A Kind of Compact Quantum Semigroups

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.

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 𝑋 corresponds to 𝒞(𝑋), the C*-algebra of all continuous complex valued maps on 𝑋, 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 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. Since the coordinates observable of a quantum mechanical systems are noncommutative, some-times noncommutative spaces are called quantum spaces.

Woronowicz [1] and Sołtan [2] have defined a quantum space 𝔔𝐶 of all maps from 𝔔𝐵 to 𝔔𝐴 and showed that 𝐶 exists under appropriate conditions on 𝐴 and 𝐵. In [3], we considered the functorial properties of this notion. In this paper, 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), then 𝔔𝐶 has a canonical quantum semigroup structure. In the other words, we construct the noncommutative version of semigroup ℱ(𝑋,𝑆) described as follows.

Let 𝑋 be a finite space and 𝑆 be a compact semigroup. Then the space ℱ(𝑋,𝑆) of all maps from 𝑋 to 𝑆 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 𝐴, I_𝐴 and 1_𝐴 denote the identity homomorphism from 𝐴 to 𝐴, and the unit of 𝐴, respectively. 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 the *-homomorphism defined by Φ⊗Φ^′(𝑎⊗𝑎^′)=Φ(𝑎)⊗Φ^′(𝑎^′) (𝑎∈𝐴,𝑎^′∈𝐴^′).

Let 𝑋,𝑌, and 𝑍 be three compact Hausdorff spaces and 𝒞(𝑌,𝑋) be the space of all continuous maps from 𝑌 to 𝑋 with compact open topology. Consider a continuous map 𝑓∶𝑍→𝒞(𝑌,𝑋). Then the pair (𝑍,𝑓) is a continuous family of maps from 𝑌 to 𝑋 indexed by 𝑓 with parameters in 𝑍. On the other hand, by topological exponential law we know that 𝑓 is characterized by a continuous map 𝑓∶𝑌×𝑍→𝑋 defined by 𝑓(𝑦,𝑧)=𝑓(𝑧)(𝑦). Thus (𝑍,𝑓) can be considered as a family of maps from 𝑌 to 𝑋. Now, by Gelfand's duality we can simply translate this system to noncommutative language.

Definition 2.1 (see [1, 2]). Let 𝐴 and 𝐵 be unital C*-algebras. By a quantum family of maps from 𝔔𝐵 to 𝔔𝐴, we mean a pair (𝐶,Φ), containing a unital C*-algebra 𝐶 and a unital *-homomorphism Φ∶𝐴→𝐵⊗𝐶.

Now, suppose instead of parameter space 𝑍 we use 𝒞(𝑌,𝑋) (note that in general this space is not locally compact). Then the family Id∶𝒞(𝑌,𝑋)→𝒞(𝑌,𝑋)Id∶𝒞(𝑌,𝑋)×𝑌→𝑋 (2.1) of all maps from 𝑌 to 𝑋 has the following universal property.

For every family 𝑓∶𝑍×𝑌→𝑋 of maps from 𝑌 to 𝑋, there is a unique map 𝑓∶𝑍→𝒞(𝑌,𝑋) 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, (𝐶,Φ) is called a quantum family of all maps from 𝔔𝐵 to 𝔔𝐴 if for every unital C*-algebra 𝐷 and any unital *-homomorphism Ψ∶𝐴→𝐵⊗𝐷, there is a unique unital *-homomorphism Γ∶𝐶→𝐷 such that the following diagram is commutative: (2.3)

By the universal property of Definition 2.2, it is clear that if (𝐶,Φ) and (𝐶^′,Φ^′) are two quantum families of all maps from 𝔔𝐵 to 𝔔𝐴, then there is a *-isometric isomorphism between 𝐶 and 𝐶^′.

Proposition 2.3. Let 𝐴 be a unital finitely generated C*-algebra and 𝐵 be a finite dimensional C*-algebra. Then the quantum family of all maps from 𝔔𝐵 to 𝔔𝐴 exists.

Proof . See [1] or [2].

Definition 2.4 (see [2, 4–6]). A pair (𝐴,Δ) consisting of a unital C*-algebra 𝐴 and a unital *-homomorphism Δ∶𝐴→𝐴⊗𝐴 is called a compact quantum semigroup if Δ is a coassociative comultiplication: (Δ⊗I_𝐴)Δ=(I_𝐴⊗Δ)Δ.

A *-homomorphism Δ∶𝐴→𝐴⊗𝐴 induces a binary operation on the dual space 𝐴^∗ defined by 𝜏𝜎=(𝜏⊗𝜎)Δ for 𝜏,𝜎∈𝐴^∗. Now, suppose that 𝑆 is a compact Hausdorff topological semigroup. Using the canonical identity 𝒞(𝑆)⊗𝒞(𝑆)≅𝒞(𝑆×𝑆), we define a *-homomorphism Δ∶𝒞(𝑆)→𝒞(𝑆)⊗𝒞(𝑆) by Δ(𝑓)(𝑠,𝑠^′)=𝑓(𝑠𝑠^′) for 𝑓∈𝒞(𝑆) and 𝑠,𝑠^′∈𝑆. Then Δ is a coassociative comultiplication on 𝒞(𝑆) and thus (𝒞(𝑆),Δ) is a compact quantum semigroup. Conversely, if (𝐴,Δ) is a compact quantum semigroup such that 𝐴 is abelian, then the character space of 𝐴, with the binary operation induced by Δ, is a compact Hausdorff topological semigroup [7]. 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 (𝐴,Δ) be a compact quantum semigroup. (i)(see [5]) (𝐴,Δ) has left (resp., right) cancellation property if the linear span of {(𝑏⊗1)Δ(𝑎)∶𝑎,𝑏∈𝐴} (resp., {(1⊗𝑏)Δ(𝑎)∶𝑎,𝑏∈𝐴}) is dense in 𝐴⊗𝐴. (ii)(see [5]) (𝐴,Δ) has weak left cancellation property if, whenever 𝜏,𝜎∈𝐴^∗ are such that (𝜏𝑎)𝜎=0 for all 𝑎∈𝐴, we must have 𝜏=0 or 𝜎=0. Similarly, (𝐴,Δ) has weak right cancellation property if, whenever 𝜏(𝜎𝑎)=0 for all 𝑎∈𝐴, we must have 𝜏=0 or 𝜎=0. (iii) (see [2]) A left (resp., right) counit for (𝐴,Δ), is a character 𝜖 on 𝐴 (a unital *-homomorphism 𝜖∶𝐴→ℂ), satisfying (𝜖⊗I_𝐴)Δ=I_𝐴 (resp., (I_𝐴⊗𝜖)Δ=I_𝐴). A left and right counit is called (two-sided) counit.

In the above definition the functionals 𝜏𝑎 and 𝑎𝜏 are defined by 𝜏𝑎(𝑥)=𝜏(𝑎𝑥) and 𝑎𝜏(𝑥)=𝜏(𝑥𝑎).

Remark 2.6. In [4], 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 (𝐴,Δ) be a compact quantum semigroup. Then (𝐴,Δ) 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 𝑆 and its corresponding compact quantum semigroup (𝒞(𝑆),Δ) defined above. Using Proposition 3.2 of [8], it is easily proved that 𝑆 is a compact group if and only if (𝒞(𝑆),Δ) is a compact quantum group.

3. The Results

In this section, we state and prove the main result.

Theorem 3.1. Let (𝐴,Δ) be a compact quantum semigroup with finitely generated 𝐴, 𝐵 be a finite dimensional commutative C*-algebra, and (𝐶,Φ) be the quantum family of all maps from 𝔔𝐵 to 𝔔𝐴. Consider the unique unital *-homomorphism Γ∶𝐶→𝐶⊗𝐶 such that the diagram (3.1) is commutative, where 𝐹∶𝐶⊗𝐵→𝐵⊗𝐶 is the flip map, that is, 𝑐⊗𝑏↦𝑏⊗𝑐 (𝑏∈𝐵,𝑐∈𝐶), and 𝑚∶𝐵⊗𝐵→𝐵 is the multiplication *-homomorphism of 𝐵, that is, 𝑚(𝑏⊗𝑏^′)=𝑏𝑏^′ (𝑏,𝑏^′∈𝐵). Then (𝐶,Γ) is a compact quantum semigroup.

Proof. We must prove that (I_𝐶⊗Γ)Γ=(Γ⊗I_𝐶)Γ, and for this, by the universal property of quantum families of maps, it is enough to prove that I_𝐵⊗I_𝐶⊗ΓI_𝐵⊗ΓΦ=I_𝐵⊗Γ⊗I_𝐶I_𝐵⊗ΓΦ. (3.2) Note that by the commutativity of (3.1), we have I_𝐵⊗ΓΦ=𝑚⊗I_𝐶⊗𝐶I_𝐵⊗𝐹⊗I_𝐶(Φ⊗Φ)Δ. (3.3) Let us begin from the left hand side of (3.2): I_𝐵⊗I_𝐶⊗ΓI_𝐵Φ=I⊗Γ_𝐵⊗I_𝐶⊗Γ𝑚⊗I_𝐶⊗𝐶I_𝐵⊗𝐹⊗I_𝐶)Δ=(Φ⊗Φ𝑚⊗I_𝐶⊗ΓI_𝐵⊗𝐹⊗I_𝐶()Δ=Φ⊗Φ𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗𝐶I_{𝐵⊗𝐶⊗𝐵}()Δ=⊗ΓΦ⊗Φ𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗𝐶I_{𝐵⊗𝐶⊗𝐵}⊗ΓΦ⊗I_𝐵⊗𝐶I_𝐴Δ=⊗Φ𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗𝐶Φ⊗I_{𝐵⊗𝐶⊗𝐶}I_𝐴⊗I_𝐵⊗ΓI_𝐴Δ=⊗Φ𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗𝐶Φ⊗I_𝐵⊗ΓI_𝐴Δ=⊗Φ𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗𝐶Φ⊗I_𝐵ΦΔ=⊗Γ𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗𝐶Φ⊗𝑚⊗I_𝐶⊗𝐶I_𝐵⊗𝐹⊗I_𝐶()ΔΔ=Φ⊗Φ𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗𝐶I_𝐵⊗𝐶⊗𝑚⊗I_𝐶⊗𝐶I_{𝐵⊗𝐶⊗𝐵}⊗𝐹⊗I_𝐶(Φ⊗Φ⊗Φ)I_𝐴⊗ΔΔ. (3.4) For the right hand side of (3.2), we have I_𝐵⊗Γ⊗I_𝐶I_𝐵Φ=I⊗Γ_𝐵⊗Γ⊗I_𝐶𝑚⊗I_𝐶⊗𝐶I_𝐵⊗𝐹⊗I_𝐶)Δ=(Φ⊗Φ𝑚⊗Γ⊗I_𝐶I_𝐵⊗𝐹⊗I_𝐶()Δ=Φ⊗Φ𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗I_𝐶I_𝐵⊗I_𝐶⊗𝐹⊗I_𝐶I_𝐵⊗Γ⊗I_𝐵⊗I_𝐶()Δ=Φ⊗Φ𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗𝐶I_𝐵⊗𝐶⊗𝐹⊗I_𝐶I_𝐵ΦΔ=⊗Γ⊗Φ𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗𝐶I_𝐵⊗𝐶⊗𝐹⊗I_𝐶𝑚⊗I_𝐶⊗𝐶I_𝐵⊗𝐹⊗I_𝐶(Φ⊗Φ)Δ⊗ΦΔ, (3.5) and thus if 𝑊=(I_𝐵⊗𝐹⊗I_𝐶⊗𝐶)(I_𝐵⊗𝐶⊗𝐹⊗I_𝐶), then I_𝐵⊗Γ⊗I_𝐶I_𝐵Φ=⊗Γ𝑚⊗I_{𝐶⊗𝐶⊗𝐶}𝑊𝑚⊗I_{𝐶⊗𝐶⊗𝐵⊗𝐶}I_𝐵⊗𝐹⊗I_{𝐶⊗𝐵⊗𝐶}(Φ⊗Φ⊗Φ)Δ⊗I_𝐴Δ. (3.6) Thus, since (I_𝐴⊗Δ)Δ=(Δ⊗I_𝐴)Δ, to prove (3.2), it is enough to show that 𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗𝐶I_𝐵⊗𝐶⊗𝑚⊗I_𝐶⊗𝐶I_{𝐵⊗𝐶⊗𝐵}⊗𝐹⊗I_𝐶=𝑚⊗I_{𝐶⊗𝐶⊗𝐶}𝑊𝑚⊗I_{𝐶⊗𝐶⊗𝐵⊗𝐶}I_𝐵⊗𝐹⊗I_{𝐶⊗𝐵⊗𝐶}. (3.7) Let 𝑏₁,𝑏₂,𝑏₃∈𝐵 and 𝑐₁,𝑐₂,𝑐₃∈𝐶. Then for the left hand side of (3.7), we have 𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗𝐶I_𝐵⊗𝐶⊗𝑚⊗I_𝐶⊗𝐶I_{𝐵⊗𝐶⊗𝐵}⊗𝐹⊗I_𝐶𝑏₁⊗𝑐₁⊗𝑏₂⊗𝑐₂⊗𝑏₃⊗𝑐₃=𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗𝐶I_𝐵⊗𝐶⊗𝑚⊗I_𝐶⊗𝐶𝑏₁⊗𝑐₁⊗𝑏₂⊗𝑏₃⊗𝑐₂⊗𝑐₃=𝑚⊗I_{𝐶⊗𝐶⊗𝐶}I_𝐵⊗𝐹⊗I_𝐶⊗𝐶𝑏₁⊗𝑐₁⊗𝑏₂𝑏₃⊗𝑐₂⊗𝑐₃=𝑚⊗I_{𝐶⊗𝐶⊗𝐶}𝑏₁⊗𝑏₂𝑏₃⊗𝑐₁⊗𝑐₂⊗𝑐₃=𝑏₁𝑏₂𝑏₃⊗𝑐₁⊗𝑐₂⊗𝑐₃=𝑏₁𝑏₂𝑏₃⊗𝑐₁⊗𝑐₂⊗𝑐₃, (3.8) and for the right hand side of (3.7), 𝑚⊗I_{𝐶⊗𝐶⊗𝐶}𝑊𝑚⊗I_{𝐶⊗𝐶⊗𝐵⊗𝐶}I_𝐵⊗𝐹⊗I_{𝐶⊗𝐵⊗𝐶}𝑏₁⊗𝑐₁⊗𝑏₂⊗𝑐₂⊗𝑏₃⊗𝑐₃=𝑚⊗I_{𝐶⊗𝐶⊗𝐶}𝑊𝑚⊗I_{𝐶⊗𝐶⊗𝐵⊗𝐶}𝑏₁⊗𝑏₂⊗𝑐₁⊗𝑐₂⊗𝑏₃⊗𝑐₃=𝑚⊗I_{𝐶⊗𝐶⊗𝐶}𝑊𝑏₁𝑏₂⊗𝑐₁⊗𝑐₂⊗𝑏₃⊗𝑐₃=𝑚⊗I_{𝐶⊗𝐶⊗𝐶}𝑏₁𝑏₂⊗𝑏₃⊗𝑐₁⊗𝑐₂⊗𝑐₃=𝑏₁𝑏₂𝑏₃⊗𝑐₁⊗𝑐₂⊗𝑐₃. (3.9) Therefore, (3.7) is satisfied and the proof is complete.

Theorem 3.2. Let (𝐴,Δ) be a compact quantum semigroup with a left counit. Suppose that 𝐵,𝐶,Φ, and Γ are as in Theorem 3.1. Then the compact quantum semigroup (𝐶,Γ) has a left counit.

Proof. Let 𝜖∶𝐴→ℂ be a left counit for (𝐴,Δ). Define the unital *-algebra homomorphism 𝜔∶𝐴→𝐵⊗ℂ=𝐵 by 𝜔(𝑎)=1_𝐵⊗𝜖(𝑎)=𝜖(𝑎)1_𝐵 (𝑎∈𝐴). Then the universal property of (𝐶,Φ) shows that there is a character 𝜖̂∶𝐶→ℂ such that the following diagram is commutative: (3.10) We show that (𝜖̂⊗I_𝐶)Γ=I_𝐶, and thus 𝜖̂ is a counit for (𝐶,Γ). By the universal property of (𝐶,Φ), it is enough to show that I_𝐵⊗𝜖̂⊗I_𝐶ΓΦ=Φ. (3.11) We have I_𝐵⊗𝜖̂⊗I_𝐶ΓΦ=I_𝐵⊗𝜖̂⊗I_𝐶I_𝐵Φ=I⊗Γ_𝐵⊗𝜖̂⊗I_𝐶𝑚⊗I_𝐶⊗𝐶I_𝐵⊗𝐹⊗I_𝐶)Δ=̂(Φ⊗Φ𝑚⊗𝜖⊗I_𝐶I_𝐵⊗𝐹⊗I_𝐶()Δ=Φ⊗Φ𝑚⊗I_𝐶I_𝐵⊗𝜖̂⊗I_𝐵⊗I_𝐶()Δ=Φ⊗Φ𝑚⊗I_𝐶I_𝐵⊗𝜖̂ΦΔ=⊗Φ𝑚⊗I_𝐶()Δ=𝜔⊗Φ𝑚⊗I_𝐶I_𝐵⊗Φ𝜔⊗I_𝐴Δ. (3.12) Since 𝜖 is a left counit for (𝐴,Δ), we have 𝜔⊗I_𝐴Δ(𝑎)=1_𝐵⊗𝑎, (3.13) for every 𝑎∈𝐴. This implies that 𝑚⊗I_𝐶I_𝐵⊗Φ𝜔⊗I_𝐴Δ(𝑎)=𝑚⊗I_𝐶I_𝐵⊗Φ1_𝐵(𝑎),⊗𝑎=Φ (3.14) for every 𝑎 in 𝐴. 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 (𝐶,Γ) arise.

Question 1. Let (𝐴,Δ) and (𝐶,Γ) be as in Theorem 3.1. (i)Suppose that (𝐴,Δ) has one of the left or weak left cancellation properties. Does this hold for (𝐶,Γ)? In particular, suppose the following.(ii)Suppose that (𝐴,Δ) is a compact quantum group. Is (𝐶,Γ) 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 𝐴=ℂ^𝑛 be the C*-algebra of functions on the commutative finite space {1,…,𝑛}, and let (𝐶,Φ) be the quantum family of all maps from𝔔𝐴 to 𝔔𝐴.A direct computation shows that 𝐶 is the universal C*-algebra generated by 𝑛² elements {𝑐_𝑖𝑗∶1≤𝑖,𝑗≤𝑛} that satisfy the relations (1)𝑐²_𝑖𝑗=𝑐_𝑖𝑗=𝑐^∗_𝑖𝑗 for every 𝑖,𝑗=1,…,𝑛, (2)∑^𝑛_𝑗=1𝑐_𝑖𝑗=1 for every 𝑖=1,…,𝑛, and (3)𝑐_𝑖𝑗𝑐_𝑖𝑘=0 for every 𝑖,𝑘,𝑗=1,…,𝑛.

Also, Φ∶𝐴→𝐴⊗𝐶 is defined by Φ(𝑒_𝑘)=∑^𝑛_𝑖=1𝑒_𝑖⊗𝑐_𝑖𝑘, where 𝑒₁,…,𝑒_𝑛 is the standard basis for 𝐴. Suppose that 𝜉∶{1,…,𝑛}×{1,…,𝑛}→{1,…,𝑛} (4.1) is a semigroup multiplication. Then 𝜉 induces a comultiplication Δ∶𝐴→𝐴⊗𝐴: Δ𝑒_𝑘=^𝑛_𝑟,𝑠=1Δ_𝑘^𝑟𝑠𝑒_𝑟⊗𝑒_𝑠, (4.2) defined by Δ_𝑘^𝑟𝑠=𝛿_{𝑘𝜉(𝑟,𝑠)}, where 𝛿 is the Kronecker delta. We compute the comultiplication Γ∶𝐶→𝐶⊗𝐶, induced by Δ as in Theorem 3.1. We have (Φ⊗Φ)Δ𝑒_𝑘=(Φ⊗Φ)^𝑛_𝑟,𝑠=1Δ_𝑘^𝑟𝑠𝑒_𝑟⊗𝑒_𝑠=^𝑛_𝑟,𝑠=1Δ_𝑘^𝑟𝑠Φ𝑒_𝑟⊗Φ𝑒_𝑠=^𝑛𝑛_𝑟,𝑠=1^𝑛_𝑗=1𝑖=1Δ_𝑘^𝑟𝑠𝑒_𝑗⊗𝑐_𝑗𝑟⊗𝑒_𝑖⊗𝑐_𝑖𝑠, (4.3) and therefore 𝑚⊗I_𝐶⊗𝐶I_𝐵⊗𝐹⊗I_𝐶(Φ⊗Φ)Δ𝑒_𝑘=𝑚⊗I_𝐶⊗𝐶^𝑛𝑛_𝑗=1^𝑛_{𝑖=1𝑟,𝑠=1}Δ_𝑘^𝑟𝑠𝑒_𝑗⊗𝑒_𝑖⊗𝑐_𝑗𝑟⊗𝑐_𝑖𝑠=^𝑛𝑛_{𝑙=1𝑟,𝑠=1}Δ_𝑘^𝑟𝑠𝑒_𝑙⊗𝑐_𝑙𝑟⊗𝑐_𝑙𝑠=^𝑛_𝑙=1𝑒_𝑙⊗^𝑛_𝑟,𝑠=1Δ_𝑘^𝑟𝑠𝑐_𝑙𝑟⊗𝑐_𝑙𝑠. (4.4) This equals to (I_𝐴⊗Γ)Φ(𝑒_𝑘)=(I_𝐴⊗Γ)∑^𝑛_𝑖=1𝑒_𝑖⊗𝑐_𝑖𝑘=∑^𝑛_𝑙=1𝑒_𝑙⊗Γ(𝑐_𝑙𝑘). Thus Γ is defined by Γ𝑐_𝑙𝑘=^𝑛_𝑟,𝑠=1Δ_𝑘^𝑟𝑠𝑐_𝑙𝑟⊗𝑐_𝑙𝑠. (4.5) We now consider the special case 𝑛=2, in more details. There are only four semigroup structures (up to isomorphism and anti-isomorphism) on the set {1,2}:

𝜉₁∶11=1,12=2,21=2,22=1.

𝜉₂∶11=1,12=2,21=2,22=2.

𝜉₃∶11=1,12=1,21=1,22=1.

𝜉₄∶11=1,12=1,21=2,22=2.

The semigroup structure 𝜉₁ is a group structure and 𝜉₄ has right cancellation property. In Semigroup Theory, 𝜉₂,𝜉₃, and 𝜉₄, are called semilattice, null, and left-zero band structures, respectively. For semigroup ({1,2},𝜉_𝑖), let (ℂ²,Δ_𝑖) and (𝐶,Γ_𝑖) be the corresponding quantum semigroups, as above. A simple computation shows that: Γ₁𝑐₁₁=𝑐₁₁⊗𝑐₁₁+𝑐₁₂⊗𝑐₁₂,Γ₁𝑐₁₂=𝑐₁₁⊗𝑐₁₂+𝑐₁₂⊗𝑐₁₁,Γ₁𝑐₂₁=𝑐₂₁⊗𝑐₂₁+𝑐₂₂⊗𝑐₂₂,Γ₁𝑐₂₂=𝑐₂₁⊗𝑐₂₂+𝑐₂₂⊗𝑐₂₁,Γ₂𝑐₁₁=𝑐₁₁⊗𝑐₁₁,Γ₂𝑐₁₂=𝑐₁₂⊗𝑐₁₂+𝑐₁₁⊗𝑐₁₂+𝑐₁₂⊗𝑐₁₁,Γ₂𝑐₂₁=𝑐₂₁⊗𝑐₂₁,Γ₂𝑐₂₂=𝑐₂₂⊗𝑐₂₂+𝑐₂₁⊗𝑐₂₂+𝑐₂₂⊗𝑐₂₁,Γ₃𝑐₁₁=1,Γ₃𝑐₁₂Γ=0,₃𝑐₂₁=1,Γ₃𝑐₂₂Γ=0,₄(𝑐)=𝑐⊗1(∀𝑐∈𝐶). (4.6) As we have explained in Section 2, (ℂ²,Δ₁) is a compact quantum group and (ℂ²,Δ₄) is a compact quantum semigroup with right cancellation property. From the above computations, it is clear that the compact quantum semigroup (𝐶,Γ₄) has right cancellation property. Now, we show that (𝐶,Γ₁) is also a compact quantum group: the unital C*-algebra 𝐶 is generated by the two unitary elements 𝑥=𝑐₁₁−𝑐₁₂ and 𝑦=𝑐₂₁−𝑐₂₂ (see the following remark for more details). A simple computation shows that Γ₁(𝑥)=𝑥⊗𝑥,Γ₁𝑦=𝑦⊗𝑦. (4.7) This easily implies that (𝐶,Γ₁) has left and right cancellation properties, and therefore (𝐶,Γ₁) is a compact quantum group.

Remark 4.1. (1) The algebra 𝐴=ℂ² is the universal C*-algebra generated by a unitary self-adjoint element, say (1,−1). It follows from the proof of Theorem 3.3 of [2], that 𝐶 becomes the universal C*-algebra generated by two unitary self-adjoint elements. A model for 𝐶 is the C*-algebra of all continuous maps from closed unit interval to 2×2 matrix algebra, which take diagonal matrices at the endpoints of the interval, equivalently 𝑓𝐶=¹¹𝑓₁₂𝑓₂₁𝑓₂₂∶𝑓_𝑖𝑗∈𝒞[0,1],𝑓₁₂(0)=𝑓₁₂(1)=𝑓₂₁(0)=𝑓₂₁(1)=0, (4.8) with unitary self-adjoint generators ()()()()()()()()𝑥=cos𝜋𝑡sin𝜋𝑡sin𝜋𝑡−cos𝜋𝑡,𝑦=−cos𝜋𝑡sin𝜋𝑡sin𝜋𝑡cos𝜋𝑡. (4.9) In this representation of 𝐶, the generators 𝑐_𝑖𝑗’s become: 𝑐₁₁=(1+𝑥)/2, 𝑐₁₂=(1−𝑥)/2, 𝑐₂₁=(1+𝑦)/2 and 𝑐₂₂=(1−𝑦)/2. Also, the homomorphism Φ∶ℂ²→ℂ²⊗𝐶=𝐶⊕𝐶 is defined by Φ(1,−1)=(𝑥,𝑦). This representation of the C*-algebra 𝐶 is one of the elementary examples of noncommutative spaces; see Section II.2.𝛽 of [9].
(2) There is another quantum semigroup structure on quantum families of all maps from any finite quantum space to itself introduced by Sołtan [2].

Acknowledgment

The author is grateful to the referee for his/her valuable suggestions.