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.


Introduction
Gelfand's duality Theorem says that the category of locally compact Hausdorff spaces and continuous maps and the category of commutative C*-algebras and *homomorphisms are dual to each other. In this duality any space X corresponds to C 0 (X), the C*-algebra of all continuous complex valued maps on X vanishing at infinity (note that X is compact if and only if C 0 (X) = C(X) is unital). Thus one can consider a non commutative C*-algebra A as the algebra of functions on a symbolic quantum space QA. In this correspondence, *-homomorphisms Φ : A −→ B interpret as symbolic continuous maps QΦ : QB −→ QA.
Woronowicz [5] and Soltan [3] 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 [2], we considered the functorial properties of this notion. In this short note, 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 non commutative 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.

Quantum families of maps and quantum semigroups
For any C*-algebra A, I A denotes the identity homomorphism from A to A. If A is unital then 1 A 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 ′ : 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 byf (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 non commutative language: Definition 1. ( [5], [3]) 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 of all maps from Y to X has the following universal property: Thus, we can make the following Definition in non commutative setting: [3]) With the assumptions of Definition 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.
By the universal property of Definition 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 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 [5] or [3]. Let S be a compact Hausdorff semigroup. Using the canonical identity we let the *-homomorphisms ∆ : C(S) −→ C(S) ⊗ C(S) be defined by Then ∆ is a coassociative comultiplication on C(S) and thus (C(S), ∆) is a compact quantum semigroup. Conversely, if (A, ∆) is a compact quantum semigroup with commutative A, then there is a compact Hausdorff semigroup such that its corresponding compact quantum semigroup is (A, ∆), [4].

The result
Now, we state and prove the main result.
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 Let b 1 , b 2 , b 3 ∈ B and c 1 , c 2 , c 3 ∈ C. Then for the left hand side of (3), we have, and for the right hand side of (3), (m⊗I C⊗C⊗C )W (m⊗I C⊗C⊗B⊗C )(I B ⊗F ⊗I C⊗B⊗C )(b 1 ⊗c 1 ⊗b 2 ⊗c 2 ⊗b 3 ⊗c 3 ) =(m⊗I C⊗C⊗C )W (m⊗I C⊗C⊗B⊗C )(b 1 ⊗b 2 ⊗c 1 ⊗c 2 ⊗b 3 ⊗c 3 ) Therefore, (3) is satisfied and the proof is complete. Now, 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 from QA to QA. Then, as is indicated in Section 7 of [3], C is the universal C*-algebra generated by n 2 self-adjoint elements {c ij : 1 ≤ i, j ≤ n} that satisfy the relations where e 1 , · · · , e n is the standard basis for A. Suppose that ξ : {1, · · · , n} × {1, · · · , n} −→ {1, · · · , n} is a semigroup multiplication. Then ξ induces a comultiplication ∆ : A −→ A⊗A ∆(e k ) = n r,s=1 ∆ rs k e r ⊗ e s , defined by ∆ rs k = δ kξ(r,s) , where δ is the Cronecker delta. Now, we compute the comultiplication Γ : C −→ C ⊗ C that ∆ induce as in Theorem 5. we have, Some natural questions arise about relations between properties of the compact quantum semigroups (A, ∆) and (C, Γ) described in Theorem 5: (i) Suppose that (A, ∆) has a counite or antipode ( [6]). Then is (C, Γ) so? (ii) Let (A, ∆) be a compact quantum group ( [6]). Is (C, Γ) a compact quantum group? (iii) Are the converse of (i) and (ii) satisfied? It seems that it is not easy to answer these questions even in special case of the above example. We end with two remarks: 1) In the above example, if A = C 2 , then one can consider C as the pointwise multiplication C*-algebra of all continuous maps f from closed interval [0, 1] to the matrix algebra M 2 (C), such that f (0) and f (1) are diagonal matrix, see section II.2.β of [1]. This is one of the basic examples of noncommutative spaces. 2) There is another quantum semigroup structure on quantum families of all maps from any finite quantum space to itself introduced by Soltan [3].