# C

Authors: Kengo Matsumoto
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

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## Abstract

A HilbertC∗-quad module of finite type has a multistructure of
Hilbert C∗-bimodules with two finite bases. We will construct a C∗-algebra
from a Hilbert C∗-quad module of finite type and prove its universality subject
to certain relations among generators. Some examples of the C∗-algebras from
Hilbert C∗-quad modules of finite type will be presented.


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## Body

## 1. Introduction

Robertson and Steger [1] have initiated a certain study of higher-dimensional analogue of Cuntz-Krieger algebras from the view point of tiling systems of 2-dimensional plane. After their work, Kumjian and Pask [2] have generalized their construction to introduce the notion of higher-rank graphs and its C*-algebras. Since then, there have been many studies on these C*-algebras by many authors (see, e.g., [1–6], etc.).In [7], the author has introduced a notion of C*-symbolic dynamical system, which is a generalization of a finite labeled graph, a λ-graph system, and an automorphism of a unital C*-algebra. It is denoted by (𝒜,ρ,Σ) and consists of a finite family {ρα}α∈Σ of endomorphisms of a unital C*-algebra 𝒜 such that ρα(Z𝒜)⊂Z𝒜, α∈Σ and ∑α∈Σ‍ρα(1)≥1 where Z𝒜 denotes the center of 𝒜. It provides a subshift Λρ over Σ and a Hilbert C*-bimodule ℋ𝒜ρ over 𝒜 which gives rise to a C*-algebra 𝒪ρ as a Cuntz-Pimsner algebra ([7] cf. [8–10]). In [11, 12], the author has extended the notion of C*-symbolic dynamical system to C*-textile dynamical system which is a higher-dimensional analogue of C*-symbolic dynamical system. A C*-textile dynamical system (𝒜,ρ,η,Σρ,Ση,κ) consists of two C*-symbolic dynamical systems (𝒜,ρ,Σρ) and (𝒜,η,Ση) with common unital C*-algebra 𝒜 and commutation relations κ between the endomorphisms ρα,α∈Σρ and ηa,a∈Ση. A C*-textile dynamical system provides a two-dimensional subshift and a multistructure of Hilbert C*-bimodules that has multi right actions and multi left actions and multi inner products. Such a multi structure of Hilbert C*-bimodule is called a Hilbert C*-quad module. In [12], the author has introduced a C*-algebra associated with the Hilbert C*-quad module of C*-textile dynamical system. It is generated by the quotient images of creation operators on two-dimensional analogue of Fock Hilbert module by module maps of compact operators. As a result, the C*-algebra has been proved to have a universal property subject to certain operator relations of generators encoded by structure of C*-textile dynamical system [12].In this paper, we will generalize the construction of theC*-algebras of Hilbert C*-quad modules of C*-textile dynamical systems. Let 𝒜, ℬ1, and ℬ2 be unital C*-algebras. Assume that 𝒜 has unital embeddings into both ℬ1 and ℬ2. A Hilbert C*-quad module ℋ over (𝒜;ℬ1,ℬ2) is a Hilbert C*-bimodule over 𝒜 with 𝒜-valued right inner product 〈·∣·〉𝒜 which has a multi structure of Hilbert C*-bimodules over ℬi with right actions φi of ℬi and left actions ϕi of ℬi and ℬi-valued inner products 〈·∣·〉ℬi for i=1,2 satisfying certain compatibility conditions. A Hilbert C*-quad module ℋ is said to be of finite type if there exists a finite basis {u1,…,uM} of ℋ as a Hilbert C*-right module over ℬ1 and a finite basis {v1,…,vN} of ℋ as a Hilbert C*-right module over ℬ2 such that(1)∑i=1M〈ui∣ϕ2(〈ξ∣η〉ℬ2)ui〉ℬ1=∑k=1N〈vk∣ϕ1(〈ξ∣η〉ℬ1)vk〉ℬ2=〈ξ∣η〉𝒜
for ξ,η∈ℋ (see [13] for the original definition of finite basis of Hilbert module). For a Hilbert C*-quad module, we will construct a Fock space F(ℋ) from ℋ, which is a 2-dimensional analogue to the ordinary Fock space of Hilbert C*-bimodules (cf. [10, 14]). We will then define two kinds of creation operators sξ, tξ for ξ∈ℋ on F(ℋ). The C*-algebra on F(ℋ) generated by them is denoted by 𝒯F(ℋ) and called the Toeplitz quad module algebra. We then define the C*-algebra 𝒪F(ℋ) associated with the Hilbert C*-quad module ℋ by the quotient C*-algebra of 𝒯F(ℋ) by the ideal generated by the finite-rank operators. We will then prove that the C*-algebra 𝒪F(ℋ) for a C*-quad module ℋ of finite type has a universal property in the following way.Theorem 1 (Theorem34).
Letℋ be a Hilbert C*-quad module over (𝒜;ℬ1,ℬ2) of finite type with a finite basis {u1,…,uM} of ℋ as a Hilbert C*-right module over ℬ1 and a finite basis {v1,…,vN} of ℋ as a Hilbert C*-right module over ℬ2. Then, the C*-algebra 𝒪F(ℋ) generated by the quotients [sξ],[tξ] of the creation operators sξ,tξ for ξ∈ℋ on the Fock spaces F(ℋ) is canonically isomorphic to the universal C*-algebra 𝒪ℋ generated by operators S1,…,SM,  T1,…,TN and elements z∈ℬ1,  w∈ℬ2 subject to the relations
(2)∑i=1MSiSi*+∑k=1NTkTk*=1,Sj*Tl=0,Si*Sj=〈ui∣uj〉ℬ1,Tk*Tl=〈vk∣vl〉ℬ2,zSj=∑i=1MSi〈ui∣ϕ1(z)uj〉ℬ1,zTl=∑k=1NTk〈vk∣ϕ1(z)vl〉ℬ2,wSj=∑i=1MSi〈ui∣ϕ2(w)uj〉ℬ1,wTl=∑k=1NTk〈vk∣ϕ2(w)vl〉ℬ2
for z∈ℬ1, w∈ℬ2, i,j=1,…,M,k, l=1,…,N.The eight relations of the operators above are called the relations(ℋ). As a corollary, we have the following.Corollary 2 (Corollary35).
For aC*-quad module ℋ of finite type, the universal C*-algebra 𝒪ℋ generated by operators S1,…,SM,  T1,…,TN and elements z∈ℬ1, w∈ℬ2 subject to the relations (ℋ) does not depend on the choice of the finite bases {u1,…,uM} and {v1,…,vN}.The paper is organized in the following way. In Section2, we will define Hilbert C*-quad module and present some basic properties. In Section 3, we will define a C*-algebra 𝒪F(ℋ) from Hilbert C*-quad module ℋ of general type by using creation operators on Fock Hilbert C*-quad module. In Section 4, we will study algebraic structure of the C*-algebra 𝒪F(ℋ) for a Hilbert C*-quad module ℋ of finite type. In Section 5, we will prove, as a main result of the paper, that the C*-algebra 𝒪F(ℋ) has the universal property stated as in Theorem 1. A strategy to prove Theorem 1 is to show that the C*-algebra 𝒪F(ℋ) is regarded as a Cuntz-Pimsner algebra for a Hilbert C*-bimodule over the C*-algebra generated by ϕ1(ℬ1) and ϕ2(ℬ2). We will then prove the gauge invariant universality of the C*-algebra (Theorem 33). In Section 6, we will present K-theory formulae for the C*-algebra 𝒪ℋ. In Section 7, we will give examples. In Section 8, we will formulate higher-dimensional analogue of our situations and state a generalized proposition of Theorem 1 without proof.Throughout the paper, we will denote byℤ+ the set of nonnegative integers and by ℕ the set of positive integers.


## 2. HilbertC*-Quad Modules

Throughout the paper, we fix three unitalC*-algebras 𝒜, ℬ1, and ℬ2 such that 𝒜⊂ℬ1, 𝒜⊂ℬ2 with common units. We assume that there exists a right action ψi of 𝒜 on ℬi so that
(3)biψi(a)∈ℬiforbi∈ℬi,a∈𝒜,i=1,2,
which satisfies
(4)∥biψi(a)∥≤∥bi∥∥a∥,biψi(aa′)=biψi(a)ψi(a′)
for bi∈ℬi,  a,a′∈𝒜, i=1,2. Hence, ℬi is a right 𝒜-module through ψi for i=1,2. Suppose that ℋ is a Hilbert C*-bimodule over 𝒜, which has a right action of 𝒜, an 𝒜-valued right inner product 〈·∣·〉𝒜, and a *-homomorphism ϕ𝒜 from 𝒜 to the algebra of all bounded adjointable right 𝒜-module maps ℒ𝒜(ℋ) satisfying the following.(i)
〈·∣·〉𝒜 is linear in the second variable.(ii)
〈ξ∣ηa〉𝒜=〈ξ∣η〉𝒜a for ξ,η∈ℋ, a∈𝒜.(iii)
〈ξ∣η〉𝒜*=〈η∣ξ〉𝒜 for ξ,η∈ℋ.(iv)
〈ξ∣ξ〉𝒜≥0, and 〈ξ∣ξ〉𝒜=0 if and only if ξ=0.A Hilbert C*-bimodule ℋ over 𝒜 is called a Hilbert C*-quad module over(𝒜;ℬ1,ℬ2) if ℋ has a further structure of a Hilbert C*-bimodule over ℬi for each i=1,2 with right action φi of ℬi and left action ϕi of ℬi and ℬi-valued right inner product 〈·∣·〉ℬi such that for z∈ℬ1, w∈ℬ2,
(5)ϕ1(z),ϕ2(w)∈ℒ𝒜(ℋ),[ϕ1(z)ξ]φ2(w)=ϕ1(z)[ξφ2(w)],[ϕ2(w)ξ]φ1(z)=ϕ2(w)[ξφ1(z)],ξφ1(zψ1(a))=[ξφ1(z)]a,ξφ2(wψ2(a))=[ξφ2(w)]a,
for ξ∈ℋ, z∈ℬ1, w∈ℬ2, a∈𝒜, and
(6)ϕ𝒜(a)=ϕ1(a)=ϕ2(a)fora∈𝒜,
where 𝒜 is regarded as a subalgebra of ℬi. The left action ϕi of ℬi on ℋ means that ϕi(bi) for bi∈ℬi is a bounded adjointable operator with respect to the inner product 〈·∣·〉ℬi for each i=1,2. The operator ϕi(bi) for bi∈ℬi is also adjointable with respect to the inner product 〈·∣·〉𝒜. We assume that the adjoint of ϕi(bi) with respect to the inner product 〈·∣·〉ℬi coincides with the adjoint of ϕi(bi) with respect to the inner product 〈·∣·〉𝒜. Both of them coincide with ϕi(bi*). We assume that the left actions ϕi of ℬi on ℋ for i=1,2 are faithful. We require the following compatibility conditions between the right 𝒜-module structure of ℋ and the right 𝒜-module structure of ℬi through ψi:
(7)〈ξ∣ηa〉ℬi=〈ξ∣η〉ℬiψi(a)forξ,η∈ℋ,a∈𝒜,i=1,2.We further assume thatℋ is a full Hilbert C*-bimodule with respect to the three inner products 〈·∣·〉𝒜, 〈·∣·〉ℬ1, and 〈·∣·〉ℬ2 for each. This means that the C*-algebras generated by elements {〈ξ∣η〉𝒜∣ξ,η∈ℋ}, {〈ξ∣η〉ℬ1∣ξ,η∈ℋ} and {〈ξ∣η〉ℬ2∣ξ,η∈ℋ} coincide with 𝒜, ℬ1, and ℬ2, respectively.For a vectorξ∈ℋ, denote by ∥ξ∥𝒜, ∥ξ∥ℬ1, and ∥ξ∥ℬ2 the norms ∥〈ξ∣ξ〉𝒜∥1/2, ∥〈ξ∣ξ〉ℬ1∥1/2, and ∥〈ξ∣ξ〉ℬ2∥1/2 induced by the right inner products, respectively. By definition, ℋ is complete under the above three norms for each.Definition 3.
(i) A HilbertC*-quad module ℋ over (𝒜;ℬ1,ℬ2) is said to be of general type if there exists a faithful completely positive map λi:ℬi→𝒜 for i=1,2 such that
(8)λi(biψi(a))=λi(bi)aforbi∈ℬi,a∈𝒜,(9)λi(〈ξ∣η〉ℬi)=〈ξ∣η〉𝒜,forξ,η∈ℋ,i=1,2.
(ii) A HilbertC*-quad module ℋ over (𝒜;ℬ1,ℬ2) is said to be of finite type if there exist a finite basis {u1,…,uM} of ℋ as a right Hilbert ℬ1-module and a finite basis {v1,…,vN} of ℋ as a right Hilbert ℬ2-module; that is,
(10)∑i=1Muiφ1(〈ui∣ξ〉ℬ1)=∑k=1Nvkφ2(〈vk∣ξ〉ℬ2)=ξ,ξ∈ℋ
such that
(11)〈ui∣ϕ2(w)uj〉ℬ1∈𝒜,i,j=1,…,M,〈vk∣ϕ1(z)vl〉ℬ2∈𝒜,k,l=1,…,N
for w∈ℬ2, z∈ℬ1 and
(12)∑i=1M〈ui∣ϕ2(〈ξ∣η〉ℬ2)ui〉ℬ1=〈ξ∣η〉𝒜,∑k=1N〈vk∣ϕ1(〈ξ∣η〉ℬ1)vk〉ℬ2=〈ξ∣η〉𝒜
for all ξ,η∈ℋ. Following [13], {u1,…,uM} and {v1,…,vN} are called finite bases of ℋ, respectively.
(iii) A HilbertC*-quad module ℋ over (𝒜;ℬ1,ℬ2) is said to be of strongly finite type if it is of finite type and there exist a finite basis {e1,…,eM′} of ℬ1 as a right 𝒜-module through ψ1∘λ and a finite basis {f1,…,fN′} of ℬ2 as a right 𝒜-module through ψ2∘λ2. This means that the following equalities hold:
(13)z=∑j=1M′ejψ1(λ1(ej*z)),z∈ℬ1,w=∑l=1N′flψ2(λ2(fl*w)),w∈ℬ2.
We note that for a HilbertC*-quad module of general type, the conditions (9) imply
(14)∥〈ξ∣ξ〉𝒜∥≤∥λi(1)∥∥〈ξ∣ξ〉∥ℬi,ξ∈ℋ.
Put Ci=∥λi(1)∥1/2>0 so that ∥ξ∥𝒜≤Ci∥ξ∥ℬi. Hence, the identity operators from the Banach spaces (ℋ,∥·∥ℬi) to (ℋ,∥·∥𝒜) are bounded linear maps. By the inverse mapping theorem, there exist constants Ci′ such that ∥ξ∥ℬi≤Ci′∥ξ∥𝒜 for ξ∈ℋ. Therefore, the three norms ∥·∥𝒜, ∥·∥ℬi, and i=1,2, induced by the three inner products 〈·∣·〉𝒜,〈·∣·〉ℬi, and i=1,2 on ℋ, are equivalent to each other.Lemma 4.
Letℋ be a Hilbert C*-quad module ℋ over (𝒜;ℬ1,ℬ2). If ℋ is of finite type, then it is of general type.Proof.
Suppose thatℋ is of finite type with finite bases {u1,…,uM} of ℋ as a right Hilbert ℬ1-module and {v1,…,vN} of ℋ as a right Hilbert ℬ2-module as above. We put
(15)λ1(z)=∑k=1N〈vk∣ϕ1(z)vk〉ℬ2∈𝒜,z∈ℬ1,λ2(w)=∑i=1M〈ui∣ϕ2(w)ui〉ℬ1∈𝒜,w∈ℬ2.
They give rise to faithful completely positive maps λi:ℬi→𝒜, i=1,2. The equalities (12) imply that
(16)λi(〈ξ∣η〉ℬi)=〈ξ∣η〉𝒜,forξ,η∈ℋ,i=1,2.
It then follows that
(17)λi(〈ξ∣η〉ℬiψi(a))=λi(〈ξ∣ηa〉ℬi)=〈ξ∣η〉𝒜a=λi(〈ξ∣η〉ℬi)a,a∈𝒜.
Since ℋ is full, the equalities (8) hold.Lemma 5.
Suppose that a HilbertC*-quad module ℋ of finite type is of strongly finite type with a finite basis {e1,…,eM′} of ℬ1 as a right 𝒜-module through ψ1∘λ1 and a finite basis {f1,…,fN′} of ℬ2 as a right 𝒜-module through ψ2∘λ2. Let {u1,…,uM} and {v1,…,vN} be finite bases of ℋ satisfying (10). Then, two families {uiφ1(ej)∣i=1,…,M,  j=1,…,M′} and {vkφ2(fl)∣k=1,…,N,  l=1,…,N′} of ℋ form bases of ℋ as right 𝒜-modules, respectively.Proof.
Forξ∈ℋ, by the equalities
(18)ξ=∑i=1Muiφ1(〈ui∣ξ〉ℬ1),〈ui∣ξ〉ℬ1=∑j=1Mejψ1(λ1(ej*〈ui∣ξ〉ℬ1)),
it follows that
(19)ξ=∑i=1Muiφ1(∑j=1M′ejψ1(λ1(ej*〈ui∣ξ〉ℬ1)))=∑i=1M∑j=1M′uiφ1(ej)·λ1(ej*〈ui∣ξ〉ℬ1)=∑i=1M∑j=1M′uiφ1(ej)·λ1(〈uiφ1(ej)∣ξ〉ℬ1)=∑i=1M∑j=1M′uiφ1(ej)·〈uiφ1(ej)∣ξ〉𝒜.
We similarly have
(20)ξ=∑k=1N∑l=1N′vkφ2(fl)·〈vkφ2(fl)∣ξ〉𝒜.We present some examples.Examples.
(1) Letα, β be automorphisms of a unital C*-algebra 𝒜 satisfying α∘β=β∘α. We set ℬ1=ℬ2=𝒜. Define right actions ψi of 𝒜 on ℬi by
(21)b1ψ1(a)=b1α(a),b2ψ2(a)=b2β(a)
for bi∈ℬi, a∈𝒜. We put ℋα,β=𝒜 and equip it with Hilbert C*-quad module structure over (𝒜;𝒜,𝒜) in the following way. For ξ=x, ξ′=x′∈ℋα,β=𝒜, a∈𝒜, z∈ℬ1=𝒜, and w∈ℬ2=𝒜, define the right 𝒜-module structure and the right 𝒜-valued inner product 〈·∣·〉𝒜 by
(22)ξ·a=xa,〈ξ∣ξ′〉𝒜=x*x′.
Define the right actions φi of ℬi with right ℬi-valued inner products 〈·∣·〉ℬi and the left actions ϕi of ℬi by setting
(23)ξφ1(z)=xα(z),ξφ2(w)=xβ(w),〈ξ∣ξ′〉ℬ1=α-1(x*x′),〈ξ∣ξ′〉ℬ2=β-1(x*x′),ϕ1(z)ξ=β(α(z))x,ϕ2(w)ξ=α(β(w))x.
It is straightforward to see that ℋα,β is a Hilbert C*-quad module over (𝒜;𝒜,𝒜) of strongly finite type.
(2) We fix natural numbers1<N, M∈ℕ. Consider finite-dimensional commutative C*-algebras 𝒜=ℂ,  ℬ1=ℂN, and ℬ2=ℂM. The right actions ψi of 𝒜 on ℬi are naturally defined as right multiplications of ℂ. The algebras ℬ1,ℬ2 have the ordinary product structure and the inner product structure which we denote by 〈·∣·〉N and 〈·∣·〉M, respectively. Let us denote by ℋM,N the tensor product ℂM⊗ℂN. Define the right actions φi of ℬi with ℬi-valued right inner products 〈·∣·〉ℬi and the left actions ϕi of ℬi on ℋM,N=ℂM⊗ℂN for i=1,2 by setting
(24)(ξ⊗η)φ1(z)=ξ⊗(η·z),(ξ⊗η)φ2(w)=(ξ·w)⊗η,〈ξ⊗η∣ξ′⊗η′〉ℬ1=〈ξ∣ξ′〉Mη*·η′∈ℬ1,〈ξ⊗η∣ξ′⊗η′〉ℬ2=ξ*·ξ′〈η∣η′〉N∈ℬ2,ϕ1(z)(ξ⊗η)=ξ⊗(z·η),ϕ2(w)(ξ⊗η)=(w·ξ)⊗η
for z∈ℬ1, ξ,ξ′∈ℂN, w∈ℬ2, and η,η′∈ℂM. Let ei, i=1,…,M and fk, k=1,…,N be the standard bases of ℂM and ℂN, respectively. Put the finite bases
(25)ui=ei⊗1∈ℋM,N,i=1,…,M,vk=1⊗fk∈ℋM,N,i=1,…,N.
It is straightforward to see that ℋM,N is a Hilbert C*-quad module over (ℂ;ℂN,ℂM) of strongly finite type.
(3) Let(𝒜,ρ,η,Σρ,Ση,κ) be a C*-textile dynamical system which means that for j∈Ση, l∈Σρ endomorphisms ηj,ρl of 𝒜 are given with commutation relations ηj∘ρl=ρk∘ηi if κ(l,j)=(i,k). In [12], a Hilbert C*-quad module ℋκρ,η over (𝒜;ℬ1,ℬ2) from (𝒜,ρ,η,Σρ,Ση,κ) is constructed (see [12] for its detail construction). The two triplets (𝒜,ρ,Σρ) and (𝒜,η,Ση) are C*-symbolic dynamical systems [7], that yield C*-algebras 𝒪ρ and 𝒪η, respectively. The C*-algebras ℬ1 and ℬ2 are defined as the C*-subalgebra of 𝒪η generated by elements TjyTj*, j∈Ση, y∈𝒜 and that of 𝒪ρ generated by SkySk*, k∈Σρ, y∈𝒜, respectively. Define the maps ψi:𝒜→ℬi, i=1,2, by
(26)ψ1(a)=∑j∈ΣηTjaTj*,ψ2(a)=∑l∈ΣρSlaSl*,a∈𝒜,
which yield the right actions of 𝒜 on ℬi,  i=1,2. Define the maps λi:ℬi→𝒜,  i=1,2 by
(27)λ1(z)=∑j∈ΣηTj*zTj,λ2(w)=∑l∈ΣρSl*wSl,0000000000000000000z∈ℬ1,w∈ℬ2.
Put ej=TjTj*∈ℬ1, j∈Ση. Let z=∑j∈Ση‍TjzjTj* be an element of ℬ1 for zj∈𝒜 with Tj*TjzjTj*Tj=zj. As λ1(ej*z)=λ1(TjzjTj*)=zj, one sees that
(28)z=∑j∈ΣηTjTj*TjzjTj*=∑j∈ΣηTjTj*ψ1(zj)=∑j∈Σηejψ1(λ1(ej*z)).
We similarly have by putting fl=SlSl*∈ℬ2,
(29)w=∑l∈Σρflψ2(λ2(fl*w))forw∈ℬ2.
We see that ℋκρ,η is a Hilbert C*-quad module of strongly finite type. In particular, two nonnegative commuting matrices A,B with a specification κ coming from the equality AB=BA yield a C*-textile dynamical system and hence a Hilbert C*-quad module of strongly finite type, which are studied in [15].


## 3. Fock HilbertC*-Quad Modules and Creation Operators

In this section, we will construct aC*-algebra from a Hilbert C*-quad module ℋ of general type by using two kinds of creation operators on Fock space of Hilbert C*-quad module. We first consider relative tensor products of Hilbert C*-quad modules and then introduce Fock space of Hilbert C*-quad modules which is a two-dimensional analogue of Fock space of Hilbert C*-bimodules. We fix a Hilbert C*-quad module ℋ over (𝒜;ℬ1,ℬ2) of general type as in the preceding section. The Hilbert C*-quad module ℋ is originally a Hilbert C*-right module over 𝒜 with 𝒜-valued inner product 〈·∣·〉𝒜. It has two other structures of Hilbert C*-bimodules: the Hilbert C*-bimodule (ϕ1,ℋ,φ1) over ℬ1 and the Hilbert C*-bimodule (ϕ2,ℋ,φ2) over ℬ2, where ϕi is a left action of ℬi on ℋ and φi is a right action of ℬi on ℋ with ℬi-valued right inner product 〈·∣·〉ℬi for each i=1,2. This situation is written as in the figure
(30)
We will define two kinds of relative tensor products
(31)ℋ⊗ℬ1ℋ,ℋ⊗ℬ2ℋ
as Hilbert C*-quad modules over (𝒜;ℬ1,ℬ2). The latter one should be written vertically as
(32)ℋ⊗ℬ2ℋ
rather than horizontally ℋ⊗ℬ2ℋ. The first relative tensor product ℋ⊗ℬ1ℋ is defined as the relative tensor product as Hilbert C*-modules over ℬ1, where the left ℋ is a right ℬ1-module through φ1 and the right ℋ is a left ℬ1-module through ϕ1. It has a right ℬ1-valued inner product and a right ℬ2-valued inner product defined by
(33)〈ξ⊗ℬ1ζ∣ξ′⊗ℬ1ζ′〉ℬ1∶=〈ζ∣ϕℬ1(〈ξ∣ξ′〉ℬ1)ζ′〉ℬ1,〈ξ⊗ℬ1ζ∣ξ′⊗ℬ1ζ′〉ℬ2∶=〈ζ∣ϕℬ1(〈ξ∣ξ′〉ℬ1)ζ′〉ℬ2,
respectively. It has two right actions: id⊗φ1 from ℬ1 and id⊗φ2 from ℬ2. It also has two left actions: ϕ1⊗id from ℬ1 and ϕ2⊗id from ℬ2. By these operations, ℋ⊗ℬ1ℋ is a Hilbert C*-bimodule over ℬ1 as well as a Hilbert C*-bimodule over ℬ2. It also has a right 𝒜-valued inner product defined by
(34)〈ξ⊗ℬ1ζ∣ξ′⊗ℬ1ζ′〉𝒜∶=λ1(〈ξ⊗ℬ1ζ∣ξ′⊗ℬ1ζ′〉ℬ1)=λ2(〈ξ⊗ℬ1ζ∣ξ′⊗ℬ1ζ′〉ℬ2),
a right 𝒜-action id⊗a for a∈𝒜 and a left 𝒜-action ϕ𝒜⊗id. By these structure ℋ⊗ℬ1ℋ is a Hilbert C*-quad module over (𝒜;ℬ1,ℬ2):
(35)
We denote the above operations ϕ1⊗id, ϕ2⊗id, id⊗φ1, and id⊗φ2 still by ϕ1, ϕ2, φ1, and φ2, respectively. Similarly, we consider the other relative tensor product ℋ⊗ℬ2ℋ defined by the relative tensor product as Hilbert C*-modules over ℬ2, where the left ℋ is a right ℬ2-module through φ2 and the right ℋ is a left ℬ2-module through ϕ2. By a symmetric discussion to the above, ℋ⊗ℬ2ℋ is a Hilbert C*-quad module over (𝒜;ℬ1,ℬ2). The following lemma is routine.Lemma 6.
Letℋi=ℋ,  i=1,2,3. The correspondences
(36)(ξ1⊗ℬ1ξ2)⊗ℬ2ξ3∈(ℋ1⊗ℬ1ℋ2)⊗ℬ2ℋ3⟶ξ1⊗ℬ1(ξ2⊗ℬ2ξ3)∈ℋ1⊗ℬ1(ℋ2⊗ℬ2ℋ3),(ξ1⊗ℬ2ξ2)⊗ℬ1ξ3∈(ℋ1⊗ℬ2ℋ2)⊗ℬ1ℋ3⟶ξ1⊗ℬ2(ξ2⊗ℬ1ξ3)∈ℋ1⊗ℬ2(ℋ2⊗ℬ1ℋ3)
yield isomorphisms of Hilbert C*-quad modules, respectively.We write the isomorphism class of the former HilbertC*-quad modules as ℋ1⊗ℬ1ℋ2⊗ℬ2ℋ3 and that of the latter ones as ℋ1⊗ℬ2ℋ2⊗ℬ1ℋ3, respectively.Note that the direct sumℬ1⊕ℬ2 has a structure of a pre-Hilbert C*-right module over 𝒜 by the following operations. For b1⊕b2, b1′⊕b2′∈ℬ1⊕ℬ2, and a∈𝒜, set
(37)(b1⊕b2)ψ𝒜(a)∶=b1ψ1(a)⊕b2ψ2(a)∈ℬ1⊕ℬ2,〈b1⊕b2∣b1′⊕b2′〉𝒜∶=λ1(b1*b1′)+λ2(b2*b2′)∈𝒜.
By (8), the equality
(38)〈b1⊕b2∣(b1′⊕b2′)ψ𝒜(a)〉𝒜=〈b1⊕b2∣b1′⊕b2′〉𝒜·a
holds so that ℬ1⊕ℬ2 is a pre-Hilbert C*-right module over 𝒜. We denote by F0(ℋ) the completion of ℬ1⊕ℬ2 by the norm induced by the inner product 〈·∣·〉𝒜. It has right ℬi-actions φi and left ℬi-action ϕi by
(39)(b1⊕b2)φ1(z)=b1z⊕0,(b1⊕b2)φ2(w)=0⊕b2w,ϕ1(z)(b1⊕b2)=zb1⊕0,ϕ2(w)(b1⊕b2)=0⊕wb2
for b1⊕b2∈ℬ1⊕ℬ2, z∈ℬ1, and w∈ℬ2.We denote the relative tensor productℋ⊗ℬiℋ and elements ξ⊗ℬiη by ℋ⊗iℋ and ξ⊗iη, respectively, for i=1,2. Let us define the Fock Hilbert C*-quad module as a two-dimensional analogue of the Fock space of Hilbert C*-bimodules. Put Γ0={∅} and Γn={(i1,…,in))∣ij=1,2}, n=1,2,…. We set
(40)F1(ℋ)=ℋ,F2(ℋ)=(ℋ⊗1ℋ)⊕(ℋ⊗2ℋ),F3(ℋ)=(ℋ⊗1ℋ⊗1ℋ)⊕(ℋ⊗1ℋ⊗2ℋ)⊕(ℋ⊗2ℋ⊗1ℋ)⊕(ℋ⊗2ℋ⊗2ℋ)⋮Fn(ℋ)=⊕(i1,…,in-1)∈Γn-1ℋ⊗i1ℋ⊗i2⋯⊗in-1ℋ⋮
as Hilbert C*-bimodules over 𝒜. We will define the Fock Hilbert C*-module F(ℋ) by setting
(41)F(ℋ)∶=⊕n=0∞Fn(ℋ)¯,
which is the completion of the algebraic direct sum ⊕n=0∞Fn(ℋ) of the Hilbert C*-right module over 𝒜 under the norm ∥ξ∥𝒜 on ⊕n=0∞Fn(ℋ) induced by the 𝒜-valued right inner product on ⊕n=0∞Fn(ℋ). Then, F(ℋ) is a Hilbert C*-right module over 𝒜. It has a natural left ℬi-action defined by ϕi for i=1,2.Forξ∈ℋκ, we define two operators
(42)sξ:Fn(ℋ)⟶Fn+1(ℋ),n=0,1,2,…,tξ:Fn(ℋ)⟶Fn+1(ℋ),n=0,1,2,…
by setting for n=0,
(43)sξ(b1⊕b2)=ξφ1(b1),b1⊕b2∈ℬ1⊕ℬ2,tξ(b1⊕b2)=ξφ2(b2),b1⊕b2∈ℬ1⊕ℬ2,
and for n=1,2,…,
(44)sξ(ξ1⊗π1⋯⊗πn-1ξn)=ξ⊗1ξ1⊗π1⋯⊗πn-1ξn,tξ(ξ1⊗π1⋯⊗πn-1ξn)=ξ⊗2ξ1⊗π1⋯⊗πn-1ξn
for ξ1⊗π1⋯⊗πn-1ξn∈Fn(ℋ) with (π1,…,πn-1)∈Γn-1.Lemma 7.
Forξ∈ℋ, the two operators
(45)sξ:Fn(ℋ)⟶Fn+1(ℋ),n=0,1,2,…,tξ:Fn(ℋ)⟶Fn+1(ℋ),n=0,1,2,…,
are both right 𝒜-module maps.Proof.
We will show the assertion forsξ. For n=0, we have for b1⊕b2∈ℬ1⊕ℬ2 and a∈𝒜,
(46)sξ((b1⊕b2)ψ𝒜(a))=ξφ1(b1ψ1(a))=(ξφ1(b1))a=(sξ(b1⊕b2))a.
For n=1,2,…, one has
(47)sξ((ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn)a)=sξ(ξ1⊗i1ξ2⊗i2⋯⊗in-1(ξna))=ξ⊗1ξ1⊗i1ξ2⊗i2⋯⊗in-1(ξna)=(ξ⊗1ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn)a=[sξ(ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn)]a.It is clear that the two operatorssξ,tξ yield bounded right 𝒜-module maps on F(ℋ) having its adjoints with respect to the 𝒜-valued right inner product on F(ℋ). The operators are still denoted by sξ,tξ, respectively. The adjoints of sξ,tξ:F(ℋ)→F(ℋ) with respect to the 𝒜-valued right inner product on F(ℋ) map Fn+1(ℋ) to Fn(ℋ), n=0,1,2,….Lemma 8.
(i) Forξ,ξ′∈ℋ=F1(ℋ), one has
(48)sξ*ξ′=〈ξ∣ξ′〉ℬ1⊕0∈ℬ1⊕ℬ2,tξ*ξ′=0⊕〈ξ∣ξ′〉ℬ2∈ℬ1⊕ℬ2.
(ii) Forξ∈ℋ and ξ1⊗i1ξ2⊗i2⋯⊗inξn+1∈Fn+1(ℋ), n=1,2,…, we have
(49)sξ*(ξ1⊗i1ξ2⊗i2⋯⊗inξn+1)={ϕ1(〈ξ∣ξ1〉ℬ1)ξ2⊗i2⋯⊗inξn+1ifi1=1,0ifi1=2,tξ*(ξ1⊗i1ξ2⊗i2⋯⊗inξn+1)={0ifi1=1,ϕ2(〈ξ∣ξ1〉ℬ2)ξ2⊗i2⋯⊗inξn+1ifi1=2.Proof.
We will show the assertions (i) and (ii) forsξ*. (i) For b1⊕b2∈ℬ1⊕ℬ2, we have
(50)〈b1⊕b2∣sξ*ξ′〉𝒜=〈ξφ1(b1)∣ξ′〉𝒜=λ1(b1*〈ξ∣ξ′〉ℬ1)=〈b1⊕b2∣〈ξ∣ξ′〉ℬ1⊕0〉𝒜
so that sξ*ξ′=〈ξ∣ξ′〉ℬ1⊕0.
(ii) Forζ1⊗j1⋯⊗jn-1ζn∈Fn(ℋ) with n=1,2,…, we have
(51)〈ζ1⊗j1⋯⊗jn-1ζn∣sξ*(ξ1⊗i1ξ2⊗i2⋯⊗inξn+1)〉𝒜=〈ξ⊗1ζ1⊗j1⋯⊗jn-1ζn∣ξ1⊗i1ξ2⊗i2⋯⊗inξn+1〉𝒜=λ1(〈ξ⊗1ζ1⊗j1⋯⊗jn-1ζn∣ξ1⊗i1ξ2⊗i2⋯⊗inξn+1〉ℬ1)={〈ζ1⊗j1⋯⊗jn-1ζn∣ϕ1(〈ξ∣ξ1〉ℬ1)ξ2⊗i2⋯⊗inξn+1〉𝒜000000000000000000000000000000000000ifi1=1,000000000000000000000000000000000000ifi1=2.Denote byϕ-i the left actions of ℬi,  i=1,2 on Fn(ℋ) and hence on F(ℋ), respectively. They satisfy the following equalities
(52)ϕ-1(z)(b1⊕b2)=zb1⊕0,ϕ-2(w)(b1⊕b2)=0⊕wb2,ϕ-1(z)(ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn)=(ϕ1(z)ξ1)⊗i1ξ2⊗i2⋯⊗in-1ξn,ϕ-2(w)(ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn)=(ϕ2(w)ξ1)⊗i1ξ2⊗i2⋯⊗in-1ξn
for z∈ℬ1, w∈ℬ2, b1⊕b2∈ℬ1⊕ℬ2, and ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn∈Fn(ℋ). More generally let us denote by ℒ𝒜(ℋ) and ℒ𝒜(F(ℋ)) the C*-algebras of all bounded adjointable right 𝒜-module maps on ℋ and on F(ℋ) with respect to their right 𝒜-valued inner products, respectively. For L∈ℒ𝒜(ℋ), define L¯∈ℒ𝒜(F(ℋ)) by
(53)L¯(b1⊕b2)=0forb1⊕b2∈ℬ1⊕ℬ2⊂F0(ℋ),L¯(ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn)=(Lξ1)⊗i1ξ2⊗i2⋯⊗in-1ξn
for ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn∈Fn(ℋ).Lemma 9.
Both the mapsϕ-i:ℬi→ℒ𝒜(F(ℋ)) for i=1,2 are faithful *-homomorphisms.Proof.
By assumption, the*-homomorphisms ϕi:ℬi→ℒ𝒜(ℋ), i=1,2 are faithful, so that the *-homomorphisms ϕ-i:ℬi→ℒ𝒜(F(ℋ)), i=1,2 are both faithful.Lemma 10.
Forξ,ζ∈ℋ, z∈ℬ1,  w∈ℬ2, L∈ℒ𝒜(ℋ), and c,d∈ℂ, the following equalities hold on F(ℋ):
(54)scξ+dζ=csξ+dsζ,tcξ+dζ=ctξ+dtζ,(55)sLξφ1(z)=L¯sξϕ-1(z),tLξφ2(w)=L¯tξϕ-2(w),(56)sζ*L¯sξ=ϕ-1(〈ζ∣Lξ〉ℬ1),tζ*L¯tξ=ϕ-2(〈ζ∣Lξ〉ℬ2).Proof.
The equalities (54) are obvious. We will show the equalities (55) and (56) for sξ. We have for b1⊕b2∈ℬ1⊕ℬ2(57)sLξφ1(z)(b1⊕b2)=[Lξφ1(z)]φ1(b1)=L¯[ξφ1(zb1)]=L¯[sξ(zb1⊕0)]=L¯[sξ[ϕ-1(z)(b1⊕b2)]]=[L¯sξϕ-1(z)](b1⊕b2).
For ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn∈Fn(ℋ), n=1,2,…, we have
(58)sLξφ1(z)(ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn)=(Lξφ1(z))⊗1ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn=L¯[(ξφ1(z))⊗1ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn]=L¯[ξ⊗1(ϕ1(z)ξ1)⊗i1ξ2⊗i2⋯⊗in-1ξn]=L¯[sξ((ϕ1(z)ξ1)⊗i1ξ2⊗i2⋯⊗in-1ξn)]=L¯sξϕ-1(z)[ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn]
so that sLξφ1(z)=L¯sξϕ-1(z) on Fn(ℋ), n=0,1,…. Hence, the equalities (55) hold.
Forb1⊕b2∈ℬ1⊕ℬ2, we have
(59)sζ*L¯sξ(b1⊕b2)=sζ*(Lξφ1(b1))=〈ζ∣Lξφ1(b1)〉ℬ1⊕0=〈ζ∣Lξ〉ℬ1b1⊕0=ϕ-1(〈ζ∣Lξ〉ℬ1)(b1⊕b2).
For ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn∈Fn(ℋ), we have
(60)sζ*L¯sξ(ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn)=sζ*L¯(ξ⊗1ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn)=(ϕ1(〈ζ∣Lξ〉ℬ1)ξ1)⊗i1ξ2⊗i2⋯⊗in-1ξn=ϕ-1(〈ζ∣Lξ〉ℬ1)(ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn)
so that sζ*L¯sξ=ϕ-1(〈ζ∣Lξ〉ℬ1) on Fn(ℋ) for n=0,1,2,…. Hence, the equalities (56) hold.TheC*-subalgebra of ℒ𝒜(F(ℋ)) generated by the operators sξ,tξ for ξ∈ℋ is denoted by 𝒯F(ℋ) and is called the Toeplitz quad module algebra for ℋ.Lemma 11.
TheC*-algebra 𝒯F(ℋ) contains the operators ϕ-1(z),ϕ-2(w) for z∈ℬ1,  w∈ℬ2.Proof.
By (56) in the preceding lemma, one sees that
(61)sζ*sξ=ϕ-1(〈ζ∣ξ〉ℬ1),tζ*tξ=ϕ-2(〈ζ∣ξ〉ℬ2),00000ζ,ξ∈ℋ.
Since ℋ is a full C*-quad module, the inner products 〈ζ∣ξ〉ℬ1,〈ζ∣ξ〉ℬ2 for ζ,ξ∈ℋ generate the C*-algebras ℬ1, ℬ2, respectively. Hence, ϕ-1(ℬ1), ϕ-2(ℬ2) are contained in 𝒯F(ℋ).Lemma 12.
There exists an actionγ of ℝ/ℤ=𝕋 on 𝒯F(ℋ) such that
(62)γr(sξ)=e2π-1rsξ,γr(tξ)=e2π-1rtξ,ξ∈ℋ,γr(ϕ-1(z))=ϕ-1(z),z∈ℬ1,γr(ϕ-2(w))=ϕ-2(w),w∈ℬ2
for r∈ℝ/ℤ=𝕋.Proof.
We will first define a one-parameter unitary groupur,  r∈ℝ/ℤ=𝕋 on F(ℋ) with respect to the right 𝒜-valued inner product as in the following way.
Forn=0:ur:F0(ℋ)→F0(ℋ) is defined by
(63)ur(b1⊕b2)=b1⊕b2forb1⊕b2∈ℬ1⊕ℬ2.
For n=1,2,…:ur:Fn(ℋ)→Fn(ℋ) is defined by
(64)ur(ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn)=e2π-1nrξ1⊗i1ξ2⊗i2⋯⊗in-1ξn
for ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn∈Fn(ℋ). We therefore have a one-parameter unitary group ur on F(ℋ). We then define an automorphism γr on ℒ𝒜(F(ℋ)) for r∈ℝ/ℤ by
(65)γr(T)=urTur*
for  T∈ℒ𝒜(F(ℋ)),  r∈ℝ/ℤ.
It then follows that forb1⊕b2∈ℬ1⊕ℬ2(66)γr(sξ)(b1⊕b2)=ursξur*(b1⊕b2)=ur(ξ(φ1(b1)))=e2π-1rsξ(b1⊕b2),
and for ξ1⊗i1⋯⊗in-1ξn∈Fn(ℋ), n=1,2,…,
(67)γr(sξ)(ξ1⊗i1⋯⊗in-1ξn)=e-2π-1nrur(ξ⊗1ξ1⊗i1⋯⊗in-1ξn)=e2π-1rsξ(ξ1⊗i1⋯⊗in-1ξn).
Therefore, we conclude that γr(sξ)=e2π-1rsξ on F(ℋ) and similarly γr(tξ)=e2π-1rtξ on F(ℋ). It is direct to see that
(68)γr(ϕ-1(z))=ϕ-1(z),γr(ϕ-2(w))=ϕ-2(w),000000000000000000000forz∈ℬ1,w∈ℬ2.
It is also obvious that γr(𝒯F(ℋ))=𝒯F(ℋ) for r∈ℝ/ℤ.Denote byJ(ℋ) the C*-subalgebra of ℒ𝒜(F(ℋ)) generated by the elements
(69)ℒ𝒜(⨁n=0finiteFn(ℋ)).
The algebra J(ℋ) is a closed two-sided ideal of ℒ𝒜(F(ℋ)).Definition 13.
TheC*-algebra 𝒪F(ℋ) associated with the Hilbert C*-quad module ℋ of general type is defined by the quotient C*-algebra of 𝒯F(ℋ) by the ideal 𝒯F(ℋ)∩J(ℋ).
We denote by[x] the quotient image of an element x∈𝒯F(ℋ) under the ideal 𝒯F(ℋ)∩J(ℋ). We set the elements of 𝒪F(ℋ)(70)Sξ=[sξ],Tξ=[tξ],Φ1(z)=[ϕ-1(z)],Φ2(w)=[ϕ-2(w)]
for ξ∈ℋ and z∈ℬ1, w∈ℬ2. By the preceding lemmas, we have the following.Proposition 14.
TheC*-algebra 𝒪F(ℋ) is generated by the family of operators Sξ, Tξ for ξ∈ℋ. It contains the operators Φ1(z),Φ2(w) for z∈ℬ1, w∈ℬ2. They satisfy the following equalities,
(71)Scξ+dζ=cSξ+dSζ,Tcξ+dζ=cTξ+dTζ,(72)Sϕ1(z′)ξφ1(z)=Φ1(z′)SξΦ1(z),Tϕ1(z′)ξφ2(w)=Φ1(z′)TξΦ2(w),(73)Sϕ2(w′)ξφ1(z)=Φ2(w′)SξΦ1(z),Tϕ2(w′)ξφ2(w)=Φ2(w′)TξΦ2(w),(74)Sζ*Sξ=Φ1(〈ζ∣ξ〉ℬ1),Tζ*Tξ=Φ2(〈ζ∣ξ〉ℬ2)
for ξ,ζ∈ℋ, c,d∈C and z,z′∈ℬ1,  w,w′∈ℬ2.


## 4. TheC*-Algebras of Hilbert C*-Quad Modules of Finite Type

In what follows, we assume that a HilbertC*-quad module ℋ is of finite type. In this section, we will study the C*-algebra 𝒪F(ℋ) for a Hilbert C*-quad module ℋ of finite type. Let {u1,…,uM} be a finite basis of ℋ as a Hilbert C*-right module over ℬ1 and {v1,…,vN} a finite basis of ℋ as a Hilbert C*-right module over ℬ2. Keep the notations as in the previous section. We set
(75)si=suifori=1,…,M,tk=tvkfork=1,…,N.
By (10) and Lemma 10, we have for ξ∈ℋ(76)sξ=∑i=1M‍siϕ-1(〈ui∣ξ〉ℬ1),tξ=∑k=1N‍tkϕ-2(〈vk∣ξ〉ℬ2).
Let Pn be the projection on F(ℋ) onto Fn(ℋ) for n=0,1,… so that ∑n=0∞‍Pn=1 on F(ℋ).Lemma 15.
Forξ,ζ∈ℋ, one has(i)
sξ*tζPn=0 for n=1,2,… and hence sξ*tζ=sξ*tζP0.(ii)
tζ*sξPn=0 for n=1,2,… and hence tζ*sξ=tζ*sξP0.Proof.
(i) Forn=1,2,…, we have
(77)sξ*tζ(ξ1⊗i1⋯⊗in-1ξn)=sξ*(ζ⊗2ξ1⊗i1⋯⊗in-1ξn)=0.
(ii) Is similar to (i).Define two projections onF(ℋ) by
(78)Ps=Theprojectiononto⨁n=0∞∑(i1,…,in)∈Γn‍ℋ⊗1ℋ⊗i1ℋ⊗i2⋯⊗inℋ,Pt=Theprojectiononto⨁n=0∞∑(i1,…,in)∈Γn‍ℋ⊗2ℋ⊗i1ℋ⊗i2⋯⊗inℋ.Lemma 16.
Keep the above notations.(79)∑i=1M‍sisi*=P1+Ps,∑k=1N‍tktk*=P1+Pt.
Hence,
(80)∑i=1M‍sisi*+∑k=1N‍tktk*+P0=1F(ℋ)+P1.Proof.
Forξ1⊗i1ξ2⊗i2⋯⊗in-1ξn∈Fn(ℋ) with 2≤n∈N, we have
(81)sisi*(ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn)={ui⊗1ϕ1(〈ui∣ξ1〉η)ξ2⊗i2⋯⊗in-1ξnifi1=1,0ifi1=2.
As ui⊗1ϕ1(〈ui∣ξ1〉ℬ1)ξ2=uiφ1(〈ui∣ξ1〉ℬ1)⊗1ξ2, and ∑i=1M‍uiφ1(〈ui∣ξ1〉ℬ1)=ξ1, we have
(82)∑i=1M‍sisi*(ξ1⊗i1ξ2⊗i2⋯⊗in-1ξn)={ξ1⊗1ξ2⊗i2⋯⊗in-1ξnifi1=1,0ifi1=2
and hence
(83)∑i=1M‍sisi*|⊕n=2∞Fn(ℋ)=Ps|⊕n=2∞Fn(ℋ).
For ξ∈F1(ℋ)=ℋ, we have sisi*ξ=si(〈ui∣ξ〉ℬ1⊕0)=uiφ1(〈ui∣ξ〉ℬ1) so that
(84)∑i=1M‍sisi*ξ=∑i=1M‍uiφ1(〈ui∣ξ〉ℬ1)=ξ
and hence
(85)∑i=1M‍sisi*|F1(ℋ)=1F1(ℋ).
As sisi*(b1⊕b2)=0 for b1⊕b2∈ℬ1⊕ℬ2, we have
(86)∑i=1M‍sisi*|F0(ℋ)=0.
Therefore, we conclude that
(87)∑i=1M‍sisi*=Ps+P1andsimilarly∑k=1N‍tktk*=Pt+P1.
As Ps+Pt+P0+P1=1F(ℋ), one obtains (80).We set the operators(88)Si=Sui(=[si])fori=1,…,M,Tk=Tvk(=[ti])fork=1,…,N
in the C*-algebra 𝒪F(ℋ). As two operators ∑i=1M‍sisi* and ∑k=1N‍tktk* are projections by (79), so are ∑i=1M‍SiSi* and ∑k=1N‍TkTk*. Since P1-P0∈J(ℋ), the identity (80) implies
(89)∑i=1M‍SiSi*+∑k=1N‍TkTk*=1.
Therefore, we have the following.Theorem 17.
Letℋ be a Hilbert C*-quad module over (𝒜;ℬ1,ℬ2) of finite type with finite basis {u1,…,uM} as a right ℬ1-module and {v1,…,vN} as a right ℬ2-module. Then, one has the following. (i)
TheC*-algebra 𝒪F(ℋ) is generated by the operators S1,…,SM,T1,…,TN and the elements Φ1(z),Φ2(w) for z∈ℬ1,  w∈ℬ2.(ii)
They satisfy the following operator relations:(90)∑i=1M‍SiSi*+∑k=1N‍TkTk*=1,Sj*Tl=0,(91)Si*Sj=Φ1(〈ui∣uj〉ℬ1),Tk*Tl=Φ2(〈vk∣vl〉ℬ2),(92)Φ1(z)Sj=∑i=1M‍SiΦ1(〈ui∣ϕ1(z)uj〉ℬ1),Φ1(z)Tl=∑k=1N‍TkΦ2(〈vk∣ϕ1(z)vl〉ℬ2),(93)Φ2(w)Sj=∑i=1M‍SiΦ1(〈ui∣ϕ2(w)uj〉ℬ1),Φ2(w)Tl=∑k=1N‍TkΦ2(〈vk∣ϕ2(w)vl〉ℬ2)forz∈ℬ1,  w∈ℬ2,  i,j=1,…,M,  k,l=1,…,N.(iii)
There exists an actionγ of  ℝ/ℤ=𝕋 on 𝒪F(ℋ) such that
(94)γr(Si)=e2π-1rSi,γr(Tk)=e2π-1rTk,γr(Φ1(z))=Φ1(z),γr(Φ2(w))=Φ2(w)forr∈ℝ/ℤ=𝕋, i=1,…,M,  k=1,…,N, and z∈ℬ1,  w∈ℬ2.Proof.
(i) The assertion comes from the equalities (76).
(ii) The first equality of (90) is (89). As the projection P0 belongs to J(ℋ), Lemma 15 ensures us the second equality of (90). The equalities (91) come from (74). For z∈ℬ1 and j=1,…,M, we have ϕ1(z)uj=∑i=1M‍uiφ1(〈ui∣ϕ1(z)uj〉ℬ1) so that
(95)ϕ-1(z)sj=sϕ1(z)uj=∑i=1M‍suiϕ-1(〈ui∣ϕ1(z)uj〉ℬ1)
which goes to the first equality of (92). The other equalities of (92) and (93) are similarly shown.
(iii) The assertion is direct from Lemma12.The actionγ of 𝕋 on 𝒪F(ℋ) defined in the above theorem (iii) is called the gauge action.


## 5. The UniversalC*-Algebras Associated with Hilbert C*-Quad Modules

In this section, we will prove that theC*-algebra 𝒪F(ℋ) associated with a Hilbert C*-quad module of finite type is the universal C*-algebra subject to the operator relations stated in Theorem 17 (ii). Throughout this section, we fix a Hilbert C*-quad module ℋ over (𝒜;ℬ1,ℬ2) of finite type with finite basis {u1,…,uM} as a right Hilbert ℬ1-module and {v1,…,vN} as a right Hilbert ℬ2-module as in the previous section.Let𝒫ℋ be the universal *-algebra generated by operators S1,…,SM,  T1,…,TN and elements z∈ℬ1, w∈ℬ2 subject to the relations
(96)∑i=1M‍SiSi*+∑k=1N‍TkTk*=1,Sj*Tl=0,(97)Si*Sj=〈ui∣uj〉ℬ1,Tk*Tl=〈vk∣vl〉ℬ2,(98)zSj=∑i=1M‍Si〈ui∣ϕ1(z)uj〉ℬ1,zTl=∑k=1N‍Tk〈vk∣ϕ1(z)vl〉ℬ2,(99)wSj=∑i=1M‍Si〈ui∣ϕ2(w)uj〉ℬ1,wTl=∑k=1N‍Tk〈vk∣ϕ2(w)vl〉ℬ2
for z∈ℬ1, w∈ℬ2, i,j=1,…,M, k,l=1,…,N. The above four relations (96), (97), (98), and (99) are called the relations (ℋ). In what follows, we fix operators S1,…,SM,  T1,…,TN satisfying the relations (ℋ).Lemma 18.
The sums∑i=1M‍SiSi* and ∑k=1N‍TkTk* are both projections.Proof.
PutP=∑i=1M‍SiSi* and Q=∑k=1N‍TkTk*. By the relations (96), one sees that 0≤P, Q≤1, P+Q=1, and PQ=0. It is easy to see that both P and Q are projections.Lemma 19.
Keep the above notations.(i)
Fori,j=1,…,M and z∈ℬ1, w∈ℬ2, one has
(100)Si*zSj=〈ui∣ϕ1(z)uj〉ℬ1,Si*wSj=〈ui∣ϕ2(w)uj〉ℬ1.(ii)
Fork,l=1,…,N and z∈ℬ1, w∈ℬ2, one has
(101)Tk*zTl=〈vk∣ϕ1(z)vl〉ℬ2,Tk*wTl=〈vk∣ϕ2(w)vl〉ℬ2.Proof.
(i) By (98), we have
(102)Si*zSj=∑h=1M‍Si*Sh〈uh∣ϕ1(z)uj〉ℬ1=∑h=1M‍〈ui∣uh〉ℬ1〈uh∣ϕ1(z)uj〉ℬ1=〈ui∣∑h=1M‍uh〈uh∣ϕ1(z)uj〉ℬ1〉ℬ1=〈ui∣ϕ1(z)uj〉ℬ1.
The other equality Si*wSj=〈ui∣ϕ2(w)uj〉ℬ1 is similar to the above equalities.
(ii) Is similar to (i).By the equalities (12), we have the following.Lemma 20.
Keep the above notations.(i)
Forw∈ℬ2, j=1,…M, the element Sj*wSj belongs to 𝒜 and the formula holds:
(103)∑j=1M‍Sj*〈ξ∣η〉ℬ2Sj=〈ξ∣η〉𝒜forξ,η∈ℋ.(ii)
Forz∈ℬ1, l=1,…N, the element Tl*zTl belongs to 𝒜 and the formula holds:
(104)∑l=1N‍Tl*〈ξ∣η〉ℬ1Tl=〈ξ∣η〉𝒜forξ,η∈ℋ.Lemma 21.
The following equalities forz∈ℬ1 and w∈ℬ2 hold: (i)
(105)z=∑i,j=1M‍Si〈ui∣ϕ1(z)uj〉ℬ1Sj*+∑k,l=1N‍Tk〈vk∣ϕ1(z)vl〉ℬ2Tl*,(106)w=∑i,j=1M‍Si〈ui∣ϕ2(w)uj〉ℬ1Sj*+∑k,l=1N‍Tk〈vk∣ϕ2(w)vl〉ℬ2Tl*.(ii)
(107)z*=∑i,j=1M‍Si〈ui∣ϕ1(z)*uj〉ℬ1Sj*+∑k,l=1N‍Tk〈vk∣ϕ1(z)*vl〉ℬ2Tl*,(108)w*=∑i,j=1M‍Si〈ui∣ϕ2(w)*uj〉ℬ1Sj*+∑k,l=1N‍Tk〈vk∣ϕ2(w)*vl〉ℬ2Tl*.(iii)
(109)zw=∑i,j=1M‍Si〈ui∣ϕ1(z)ϕ2(w)uj〉ℬ1Sj*+∑k,l=1N‍Tk〈vk∣ϕ1(z)ϕ2(w)vl〉ℬ2Tl*,wz=∑i,j=1M‍Si〈ui∣ϕ2(w)ϕ1(z)uj〉ℬ1Sj*+∑k,l=1N‍Tk〈vk∣ϕ2(w)ϕ1(z)vl〉ℬ2Tl*.Proof.
(i) By (98) and (99), we have
(110)zSjSj*=∑i=1M‍Si〈ui∣ϕ1(z)uj〉ℬ1Sj*,zTlTl*=∑k=1N‍Tk〈vk∣ϕ1(z)vl〉ℬ2Tl*
so that by (96)
(111)z=∑i,j=1M‍Si〈ui∣ϕ1(z)uj〉ℬ1Sj*+∑k,l=1N‍Tk〈vk∣ϕ1(z)vl〉ℬ2Tl*.
Similarly, we have (106).
(ii) All the adjoints ofϕ1(z), ϕ2(w) for z∈ℬ1, w∈ℬ2 by the three inner products 〈·∣·〉ℬ1, 〈·∣·〉ℬ2, and 〈·∣·〉𝒜 on ℋ coincide with ϕ1(z*), ϕ2(w*), respectively. Hence, the assertions are clear.
(iii) By (i), we have(112)zw=(∑i,j=1M‍Si〈ui∣ϕ1(z)uj〉ℬ1Sj*0+∑k,l=1N‍Tk〈vk∣ϕ1(z)vl〉ℬ2Tl*)·(∑g,h=1M‍Sg〈ug∣ϕ2(w)uh〉ℬ1Sh*00+∑m,n=1N‍Tm〈vm∣ϕ2(w)vn〉ℬ2Tn*).
As Sj*Tm=Tl*Sg=0 for any j,g=1,…,M,  l,m=1,…,N, it follows that
(113)zw=∑i,j,g,h=1M‍Si〈ui∣ϕ1(z)uj〉ℬ1000000×Sj*Sg〈ug∣ϕ2(w)uh〉ℬ1Sh*+∑k,l,m,n=1N‍Tk〈vk∣ϕ1(z)vl〉ℬ20000000×Tl*Tm〈vm∣ϕ2(w)vn〉ℬ2Tn*=∑i,j,g,h=1M‍Si〈ui∣ϕ1(z)uj〉ℬ100000×〈uj∣ug〉ℬ1〈ug∣ϕ2(w)uh〉ℬ1Sh*+∑k,l,m,n=1N‍Tk〈vk∣ϕ1(z)vl〉ℬ20000000×〈vl∣vm〉ℬ2〈vm∣ϕ2(w)vn〉ℬ2Tn*=∑i,g,h=1M‍Si〈ui∣ϕ1(z)ug〉ℬ1〈ug∣ϕ2(w)uh〉ℬ1Sh*+∑k,m,n=1N‍Tk〈vk∣ϕ1(z)∑l=1N‍vl〈vl∣vm〉ℬ2〉ℬ20000000×〈vm∣ϕ2(w)vn〉ℬ2Tn*=∑i,g,h=1M‍Si〈ui∣ϕ1(z)ug〉ℬ1〈ug∣ϕ2(w)uh〉ℬ1Sh*+∑k,m,n=1N‍Tk〈vk∣ϕ1(z)vm〉ℬ2〈vm∣ϕ2(w)vn〉ℬ2Tn*=∑i,h=1M‍Si〈ui∣ϕ1(z)ϕ2(w)uh〉ℬ1Sh*+∑k,n=1N‍Tk〈vk∣ϕ1(z)ϕ2(w)vn〉ℬ2Tn*.Lemma 22.
Letp(z,w) be a polynomial of elements of ℬ1 and ℬ2. Then, one has (i)
p(z,w)Sj=∑i=1M‍Sizi for some zi∈ℬ1.(ii)
p(z,w)Tl=∑k=1N‍Tkwk for some wk∈ℬ2.Proof.
Forz∈ℬ1, w∈ℬ2, and i,j=1,…,M, by putting zi,j=〈ui∣ϕ1(z)uj〉ℬ1∈ℬ1 and wi,j=〈ui∣ϕ2(w)uj〉ℬ1∈ℬ1, the relations (98), (99) imply
(114)zSj=∑i=1M‍Sizi,j,wSj=∑i=1M‍Siwi,j
so that the assertion of (i) holds. (ii) is similar to (i).Lemma 23.
Letp(z,w) be a polynomial of elements of ℬ1 and ℬ2. Then, one has (i)
Si*p(z,w)Sj belongs to ℬ1 for all i,j=1,…,M.(ii)
Tk*p(z,w)Tl belongs to ℬ2 for all k,l=1,…,N.(iii)
Si*p(z,w)Tl=0 for all i=1,…,M, l=1,…,N.(iv)
Tk*p(z,w)Sj=0 for all k=1,…,N, j=1,…,M.Proof.
(i) By the previous lemma, we know(115)p(z,w)Sj=∑h=1M‍Shzhforsomezh∈ℬ1
so that
(116)Si*p(z,w)Sj=∑h=1M‍Si*Shzh=∑h=1M‍〈ui∣uh〉ℬ1zh.
As 〈ui∣uh〉ℬ1zh belongs to ℬ1, we see the assertion.
(ii) Is similar to (i).
(iii) AsTk*Si=0, we have
(117)Tk*p(z,w)Sj=∑i=1M‍Tk*Sizi=0.
(iv) Is similar to (i).We set(118)S1,i≔Si,i=1,…,M,S2,k:=Tk,k=1,…,N.
Put
(119)Σ1={(1,i)∣i=1,…,M},Σ2={(2,k)∣k=1,…,N}.Lemma 24.
Every element of𝒫ℋ can be written as a linear combination of elements of the form
(120)Sg1,i1Sg2,i2⋯Sgm,imbShn,jn*⋯Sh2,j2*Sh1,j1*
for some (g1,i1),  (g2,i2),…,(gm,im),  (h1,j1),  (h2,j2),…,(hn,jn)∈Σ1∪Σ2 where b is a polynomial of elements of ℬ1 and ℬ2.Proof.
The assertion follows from the preceding lemmas.By construction, every representation ofℬ1 and ℬ2 on a Hilbert space H together with operators Si,  i=1,…,M, Tk,  k=1,…,N satisfying the relations (ℋ) extends to a representation of 𝒫ℋ on B(H). We will endow 𝒫ℋ with the norm obtained by taking the supremum of the norms in B(H) over all such representations. Note that this supremum is finite for every element of 𝒫ℋ because of the inequalities ∥Si∥,∥Tk∥≤1, which come from (96). The completion of the algebra 𝒫ℋ under the norm becomes a C*-algebra denoted by 𝒪ℋ, which is called the universal C*-algebra subject to the relations (ℋ).Denote byC*(ϕ1(ℬ1),ϕ2(ℬ2)) the C*-subalgebra of ℒ𝒜(ℋ) generated by ϕ1(ℬ1) and ϕ2(ℬ2).Lemma 25.
An elementL of the C*-algebra C*(ϕ1(ℬ1),ϕ2(ℬ2)) is both a right ℬ1-module map and a right ℬ2-module map. This means that the equalities
(121)[Lξ]φi(bi)=L[ξφi(bi)]forξ∈ℋ,bi∈ℬi
hold.Proof.
Since both the operatorsϕ1(z) for z∈ℬ1 and ϕ2(w) for w∈ℬ2 are right ℬi-module maps for i=1,2, any element of the *-algebra algebraically generated by ϕ1(ℬ1) and ϕ2(ℬ2) is both a right ℬ1-module map and a right ℬ2-module map. Hence, it is easy to see that any element L of the C*-algebra C*(ϕ1(ℬ1),ϕ2(ℬ2)) is both a right ℬ1-module map and a right ℬ2-module map.Denote byℬ∘ the C*-subalgebra of 𝒪ℋ generated by ℬ1 and ℬ2.Lemma 26.
The correspondence(122)z,w∈ℬ∘⟶ϕ1(z),ϕ2(w)∈C*(ϕ1(ℬ1),ϕ2(ℬ2))
gives rise to an isomorphism from ℬ∘ onto C*(ϕ1(ℬ1),ϕ2(ℬ2)) as C*-algebras.Proof.
We note that by hypothesis both the maps(123)ϕ1:z∈ℬ1⟶ϕ1(z)∈ℒ𝒜(ℋ),ϕ2:w∈ℬ2⟶ϕ2(w)∈ℒ𝒜(ℋ)
are injective. Denote by P(ϕ1(ℬ1),ϕ2(ℬ2)) the *-algebra on ℋ algebraically generated by ϕ1(z), ϕ2(w) for z∈ℬ1, w∈ℬ2. Define an operator π(L)∈𝒪ℋ for L∈P(ϕ1(ℬ1),ϕ2(ℬ2)) by
(124)π(L)=∑i,j=1M‍Si〈ui∣Luj〉ℬ1Sj*+∑k,l=1N‍Tk〈vk∣Lvl〉ℬ2Tl*.
Let P∘ be the *-subalgebra of 𝒫ℋ algebraically generated by ℬ1 and ℬ2. Since π(ϕ1(z))=z for z∈ℬ1 and π(ϕ2(w))=w for w∈ℬ2 and by Lemma 21, the map
(125)π:P(ϕ1(ℬ1),ϕ2(ℬ2))⟶P∘⊂ℬ∘
yields a *-homomorphism. As Si*Tk=0 for i=1,…,M,  k=1,…,N, we have
(126)∥π(L)∥=Max{∥∑i,j=1M‍Si〈ui∣Luj〉ℬ1Sj*∥,000000∥∑k,l=1N‍Tk〈vk∣Lvl〉ℬ2Tl*∥∥∑i,j=1M‍Si〈ui∣Luj〉ℬ1Sj*∥}.
We then have
(127)∥∑i,j=1M‍Si〈ui∣Luj〉ℬ1Sj*∥≤∑i,j=1M‍∥〈ui∣Luj〉ℬ1∥≤(∑i,j=1M‍∥ui∥ℬ1∥uj∥ℬ1)∥L∥
and similarly
(128)∥∑k,l=1N‍Tk〈vk∣Lvl〉ℬ2Tl*∥≤(∑k,l=1N‍∥vk∥ℬ2∥vl∥ℬ2)∥L∥.
By puttingC=Max{∑i,j=1M‍∥ui∥ℬ1∥uj∥ℬ1,∑k,l=1N‍∥vk∥ℬ2∥vl∥ℬ2},one has
(129)∥π(L)∥≤C∥L∥,∀L∈P(ϕ1(ℬ1),ϕ2(ℬ2)).
Hence, π extends to the C*-algebra C*(ϕ1(ℬ1),ϕ2(ℬ2)) such that π(C*(ϕ1(ℬ1),ϕ2(ℬ2)))=ℬ∘. The equality (124) holds for L∈C*(ϕ1(ℬ1),ϕ2(ℬ2)).
We will next show thatπ:C*(ϕ1(ℬ1),ϕ2(ℬ2))→ℬ∘ is injective. By (124), we have for L∈C*(ϕ1(ℬ1),ϕ2(ℬ2)) and h,h′=1,…,M,
(130)Sh*π(L)Sh′=∑i,j=1M‍Sh*Si〈ui∣Luj〉ℬ1Sj*Sh′=∑i,j=1M‍〈uh∣ui〉ℬ1〈ui∣Luj〉ℬ1〈uj∣uh′〉ℬ1=∑j=1M‍〈uh∣∑i=1M‍ui〈ui∣Luj〉ℬ1〉ℬ1〈uj∣uh′〉ℬ1=∑j=1M‍〈uh∣Luj〉ℬ1〈uj∣uh′〉ℬ1=〈uh∣Luh′〉ℬ1.
Suppose that π(L)=0 so that 〈uh∣Luh′〉ℬ1=0. Since
(131)Luh′=∑h=1M‍uh〈uh∣Luh′〉ℬ1,
we see that Luh′=0 so that L=0. We thus conclude that π is injective and hence isomorphic.Denote byϕ∘:ℬ∘→C*(ϕ1(ℬ1),ϕ2(ℬ2)) the inverse π-1 of the *-isomorphism π given in the proof of the above lemma which satisfies
(132)ϕ∘(z)=ϕ1(z)forz∈ℬ1,ϕ∘(w)=ϕ2(w)forw∈ℬ2.
We put ℱℋ0=ℬ∘. For n∈ℕ, we denote by ℱℋn the closed linear span of elements of the form
(133)Sg1,i1Sg2,i2⋯Sgn,inbShn,jn*⋯Sh2,j2*Sh1,j1*
for some (g1,i1),(g2,i2),…,(gn,in),(h1,j1),(h2,j2),…,(hn,jn)∈Σ1∪Σ2 and b∈ℬ∘. Let us denote by ℱℋ the C*-subalgebra of 𝒪ℋ generated by ∪n=0∞ℱℋn. By the relations (105) and (106), we see the following.Lemma 27.
Forx∈ℬ∘, the following identity holds:
(134)x=∑i,j=1M‍Si〈ui∣ϕ∘(x)uj〉ℬ1Sj*+∑k,l=1N‍Tk〈vk∣ϕ∘(x)vl〉ℬ2Tl*.Hence, by putting forb∈ℬ∘(135)b1,ij=〈ui∣ϕ∘(b)uj〉ℬ1,i,j=1,…,M,b2,kl=〈vk∣ϕ∘(b)vl〉ℬ2,k,l=1,…,N,
we  have the following.Lemma 28.
Forb∈ℬ∘, the identity
(136)Sg1,i1Sg2,i2⋯Sgn,inbShn,jn*⋯Sh2,j2*Sh1,j1*=∑i,j=1M‍Sg1,i1Sg2,i2⋯Sgn,inS1,ib1,ijS1,j*Shn,jn*⋯Sh2,j2*Sh1,j1*+∑k,l=1N‍Sg1,i1Sg2,i2⋯Sgn,inS2,kb2,klS2,l*Shn,jn*⋯Sh2,j2*Sh1,j1*
holds and induces an embedding of ℱℋn↪ℱℋn+1 for n∈ℤ+.Lemma 29.
TheC*-algebra ℱℋ is the inductive limit limn→∞ℱℋn of the sequence of the inclusions
(137)ℱℋ0↪ℱℋ1↪ℱℋ2↪⋯↪ℱℋn↪ℱℋn+1↪⋯↪ℱℋ.Lete2π-1r∈𝕋 be a complex number of modulus one for r∈ℝ/ℤ. The elements
(138)e2π-1rSi,i=1,…,M,e2π-1rTk,k=1,…,N,z∈ℬ1,w∈ℬ2
in 𝒪ℋ instead of
(139)Si,i=1,…,M,Tk,k=1,…,N,z∈ℬ1,w∈ℬ2
satisfy the relations (ℋ). This implies the existence of an action on 𝒫ℋ by automorphisms of the one-dimensional torus 𝕋 that acts on the generators by
(140)hr(Si)=e2π-1rSi,hr(Tk)=e2π-1rTk,hr(z)=z,hr(w)=w
for i=1,…,M,  k=1,…,N,  z∈ℬ1,  w∈ℬ2, and r∈ℝ/ℤ=𝕋. As the C*-algebra 𝒪ℋ has the largest norm on 𝒫ℋ, the action (hr)r∈𝕋 on 𝒫ℋ extends to an action of 𝕋 on 𝒪ℋ, still denoted by h. The formula
(141)a∈𝒪ℋ⟶∫r∈𝕋‍hr(a)dr∈𝒪ℋ,
where dr is the normalized Lebesgue measure on 𝕋 defines a faithful conditional expectation denoted by ℰℋ from 𝒪ℋ onto the fixed-point algebra (𝒪ℋ)h. The following lemma is routine.Lemma 30.
(
𝒪
ℋ
)
h
=
ℱ
ℋ.TheC*-algebra 𝒪ℋ satisfies the following universal property. Let 𝒟 be a unital C*-algebra and Φ1:ℬ1→𝒟, Φ2:ℬ2→𝒟 be *-homomorphisms such that Φ1(a)=Φ2(a) for a∈𝒜. Assume that there exist elements S^1,…,S^M,  T^1,…,T^N in 𝒟 satisfying the relations
(142)∑i=1M‍S^iS^i*+∑k=1N‍T^kT^k*=1,S^j*T^l=0,S^i*S^j=Φ1(〈ui∣uj〉ℬ1),T^k*T^l=Φ2(〈vk∣vl〉ℬ2),Φ1(z)S^j=∑i=1M‍S^iΦ1(〈ui∣ϕ1(z)uj〉ℬ1),Φ1(z)T^l=∑k=1N‍T^kΦ2(〈vk∣ϕ1(z)vl〉ℬ2),Φ2(w)S^j=∑i=1M‍S^iΦ1(〈ui∣ϕ2(w)uj〉ℬ1),Φ2(w)T^l=∑k=1N‍T^kΦ2(〈vk∣ϕ2(w)vl〉ℬ2),
for z∈ℬ1,  w∈ℬ2,  i,j=1,…,M,  k,l=1,…,N; then, there exists a unique *-homomorphism Φ:𝒪ℋ→𝒟 such that
(143)Φ(Si)=S^i,Φ(Tk)=T^k,Φ(z)=Φ1(z),Φ(w)=Φ2(w)
for i=1,…,M, k=1,…,N and z∈ℬ1, w∈ℬ2. We further assume that both the homomorphisms Φi:ℬi→𝒟, i=1,2 are injective. We denote by Φ∘:ℬ∘→𝒟 the restriction of Φ to the subalgebra ℬ∘. Let us denote by 𝒪^ℋ the C*-subalgebra of 𝒟 generated by S^i,T^k, i=1,…,M, k=1,…,N and Φ1(z), Φ2(w) for z∈ℬ1, w∈ℬ2.Lemma 31.
Keep the above situation. The*-homomorphism Φ∘:ℬ∘→𝒟 is injective.Proof.
Since the correspondence in Lemma26(144)ϕ∘:z,w,∈ℬ∘⟶ϕ1(z),ϕ2(w)∈C*(ϕ1(ℬ1),ϕ2(ℬ2))
yields an isomorphism of C*-algebras, it suffices to prove that the correspondence
(145)ϕ1(z),ϕ2(w)∈C*(ϕ1(ℬ1),ϕ2(ℬ2))⟶Φ1(z),Φ2(w)∈𝒟
yields an isomorphism. Let ℬ^∘ be the C*-subalgebra of 𝒪^ℋ generated by elements Φ1(z),Φ2(w)∈𝒜 for z∈ℬ1, w∈ℬ2. Define an element π^(L) of 𝒟 for L∈C*(ϕ1(ℬ1),ϕ2(ℬ2)) by setting
(146)π^(L)=∑i,j=1M‍S^iΦ1(〈ui∣Luj〉ℬ1)S^j*+∑k,l=1N‍T^kΦ2(〈vk∣Lvl〉ℬ2)T^l*∈𝒟.
As in the proof of Lemma 26, one sees that π^ gives rise to a *-homomorphism from C*(ϕ1(ℬ1),ϕ2(ℬ2)) into 𝒟. Since
(147)π^(ϕ1(z))=∑i,j=1M‍S^iΦ1(〈ui∣ϕ1(z)uj〉ℬ1)S^j*+∑k,l=1N‍T^kΦ2(〈vk∣ϕ1(z)vl〉ℬ2)T^l*=Φ1(z),
and similarly π^(ϕ2(w))=Φ2(w), it is enough to show that π^ is injective. Suppose that π^(L)=0 for some L∈C*(ϕ1(ℬ1),ϕ2(ℬ2)). By following the proof of Lemma 26, one sees that S^h*π^(L)S^h′=Φ1(〈uh∣Luh′〉ℬ1) for all h,h′=1,…,M. Hence, the condition π^(L)=0 implies that Φ1(〈uh∣Luh′〉ℬ1)=0. Since Φ1 is injective, we have 〈uh∣Luh′〉ℬ1=0 for all h,h′=1,…,M. As L is a right ℬ1-module map, we have for ξ∈ℋ,
(148)Lξ=∑h′=1M‍L(uh′〈uh′∣ξ〉ℬ1)=∑h′=1M‍(Luh′)〈uh′∣ξ〉ℬ1=0
so that L=0. Therefore, π^:C*(ϕ1(ℬ1),ϕ2(ℬ2))→𝒟 is injective. Hence, the composition
(149)Φ∘:π^∘ϕ∘:ℬ∘⟶ϕ∘C*(ϕ1(ℬ1),ϕ2(ℬ2))⟶π^𝒟
is injective.We set(150)S^1,i≔S^i,i=1,…,M,S^2,k:=T^k,k=1,…,N.
We put ℱ^ℋ0=ℬ∘. For n∈ℕ, let ℱ^ℋn be the closed linear span in the C*-algebra 𝒪^ℋ of elements of the form
(151)S^g1,i1S^g2,i2⋯S^gn,inΦ∘(b)S^hn,jn*⋯S^h2,j2*S^h1,j1*
for (g1,i1),(g2,i2),…,(gn,in),(h1,j1),(h2,j2),…,(hn,jn)∈Σ1∪Σ2 and b∈ℬ∘. Similar to the subalgebras ℱℋn,n∈ℤ+, of 𝒪ℋ, one knows that the closed linear span ℱ^ℋn is a C*-algebra and naturally regarded as a subalgebra of ℱ^ℋn+1 for each n∈ℤ+. Let us denote by ℱ^ℋ the C*-subalgebra of 𝒪^ℋ generated by ∪n=0∞ℱ^ℋn. Then, the C*-algebra ℱ^ℋ is the inductive limit limn→∞ℱ^ℋn of the sequence of the inclusions
(152)ℱ^ℋ0↪ℱ^ℋ1↪ℱ^ℋ2↪⋯↪ℱ^ℋn↪ℱ^ℋn+1↪⋯↪ℱ^ℋ.Lemma 32.
Suppose that both the*-homomorphisms Φi:ℬi→𝒪^ℋ, i=1,2 are injective. Then, the restriction of Φ to the subalgebra ℱℋ yields a *-isomorphism Φ|ℱℋ:ℱℋ→ℱ^ℋ.Proof.
By the universality of𝒪ℋ, the restriction of Φ to ℱℋ yields a surjective *-homomorphism Φ|ℱℋ:ℱℋ→ℱ^ℋ. It suffices to show that Φ|ℱℋ is injective. Suppose that Ker(Φ|ℱℋ)≠{0} and put I=Ker(Φ|ℱℋ). Since Φℱ(ℱℋn)=ℱ^ℋn and ℱℋ=limn→∞ℱℋn, there exists n∈ℤ+ such that I∩ℱℋn≠0. Let us denote by Σn the set of n-tuples of Σ1∪Σ2:
(153)Σn={(μ1,…,μn)∣μ1,…,μn∈Σ1∪Σ2}.
For μ=(μ1,…,μn)∈Σn, denote by Sμ the operator
(154)Sμ=Sμ1Sμ2⋯Sμn,
where
(155)Sμm={S1,i=Siifμm=(1,i)∈Σ1,S2,k=Tkifμm=(2,k)∈Σ2,m=1,…,n.
Any element of ℱℋn is of the form
(156)∑μ,ν∈Σn‍Sμbμ,νSν*forsomebμ,ν∈ℬ∘.
Hence, one may find a nonzero element ∑μ,ν∈Σn‍Sμbμ,νSν*∈I∩ℱℋn. Since ∑i=1M‍SiSi*+∑k=1N‍TkTk*=1, the equality ∑μ∈Σn‍SμSμ*=1 holds. For some ω,γ∈Σn, one then sees
(157)0≠Sω*(∑μ,ν∈Σn‍Sμbμ,νSν*)Sγ∈I∩ℱℋn.
As Si*Tk=0 and Si*Sj=〈ui∣uj〉ℬ1, Tk*Tl=〈vk∣vl〉ℬ2 for i,j=1,…,M,  k,l=1,…,N, the element Sω*(∑μ,ν∈Σn‍Sμbμ,νSν*)Sγ belongs to I∩ℬ∘. By the preceding lemma, the homomorphism Φ∘:ℬ∘→𝒪^ℋ is injective, so that we have Φ∘(Sω*(∑μ,ν∈Σn‍Sμbμ,νSν*)Sγ)≠0 a contradiction. Therefore we conclude that Φ|ℱℋ:ℱℋ→ℱ^ℋ is injective and hence isomorphic.The following theorem is one of the main results of the paper.Theorem 33.
Let𝒟 be a unital C*-algebra. Suppose that there exist *-homomorphisms Φ1:ℬ1→𝒟, Φ2:ℬ2→𝒟 such that Φ1(a)=Φ2(a) for a∈𝒜 and there exist elements S^1,…,S^M,  T^1,…,T^N in 𝒟 satisfying the relations
(158)∑i=1M‍S^iS^i*+∑k=1N‍T^kT^k*=1,Sj*Tl=0,S^i*S^j=Φ1(〈ui∣uj〉ℬ1),T^k*T^l=Φ2(〈vk∣vl〉ℬ2),Φ1(z)S^j=∑i=1M‍S^iΦ1(〈ui∣ϕ1(z)uj〉ℬ1),Φ1(z)T^l=∑k=1N‍T^kΦ2(〈vk∣ϕ1(z)vl〉ℬ2),Φ2(w)S^j=∑i=1M‍S^iΦ1(〈ui∣ϕ2(w)uj〉ℬ1),Φ2(w)T^l=∑k=1N‍T^kΦ2(〈vk∣ϕ2(w)vl〉ℬ2)
for z∈ℬ1, w∈ℬ2, i,j=1,…,M, k,l=1,…,N. Let us denote by 𝒪^ℋ the C*-subalgebra of 𝒟 generated by S^i,T^k, i=1,…,M, k=1,…,N and Φ1(z),Φ2(w), for z∈ℬ1,  w∈ℬ2. One further assumes that the algebra 𝒪^ℋ admits a gauge action. If both the *-homomorphisms Φi:ℬi→𝒜, i=1,2 are injective, then there exists a *-isomorphism Φ:𝒪ℋ→𝒪^ℋ satisfying
(159)Φ(Si)=S^i,Φ(Tk)=T^k,Φ(z)=Φ1(z),Φ(w)=Φ2(w)
for i=1,…,M, k=1,…,N and z∈ℬ1, w∈ℬ2.Proof.
By assumption,𝒪^ℋ admits a gauge action, which we denote by h^. Let us denote by (𝒪^ℋ)h^ the fixed-point algebra of 𝒪^ℋ under the gauge action h^ and by ℱ^ℋ the C*-subalgebra of 𝒪^ℋ defined by the inductive limit (152). Then, it is routine to check that (𝒪^ℋ)h^ is canonically *-isomorphic to ℱ^ℋ. There exists a conditional expectation
(160)ℰ^ℋ:𝒪^ℋ⟶ℱ^ℋ
defined by
(161)ℰ^ℋ(x)=∫r∈𝕋‍h^r(x)drforx∈𝒪^ℋ.
By the universality of the algebra 𝒪ℋ, there exists a surjective *-homomorphism Φ from 𝒪ℋ to 𝒪^ℋ such that
(162)Φ(Si)=S^i,Φ(Tk)=T^k,Φ(z)=Φ1(z),Φ(w)=Φ2(w)
for i,j=1,…,M,  k,l=1,…,N,  z∈ℬ1,  w∈ℬ2. Then, Φ(ℱℋ)=ℱ^ℋ and the following diagram
(163)
is commutative. Denote by Φ∘ the restriction of Φ to the C*-subalgebra ℬ∘ of 𝒪ℋ generated by z∈ℬ1,  w∈ℬ2. By assumption, both the maps Φi:ℬi→𝒪^ℋ, i=1,2 are injective, so that Φ∘:ℬ∘→𝒪^ℋ is injective by Lemma 31. By the preceding lemma, Φ|ℱℋ:ℱℋ→ℱ^ℋ is an isomorphism. Since the conditional expectation ℰℋ:𝒪ℋ→ℱℋ is faithful, a routine argument shows that Φ is injective and hence isomorphic.Therefore, we have the following.Theorem 34.
For aC*-quad module ℋ of finite type, the C*-algebra 𝒪F(ℋ) generated by the quotients [sξ],[tξ] of the creation operators sξ,tξ for ξ∈ℋ on the Fock spaces F(ℋ) is canonically isomorphic to the universal C*-algebra 𝒪ℋ generated by operators S1,…,SM,  T1,…,TN and elements z∈ℬ1, w∈ℬ2 subject to the relations
(164)∑i=1M‍SiSi*+∑k=1N‍TkTk*=1,Sj*Tl=0,Si*Sj=〈ui∣uj〉ℬ1,Tk*Tl=〈vk∣vl〉ℬ2,zSj=∑i=1M‍Si〈ui∣ϕ1(z)uj〉ℬ1,zTl=∑k=1N‍Tk〈vk∣ϕ1(z)vl〉ℬ2,wSj=∑i=1M‍Si〈ui∣ϕ2(w)uj〉ℬ1,wTl=∑k=1N‍Tk〈vk∣ϕ2(w)vl〉ℬ2
for i,j=1,…,M,  k,l=1,…,N and z∈ℬ1,  w∈ℬ2.Proof.
Theorem17 implies that the operators S1,…,SM,  T1,…,TN and the elements Φ1(z), Φ2(w) for z∈ℬ1, w∈ℬ2 in 𝒪F(ℋ) satisfy the eight relations of Theorem 33. By Theorem 33, we see that the correspondences
(165)Si⟶Si,Tk⟶Tk,z∈ℬ1⟶Φ1(z),w∈ℬ2⟶Φ2(w)
for i=1,…,M, k=1,…,N, and z∈ℬ1, w∈ℬ2 give rise to an isomorphism from 𝒪ℋ to 𝒪F(ℋ).The eight relations of the operators above are called the relations(ℋ). The above generating operators S1,…,SM and T1,…,TN of the universal C*-algebra 𝒪ℋ correspond to two finite bases {u1,…,uM} and {v1,…,vN} of the Hilbert C*-quad module ℋ, respectively. On the other hand, the other C*-algebra 𝒪F(ℋ) is generated by the quotients of the creation operators sξ,tξ for ξ∈ℋ on the Fock spaces F(ℋ), which do not depend on the choice of the two finite bases. Hence, we have the following.Corollary 35.
For aC*-quad module ℋ of finite type, the universal C*-algebra 𝒪ℋ generated by operators S1,…,SM,  T1,…,TN and elements z∈ℬ1,w∈ℬ2 subject to the relations (ℋ) does not depend on the choice of the finite bases {u1,…,uM} and {v1,…,vN}.


## 6.K-Theory Formulae

Letℋ be a Hilbert C*-quad module over (𝒜;ℬ1,ℬ2) of finite type as in the preceding section. In this section, we will state K-theory formulae for the C*-algebra 𝒪F(ℋ). By the previous section, the C*-algebra 𝒪F(ℋ) is regarded as the universal C*-algebra 𝒪ℋ generated by the operators S1,…,SM and T1,…,TN and the elements z∈ℬ1 and w∈ℬ2 subject to the relations (ℋ). Let us denote by ℬ∘ the C*-subalgebra of 𝒪ℋ generated by elements z∈ℬ1 and w∈ℬ2. By Lemma 26, the correspondence
(166)z,w∈ℬ∘⟶ϕ1(z),ϕ2(w)∈C*(ϕ1(ℬ1),ϕ2(ℬ2))⊂ℒ𝒜(ℋ)
gives rise to a *-isomorphism from ℬ∘ onto C*(ϕ1(ℬ1),ϕ2(ℬ2)) as C*-algebras, which is denoted by ϕ∘. We will restrict our interest to the case when (i)
S1,…,SM and T1,…,TN are partial isometries, and(ii)
S1S1*,…,SMSM*,  T1T1*,…,TNTN* commute with all elements of ℬ∘. If the bases {u1,…,uM} and {v1,…,vN} satisfy the conditions
(167)〈ui∣uj〉ℬ1=0fori≠j,〈vk∣vl〉ℬ2=0fork≠l,
the condition (i) holds. Furthermore, if ϕ1(z) acts diagonally on {u1,…,uM} for z∈ℬ1 and ϕ2(w) acts diagonally on {v1,…,vN} for w∈ℬ2, the condition (ii) holds. Recall that the gauge action is denoted by h which is an action of 𝕋 on 𝒪ℋ such that the fixed-point algebra (𝒪ℋ)h under h is canonically isomorphic to the C*-algebra ℱℋ. Denote by h^ the dual action of h which is an action of ℤ=𝕋^ on the C*-crossed product 𝒪ℋ×h𝕋 by the gauge action h of 𝕋. As in the argument of [16], 𝒪ℋ×h𝕋 is stably isomorphic to ℱℋ. Hence, we have that K*(𝒪ℋ×h𝕋) is isomorphic to K*(ℱℋ). The dual action h^ induces an automorphism on the group K*(𝒪ℋ×h𝕋) and hence on K*(ℱℋ), which is denoted by σ*. Then, by [16] (cf. [10, 17], etc.), we have the following.Proposition 36.
The following six-term exact sequence ofK-theory holds:
(168)We put forx∈ℬ∘(169)λ1,i(x)=Si*xSi,i=1,…,M,λ2,k(x)=Tk*xTk,k=1,…,N.
Both the families λ1,i, λ2,k yield endomorphisms on ℬ∘ which give rise to endomorphisms on the K-groups:
(170)λ1,i*:K0(ℬ∘)⟶K0(ℬ∘),i=1,…,M,λ2,k*:K0(ℬ∘)⟶K0(ℬ∘),k=1,…,N.
We put λ∘=∑i=1M‍λ1,i*+∑k=1N‍λ2,k* which is an endomorphism on K0(ℬ∘). Now, we further assume that K1(ℱℋ)={0}. It is routine to show that the groups Coker(id-σ*) in K0(ℱℋ) and Ker(id-σ*) in K0(ℱℋ) are isomorphic to the groups Coker(id-λ∘) in K0(ℬ∘), and Ker(id-λ∘) in K0(ℬ∘), respectively, by an argument of [17]. Therefore, we haveProposition 37.
The following formulae hold:(171)K0(𝒪ℋ)=Coker(id-λ∘)inK0(ℬ∘),K1(𝒪ℋ)=Ker(id-λ∘)inK0(ℬ∘).


## 7. Examples

In this section, we will study theC*-algebras 𝒪ℋ for the Hilbert C*-quad modules presented in Examples in Section 2.(1) Letα,β be automorphisms of a unital C*-algebra 𝒜 satisfying α∘β=β∘α. Let ℋα,β be the associated Hilbert C*-quad module of finite type as in (1) in Section 2. It is easy to see the following proposition.Proposition 38.
TheC*-algebra 𝒪ℋα,β associated with the Hilbert C*-quad module ℋα,β coming from commuting automorphisms α,β of a unital C*-algebra 𝒜 is isomorphic to the universal C*-algebra generated by two isometries U,V and elements x of 𝒜 subject to the following relations:
(172)UU*+VV*=1,UU*x=xUU*,VV*x=xVV*,α(x)=U*xU,β(x)=V*xV
for x∈𝒜.(2) We fix natural numbers1<N,  M∈ℕ. Consider finite-dimensional commutative C*-algebras 𝒜=ℂ,  ℬ1=ℂN, and  ℬ2=ℂM. The algebras ℬ1,ℬ2 have the ordinary product structure and the inner product structure which we denote by 〈·∣·〉N and 〈·∣·〉M, respectively. Let us denote by ℋM,N the Hilbert C*-quad module ℂM⊗ℂN over (ℂ;ℂN,ℂM) defined in (2) in Section 2. Put the finite bases
(173)ui=ei⊗1∈ℋM,N,i=1,…,Maaaaaaaaaaaasarightℬ1-module,vk=1⊗fk∈ℋM,N,k=1,…,Naaaaaaaaaaasarightℬ2-module.
We set Σ∘={(i,k)∣1≤i≤M,1≤k≤N} and put e(i,k)=ei⊗fk,  (i,k)∈Σ∘ the standard basis of ℋM,N. Then, the C*-algebra ℬ∘ on ℋM,N generated by ℬ1 and ℬ2 is regarded as ℂM⊗ℂN=ℬ2⊗ℬ1. Hence,
(174)ℬ∘=∑(i,k)∈Σ∘‍ℂe(i,k).Lemma 39.
TheC*-algebra 𝒪ℋM,N is generated by operators Si,Tk,e(i,k),  i=1,…,M,  k=1,…,N satisfying
(175)∑i=1M‍SiSi*+∑k=1N‍TkTk*=1,(176)Si*Sj=δi,j,Tk*Tl=δk,l,(177)e(i,k)Sj=δi,j∑h=1M‍Sje(h,k),e(i,k)Tl=δk,l∑m=1N‍Tle(i,m)
for i,j=1,…,M,  k,l=1,…,N.Proof.
It suffices to show the equalities (177). We have
(178)e(i,k)Sj=Sj〈uj∣ϕ∘(e(i,k))uj〉ℬ1=Sj〈ej⊗1∣(ei⊗fk)(ej⊗1)〉ℬ1=δi,jSj(1⊗fk)=δi,jSj∑h=1M‍e(h,k).
The other equality of (177) is similarly shown.Put(179)S(i,k)=e(i,k)Si,T(i,k)=e(i,k)Tkfor(i,k)∈Σ∘.
Then we have the following.Lemma 40.
The following equalities hold:(180)e(i,k)=S(i,k)S(i,k)*+T(i,k)T(i,k)*,(181)Si=∑k=1N‍S(i,k),Tk=∑i=1M‍T(i,k),(182)∑(i,k)∈Σ∘‍S(i,k)S(i,k)*+∑(i,k)∈Σ∘‍T(i,k)T(i,k)*=1,(183)S(i,k)*S(i,k)=∑j=1M‍(S(j,k)S(j,k)*+T(j,k)T(j,k)*),(184)T(i,k)*T(i,k)=∑l=1N‍(S(i,l)S(i,l)*+T(i,l)T(i,l)*)
for i=1,…,M,  k=1,…,N and (i,k)∈Σ∘.Proof.
Sincee(i,k)Sj=δi,je(i,k)Si, we have
(185)S(i,k)S(i,k)*=e(i,k)SiSi*e(i,k)=e(i,k)(∑j=1M‍SjSj*)e(i,k)
and similarly T(i,k)T(i,k)*=e(i,k)(∑l=1N‍TlTl*)e(i,k). Hence, we have
(186)e(i,k)=e(i,k)(∑j=1M‍SjSj*+∑l=1N‍TlTl*)e(i,k)=S(i,k)S(i,k)*+T(i,k)T(i,k)*,
so that (180) holds. As 1=∑(j,k)∈Σ∘‍e(j,k), the equality (182) holds. Since e(j,k)Si=0 for j≠i, we have
(187)Si=(∑(j,k)∈Σ∘‍e(j,k))Si=∑k=1N‍S(i,k)
and similarly Tk=∑i=1M‍T(i,k), so that (181) holds. By (177), it follows that
(188)S(i,k)*S(i,k)=Si*∑j=1M‍Sie(j,k)=∑j=1M‍e(j,k)=∑j=1M‍(S(j,k)S(j,k)*+T(j,k)T(j,k)*),
and similarly, we have
(189)T(i,k)*T(i,k)=∑l=1N‍(S(i,l)S(i,l)*+T(i,l)T(i,l)*).Theorem 41.
TheC*-algebra 𝒪ℋM,N associated with the Hilbert C*-quad module ℋM,N=ℂM⊗ℂN is generated by partial isometries S(i,k),T(i,k) for (i,k)∈Σ∘={(i,k)∣i=1,…,M,k=1,…,N} satisfying the relations
(190)∑(i,k)∈Σ∘‍S(i,k)S(i,k)*+∑(i,k)∈Σ∘‍T(i,k)T(i,k)*=1,S(i,k)*S(i,k)=∑j=1M‍(S(j,k)S(j,k)*+T(j,k)T(j,k)*),T(i,k)*T(i,k)=∑l=1N‍(S(i,l)S(i,l)*+T(i,l)T(i,l)*)
for (i,k)∈Σ∘.Proof.
By the preceding lemma, one knows thate(i,k),Si,Tk are generated by the operators S(i,k), T(i,k) so that the algebra 𝒪ℋM,N is generated by the partial isometries S(i,k), T(i,k), (i,k)∈Σ∘.LetIn be the n×n identity matrix and En the n×n matrix whose entries are all 1s. For an M×M-matrix C=[ci,j]i,j=1M and an N×N-matrix D=[dk,l]k,l=1N, denote by C⊗D the MN×MN matrix
(191)C⊗D=[c11Dc12D⋯c1MDc21Dc22D⋯c2MD⋮⋮⋱⋮cM1DcM2D⋯cMMD]
so that
(192)EM⊗IN=[ININ⋯INININ⋯IN⋮⋮⋱⋮ININ⋯IN],IM⊗EN=[EN0⋯00EN⋱⋮⋮⋱⋱00⋯0EN].
The index set {(i,k)∣i=1,…,M,k=1,…,N} of the standard basis of ℂM⊗ℂN is ordered lexicographically from left as in the following way:
(193)(1,1),…,(1,N),(2,1),…,(2,N),…,(M,1),…,(M,N).
Put the MN×MN matrices
(194)AM,N=EM⊗IN,BM,N=IM⊗EN
and the 2MN×2MN matrix
(195)HM,N=[AM,NAM,NBM,NBM,N].
Then, we have the following.Theorem 42.
TheC*-algebra 𝒪ℋM,N is isomorphic to the Cuntz-Krieger algebra 𝒪HM,N for the matrix HM,N. The algebra 𝒪HM,N is simple and purely infinite and is isomorphic to the Cuntz-Krieger algebra 𝒪AM,N+BM,N for the matrix AM,N+BM,N.Proof.
By the preceding proposition, theC*-algebra 𝒪ℋM,N is isomorphic to the Cuntz-Krieger algebra 𝒪HM,N for the matrix HM,N. Since the matrix HM,N is aperiodic, the algebra is simple and purely infinite. The nth column of the matrix HM,N coincides with the (n+N)th column for every n=1,…,M. One sees that the matrix AM,N+BM,N is obtained from HM,N by amalgamating them. The procedure is called the column amalgamation and induces an isomorphism on their Cuntz-Krieger algebras (see [15]).In [15], the abelian groups  ℤMN/(AM,N+BM,N-IMN)ℤMN,Ker(AM,N+BM,N-IMN)inℤMN have been computed by using Euclidean algorithms. For the case M=2, they are ℤ/(N2-1)ℤ,{0}, respectively, so that we see K0(𝒪ℋ2,N)=ℤ/(N2-1)ℤ,K1(𝒪ℋ2,N)=0 (see [15] for details).(3) For aC*-textile dynamical system (𝒜,ρ,η,Σρ,Ση,κ), let ℋκρ,η be the C*-quad module over (𝒜;ℬ1,ℬ2) as in (3) in Section 2. The C*-algebra 𝒪ℋκρ,η has been studied in [12].


## 8. Higher-Dimensional Analogue

In this final section, we will state a generalization of HilbertC*-quad modules to Hilbert modules with multi actions of C*-algebras.Let𝒜 be a unital C*-algebra and let ℬ1,…,ℬn be n-family of unital C*-algebras. Suppose that there exists a unital embedding
(196)ιi:𝒜↪ℬi
for each i=1,…,n. Suppose that there exists a right action ψi of 𝒜 on ℬi such that
(197)biψi(a)∈ℬiforbi∈ℬi,a∈𝒜,i=1,…,n.
Hence, ℬi is a right 𝒜-module through ψi for i=1,…,n. Let ℋ be a Hilbert C*-bimodule over 𝒜 with a right action of 𝒜, an 𝒜-valued right inner product 〈·∣·〉𝒜, and a *-homomorphism ϕ𝒜 from 𝒜 to ℒ𝒜(ℋ). It is called a Hilbert C*-multimodule over (𝒜;ℬi,i=1,…,n) if ℋ has a multistructure of Hilbert C*-bimodules over ℬi for i=1,…,n such that for each i=1,…,n there exist a right action φi of ℬi on ℋ and a left action ϕi of ℬi on ℋ and a ℬi-valued right inner product 〈·∣·〉ℬi such that ϕi(zi)∈ℒ𝒜(ℋ) and
(198)[ϕi(zi)ξ]φj(wj)=ϕi(zi)[ξφj(wj)],ξφj(zjψj(a))=[ξφj(zj)]a
for ξ∈ℋ, zi∈ℬi, wj∈ℬj, a∈𝒜, i,j=1,…,n and
(199)ϕ𝒜(a)=ϕi(ιi(a)),a∈𝒜,i=1,…,n.
The operator ϕi(zi) on ℋ is adjointable with respect to the inner product 〈·∣·〉ℬi whose adjoint ϕi(zi)* coincides with the adjoint of ϕi(zi) with respect to the inner product 〈·∣·〉𝒜 so that ϕi(zi)*=ϕi(zi*). We assume that the left actions ϕi of ℬi on ℋ for i=1,2 are faithful. We require the following compatibility conditions between the right 𝒜-module structure of ℋ and the right 𝒜-module structure of ℬi through ψi:
(200)a〈ξ∣ηa〉ℬi=〈ξ∣η〉ℬiψi(a),ξ,η∈ℋ,a∈𝒜,i=1,…,n.
We further assume that ℋ is a full Hilbert C*-bimodule with respect to the inner product 〈·∣·〉𝒜,〈·∣·〉ℬi for each. A Hilbert C*-multimodule ℋ over (𝒜;ℬi,i=1,…,n) is said to be of general type if there exists a faithful completely positive map λi:ℬi→𝒜 for i=1,…,n such that
(201)λi(biψi(a))=λi(bi)a,bi∈ℬi,a∈𝒜,λi(〈ξ∣η〉ℬi)=〈ξ∣η〉𝒜,ξ,η∈ℋ,i=1,…,n.
A Hilbert C*-multimodule ℋ over (𝒜;ℬi,i=1,…,n) is said to be of finite type if there exists a family {u1(i),…,uM(i)(i)}, i=1,…,n of finite bases of ℋ as a right Hilbert ℬi-module for each i=1,…,n such that
(202)∑j=1M(i)‍uj(i)φi(〈uj(i)∣ξ〉ℬi)=ξ,ξ∈ℋ,i=1,…,n〈uj(i)∣ϕk(wk)uh(i)〉ℬi∈𝒜,wk∈ℬk,j,h=1,…,M(i),∑j=1M(i)‍〈uj(i)∣ϕk(〈ξ∣η〉ℬk)uj(i)〉ℬi=〈ξ∣η〉𝒜
for all ξ,η∈ℋ, i,k=1,…,n with i≠k.By a generalizing argument to the preceding sections, we may construct aC*-algebra 𝒪F(ℋ) associated with the Hilbert C*-multimodule ℋ by a similar manner to the preceding sections; that is, the C*-algebra is generated by the quotients of the n-kinds of creation operators sξ(i),ξ∈ℋ, i=1,…,n on the generalized Fock space F(ℋ) by the ideal generated by the finite-rank operators. One may show the following generalization.Proposition 43.
Letℋ be a Hilbert C*-multimodule over (𝒜;ℬi,i=1,…,n) of finite type with a finite basis {u1(i),…,uM(i)(i)} of ℋ as a Hilbert C*-right module over ℬi for each i=1,…,n. Then, the C*-algebra 𝒪F(ℋ) generated by the quotients of the n-kinds of creation operators on the generalized Fock spaces F(ℋ) is canonically isomorphic to the universal C*-algebra 𝒪ℋ generated by the operators S1(i),…,SM(i)(i) and elements zi∈ℬi for i=1,…,n subject to the relations
(203)∑i=1n‍∑k=1M(i)‍Sk(i)Sk(i)*=1,Sk(i)*Sm(j)=0,i≠j,Sk(i)*Sl(i)=〈uk(i)∣ul(i)〉ℬi,zjSk(i)=∑l=1M(i)‍Sl(i)〈ul(i)∣ϕj(zj)uk(i)〉ℬi
for zj∈ℬj, i,j=1,…,n,  k,l=1,…,M(i),  m=1,…,M(j).The proof of the above proposition is similar to the proof of Theorem1.



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