cancellation properties.
Definition 2.8 (see [4, 5, 8]). A compact quantum semigroup with both left and right cancellation properties is called compact quantum group.
Again consider compact semigroup and its corresponding compact quantum semigroup defined above. Using Proposition 3.2 of , it is easily proved that is a compact group if and only if is a compact quantum group.
3. The Results
In this section, we state and prove the main result.
Theorem 3.1. Let be a compact quantum semigroup with finitely generated , be a finite dimensional commutative C*-algebra, and be the quantum family of all maps from to . Consider the unique unital *-homomorphism such that the diagram (3.1) is commutative, where is the flip map, that is, (), and is the multiplication *-homomorphism of , that is, (. Then is a compact quantum semigroup.
Proof. We must prove that , and for this, by the universal property of quantum families of maps, it is enough to prove that Note that by the commutativity of (3.1), we have Let us begin from the left hand side of (3.2): For the right hand side of (3.2), we have and thus if , then Thus, since , to prove (3.2), it is enough to show that Let and . Then for the left hand side of (3.7), we have and for the right hand side of (3.7), Therefore, (3.7) is satisfied and the proof is complete.
Theorem 3.2. Let be a compact quantum semigroup with a left counit. Suppose that , and are as in Theorem 3.1. Then the compact quantum semigroup has a left counit.
Proof. Let be a left counit for . Define the unital *-algebra homomorphism by (). Then the universal property of shows that there is a character such that the following diagram is commutative: (3.10) We show that , and thus is a counit for . By the universal property of , it is enough to show that We have Since is a left counit for , we have for every . This implies that for every in . This completes the proof.
Analogous of Theorem 3.2 is satisfied for quantum groups that have right and (two-sided) counits. Some natural questions about the structure of the compact quantum semigroup arise.
Question 1. Let and be as in Theorem 3.1. (i)Suppose that has one of the left or weak left cancellation properties. Does this hold for ? In particular, suppose the following.(ii)Suppose that is a compact quantum group. Is a compact quantum group? (iii)Are the converses of (i) and (ii) satisfied?
We consider some parts of these questions for a simple example in the next section.
4. Some Examples
In this section, we consider a class of examples. Let be the C*-algebra of functions on the commutative finite space , and let be the quantum family of all maps from to .A direct computation shows that is the universal C*-algebra generated by elements that satisfy the relations (1) for every , (2) for every , and (3) for every .
Also, is defined by , where is the standard basis for . Suppose that is a semigroup multiplication. Then induces a comultiplication : defined by , where is the Kronecker delta. We compute the comultiplication , induced by as in Theorem 3.1. We have and therefore This equals to . Thus is defined by We now consider the special case , in more details. There are only four semigroup structures (up to isomorphism and anti-isomorphism) on the set :
The semigroup structure is a group structure and has right cancellation property. In Semigroup Theory, , and , are called semilattice, null, and left-zero band structures, respectively. For semigroup , let and be the corresponding quantum semigroups, as above. A simple computation shows that: As we have explained in Section 2, is a compact quantum group and is a compact quantum semigroup with right cancellation property. From the above computations, it is clear that the compact quantum semigroup has right cancellation property. Now, we show that is also a compact quantum group: the unital C*-algebra is generated by the two unitary elements and (see the following remark for more details). A simple computation shows that This easily implies that has left and right cancellation properties, and therefore is a compact quantum group.
Remark 4.1. (1) The algebra is the universal C*-algebra generated by a unitary self-adjoint element, say . It follows from the proof of Theorem 3.3 of , that becomes the universal C*-algebra generated by two unitary self-adjoint elements. A model for is the C*-algebra of all continuous maps from closed unit interval to matrix algebra, which take diagonal matrices at the endpoints of the interval, equivalently
with unitary self-adjoint generators
In this representation of , the generators ’s become: , , and . Also, the homomorphism is defined by . This representation of the C*-algebra is one of the elementary examples of noncommutative spaces; see Section II.2. of .
(2) There is another quantum semigroup structure on quantum families of all maps from any finite quantum space to itself introduced by Sołtan .
The author is grateful to the referee for his/her valuable suggestions.
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