𝐶^∗-Algebras Associated with Hilbert 𝐶^∗-Quad Modules of Finite Type

Abstract

A Hilbert 𝐶^∗-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.

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 {𝜌_𝛼}_𝛼∈Σ of endomorphisms of a unital 𝐶^∗-algebra 𝒜 such that 𝜌_𝛼(𝑍_𝒜)⊂𝑍_𝒜, 𝛼∈Σ and ∑_𝛼∈Σ𝜌_𝛼(1)≥1 where 𝑍_𝒜 denotes the center of 𝒜. It provides a subshift Λ_𝜌 over Σ and a Hilbert 𝐶^∗-bimodule ℋ^𝜌_𝒜 over 𝒜 which gives rise to a 𝐶^∗-algebra 𝒪_𝜌 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,Σ^𝜌,Σ^𝜂,𝜅) consists of two 𝐶^∗-symbolic dynamical systems (𝒜,𝜌,Σ^𝜌) and (𝒜,𝜂,Σ^𝜂) with common unital 𝐶^∗-algebra 𝒜 and commutation relations 𝜅 between the endomorphisms 𝜌_𝛼,𝛼∈Σ^𝜌 and 𝜂_𝑎,𝑎∈Σ^𝜂. 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 the 𝐶^∗-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 ℬ_𝑖 with right actions 𝜑_𝑖 of ℬ_𝑖 and left actions 𝜙_𝑖 of ℬ_𝑖 and ℬ_𝑖-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,…,𝑢_𝑀} of ℋ as a Hilbert 𝐶^∗-right module over ℬ₁ and a finite basis {𝑣₁,…,𝑣_𝑁} of ℋ as a Hilbert 𝐶^∗-right module over ℬ₂ such that^𝑀_𝑖=1𝑢_𝑖∣𝜙₂𝜉∣𝜂_ℬ2𝑢_𝑖ℬ1=^𝑁_𝑘=1𝑣_𝑘∣𝜙₁𝜉∣𝜂_ℬ1𝑣_𝑘ℬ2=𝜉∣𝜂_𝒜 (1) 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 𝑠_𝜉, 𝑡_𝜉 for 𝜉∈ℋ on 𝐹(ℋ). The 𝐶^∗-algebra on 𝐹(ℋ) generated by them is denoted by 𝒯_𝐹(ℋ) and called the Toeplitz quad module algebra. We then define the 𝐶^∗-algebra 𝒪_𝐹(ℋ) associated with the Hilbert 𝐶^∗-quad module ℋ by the quotient 𝐶^∗-algebra of 𝒯_𝐹(ℋ) by the ideal generated by the finite-rank operators. We will then prove that the 𝐶^∗-algebra 𝒪_𝐹(ℋ) for a 𝐶^∗-quad module ℋ of finite type has a universal property in the following way.

Theorem 1 (Theorem 34). Let ℋ be a Hilbert 𝐶^∗-quad module over (𝒜;ℬ₁,ℬ₂) of finite type with a finite basis {𝑢₁,…,𝑢_𝑀} of ℋ as a Hilbert 𝐶^∗-right module over ℬ₁ and a finite basis {𝑣₁,…,𝑣_𝑁} of ℋ as a Hilbert 𝐶^∗-right module over ℬ₂. Then, the 𝐶^∗-algebra 𝒪_𝐹(ℋ) generated by the quotients [𝑠_𝜉],[𝑡_𝜉] of the creation operators 𝑠_𝜉,𝑡_𝜉 for 𝜉∈ℋ on the Fock spaces 𝐹(ℋ) is canonically isomorphic to the universal 𝐶^∗-algebra 𝒪_ℋ generated by operators 𝑆₁,…,𝑆_𝑀,  𝑇₁,…,𝑇_𝑁 and elements 𝑧∈ℬ₁,  𝑤∈ℬ₂ subject to the relations ^𝑀_𝑖=1𝑆_𝑖𝑆^∗_𝑖+^𝑁_𝑘=1𝑇_𝑘𝑇^∗_𝑘=1,𝑆^∗_𝑗𝑇_𝑙𝑆=0,_∗𝑖𝑆_𝑗=𝑢_𝑖∣𝑢_𝑗ℬ1,𝑇^∗_𝑘𝑇_𝑙=𝑣_𝑘∣𝑣_𝑙ℬ2,𝑧𝑆_𝑗=^𝑀_𝑖=1𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1,𝑧𝑇_𝑙=^𝑁_𝑘=1𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2,𝑤𝑆_𝑗=^𝑀_𝑖=1𝑆_𝑖𝑢_𝑖∣𝜙₂(𝑤)𝑢_𝑗ℬ1,𝑤𝑇_𝑙=^𝑁_𝑘=1𝑇_𝑘𝑣_𝑘∣𝜙₂(𝑤)𝑣_𝑙ℬ2 (2) 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 (Corollary 35). For a 𝐶^∗-quad module ℋ of finite type, the universal 𝐶^∗-algebra 𝒪_ℋ generated by operators 𝑆₁,…,𝑆_𝑀,  𝑇₁,…,𝑇_𝑁 and elements 𝑧∈ℬ₁, 𝑤∈ℬ₂ subject to the relations (ℋ) does not depend on the choice of the finite bases {𝑢₁,…,𝑢_𝑀} and {𝑣₁,…,𝑣_𝑁}.

The paper is organized in the following way. In Section 2, we will define Hilbert 𝐶^∗-quad module and present some basic properties. In Section 3, we will define a 𝐶^∗-algebra 𝒪_𝐹(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 𝒪_𝐹(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 𝒪_𝐹(ℋ) has the universal property stated as in Theorem 1. A strategy to prove Theorem 1 is to show that the 𝐶^∗-algebra 𝒪_𝐹(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 𝒪_ℋ. 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 by ℤ₊ the set of nonnegative integers and by ℕ the set of positive integers.

2. Hilbert 𝐶^∗-Quad Modules

Throughout the paper, we fix three unital 𝐶^∗-algebras 𝒜, ℬ₁, and ℬ₂ such that 𝒜⊂ℬ₁, 𝒜⊂ℬ₂ with common units. We assume that there exists a right action 𝜓_𝑖 of 𝒜 on ℬ_𝑖 so that 𝑏_𝑖𝜓_𝑖(𝑎)∈ℬ_𝑖for𝑏_𝑖∈ℬ_𝑖,𝑎∈𝒜,𝑖=1,2, (3) which satisfies 𝑏_𝑖𝜓_𝑖(𝑎)≤𝑏_𝑖‖𝑎‖,𝑏_𝑖𝜓_𝑖𝑎𝑎^′=𝑏_𝑖𝜓_𝑖(𝑎)𝜓_𝑖𝑎^′ (4) for 𝑏_𝑖∈ℬ_𝑖,  𝑎,𝑎^′∈𝒜, 𝑖=1,2. Hence, ℬ_𝑖 is a right 𝒜-module through 𝜓_𝑖 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 𝜙_𝒜 from 𝒜 to the algebra of all bounded adjointable right 𝒜-module maps ℒ_𝒜(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 ℬ_𝑖 for each 𝑖=1,2 with right action 𝜑_𝑖 of ℬ_𝑖 and left action 𝜙_𝑖 of ℬ_𝑖 and ℬ_𝑖-valued right inner product ⟨⋅∣⋅⟩_ℬ𝑖 such that for 𝑧∈ℬ₁, 𝑤∈ℬ₂, 𝜙₁(𝑧),𝜙₂(𝑤)∈ℒ_𝒜(ℋ),𝜙₁(𝑧)𝜉𝜑₂(𝑤)=𝜙₁(𝑧)𝜉𝜑₂(𝑤,𝜙₂)(𝑤)𝜉𝜑₁(𝑧)=𝜙₂(𝑤)𝜉𝜑₁(𝑧),𝜉𝜑₁𝑧𝜓₁(𝑎)=𝜉𝜑₁(𝑧)𝑎,𝜉𝜑₂𝑤𝜓₂(𝑎)=𝜉𝜑₂(𝑤)𝑎, (5) for 𝜉∈ℋ, 𝑧∈ℬ₁, 𝑤∈ℬ₂, 𝑎∈𝒜, and 𝜙_𝒜(𝑎)=𝜙₁(𝑎)=𝜙₂(𝑎)for𝑎∈𝒜, (6) where 𝒜 is regarded as a subalgebra of ℬ_𝑖. The left action 𝜙_𝑖 of ℬ_𝑖 on ℋ means that 𝜙_𝑖(𝑏_𝑖) for 𝑏_𝑖∈ℬ_𝑖 is a bounded adjointable operator with respect to the inner product ⟨⋅∣⋅⟩_ℬ𝑖 for each 𝑖=1,2. The operator 𝜙_𝑖(𝑏_𝑖) for 𝑏_𝑖∈ℬ_𝑖 is also adjointable with respect to the inner product ⟨⋅∣⋅⟩_𝒜. We assume that the adjoint of 𝜙_𝑖(𝑏_𝑖) with respect to the inner product ⟨⋅∣⋅⟩_ℬ𝑖 coincides with the adjoint of 𝜙_𝑖(𝑏_𝑖) with respect to the inner product ⟨⋅∣⋅⟩_𝒜. Both of them coincide with 𝜙_𝑖(𝑏^∗_𝑖). We assume that the left actions 𝜙_𝑖 of ℬ_𝑖 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 ℬ_𝑖 through 𝜓_𝑖: 𝜉∣𝜂𝑎_ℬ𝑖=𝜉∣𝜂_ℬ𝑖𝜓_𝑖(𝑎)for𝜉,𝜂∈ℋ,𝑎∈𝒜,𝑖=1,2. (7)

We further assume that ℋ 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 vector 𝜉∈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 Hilbert 𝐶^∗-quad module ℋ over (𝒜;ℬ₁,ℬ₂) is said to be of general type if there exists a faithful completely positive map 𝜆_𝑖∶ℬ_𝑖→𝒜 for 𝑖=1,2 such that 𝜆_𝑖𝑏_𝑖𝜓_𝑖(𝑎)=𝜆_𝑖𝑏_𝑖𝑎for𝑏_𝑖∈ℬ_𝑖,𝑎∈𝒜,𝜆_𝑖𝜉∣𝜂_ℬ𝑖=𝜉∣𝜂_𝒜,for𝜉,𝜂∈ℋ,𝑖=1,2. (8)(9)
(ii) A Hilbert 𝐶^∗-quad module ℋ over (𝒜;ℬ₁,ℬ₂) is said to be of finite type if there exist a finite basis {𝑢₁,…,𝑢_𝑀} of ℋ as a right Hilbert ℬ₁-module and a finite basis {𝑣₁,…,𝑣_𝑁} of ℋ as a right Hilbert ℬ₂-module; that is, ^𝑀_𝑖=1𝑢_𝑖𝜑₁𝑢_𝑖∣𝜉_ℬ1=^𝑁_𝑘=1𝑣_𝑘𝜑₂𝑣_𝑘∣𝜉_ℬ2=𝜉,𝜉∈ℋ (10) such that 𝑢_𝑖∣𝜙₂(𝑤)𝑢_𝑗ℬ1𝑣∈𝒜,𝑖,𝑗=1,…,𝑀,_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2∈𝒜,𝑘,𝑙=1,…,𝑁 (11) for 𝑤∈ℬ₂, 𝑧∈ℬ₁ and ^𝑀_𝑖=1𝑢_𝑖∣𝜙₂𝜉∣𝜂_ℬ2𝑢_𝑖ℬ1=𝜉∣𝜂_𝒜,_𝑁𝑘=1𝑣_𝑘∣𝜙₁𝜉∣𝜂_ℬ1𝑣_𝑘ℬ2=𝜉∣𝜂_𝒜 (12) for all 𝜉,𝜂∈ℋ. Following [13], {𝑢₁,…,𝑢_𝑀} and {𝑣₁,…,𝑣_𝑁} are called finite bases of ℋ, respectively.
(iii) A Hilbert 𝐶^∗-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,…,𝑒_𝑀′} of ℬ₁ as a right 𝒜-module through 𝜓₁∘𝜆 and a finite basis {𝑓₁,…,𝑓_𝑁′} of ℬ₂ as a right 𝒜-module through 𝜓₂∘𝜆₂. This means that the following equalities hold: 𝑧=^𝑀′_𝑗=1𝑒_𝑗𝜓₁𝜆₁𝑒^∗_𝑗𝑧,𝑧∈ℬ₁,𝑤=^𝑁′_𝑙=1𝑓_𝑙𝜓₂𝜆₂𝑓^∗_𝑙𝑤,𝑤∈ℬ₂. (13)
We note that for a Hilbert 𝐶^∗-quad module of general type, the conditions (9) imply 𝜉∣𝜉_𝒜≤𝜆_𝑖(1)𝜉∣𝜉_ℬ𝑖,𝜉∈ℋ. (14) Put 𝐶_𝑖=‖𝜆_𝑖(1)‖^1/2>0 so that ‖𝜉‖_𝒜≤𝐶_𝑖‖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 𝐶^′_𝑖 such that ‖𝜉‖_ℬ𝑖≤𝐶^′_𝑖‖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. Let ℋ 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 that ℋ is of finite type with finite bases {𝑢₁,…,𝑢_𝑀} of ℋ as a right Hilbert ℬ₁-module and {𝑣₁,…,𝑣_𝑁} of ℋ as a right Hilbert ℬ₂-module as above. We put 𝜆₁(𝑧)=^𝑁_𝑘=1𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑘ℬ2∈𝒜,𝑧∈ℬ₁,𝜆₂(𝑤)=^𝑀_𝑖=1𝑢_𝑖∣𝜙₂(𝑤)𝑢_𝑖ℬ1∈𝒜,𝑤∈ℬ₂. (15) They give rise to faithful completely positive maps 𝜆_𝑖∶ℬ_𝑖→𝒜, 𝑖=1,2. The equalities (12) imply that 𝜆_𝑖𝜉∣𝜂_ℬ𝑖=𝜉∣𝜂_𝒜,for𝜉,𝜂∈ℋ,𝑖=1,2. (16) It then follows that 𝜆_𝑖𝜉∣𝜂_ℬ𝑖𝜓_𝑖(𝑎)=𝜆_𝑖𝜉∣𝜂𝑎_ℬ𝑖=𝜉∣𝜂_𝒜𝑎=𝜆_𝑖𝜉∣𝜂_ℬ𝑖𝑎,𝑎∈𝒜. (17) Since ℋ is full, the equalities (8) hold.

Lemma 5. Suppose that a Hilbert 𝐶^∗-quad module ℋ of finite type is of strongly finite type with a finite basis {𝑒₁,…,𝑒_𝑀′} of ℬ₁ as a right 𝒜-module through 𝜓₁∘𝜆₁ and a finite basis {𝑓₁,…,𝑓_𝑁′} of ℬ₂ as a right 𝒜-module through 𝜓₂∘𝜆₂. Let {𝑢₁,…,𝑢_𝑀} and {𝑣₁,…,𝑣_𝑁} be finite bases of ℋ satisfying (10). Then, two families {𝑢_𝑖𝜑₁(𝑒_𝑗)∣𝑖=1,…,𝑀,  𝑗=1,…,𝑀^′} and {𝑣_𝑘𝜑₂(𝑓_𝑙)∣𝑘=1,…,𝑁,  𝑙=1,…,𝑁^′} of ℋ form bases of ℋ as right 𝒜-modules, respectively.

Proof. For 𝜉∈ℋ, by the equalities 𝜉=^𝑀_𝑖=1𝑢_𝑖𝜑₁𝑢_𝑖∣𝜉_ℬ1,𝑢_𝑖∣𝜉_ℬ1=^𝑀_𝑗=1𝑒_𝑗𝜓₁𝜆₁𝑒^∗_𝑗𝑢_𝑖∣𝜉_ℬ1, (18) it follows that 𝜉=^𝑀_𝑖=1𝑢_𝑖𝜑₁^𝑀′_𝑗=1𝑒_𝑗𝜓₁𝜆₁𝑒^∗_𝑗𝑢_𝑖∣𝜉_ℬ1=^𝑀𝑀_𝑖=1^′_𝑗=1𝑢_𝑖𝜑₁𝑒_𝑗⋅𝜆₁𝑒^∗_𝑗𝑢_𝑖∣𝜉_ℬ1=^𝑀𝑀_𝑖=1^′_𝑗=1𝑢_𝑖𝜑₁𝑒_𝑗⋅𝜆₁𝑢_𝑖𝜑₁𝑒_𝑗∣𝜉_ℬ1=^𝑀𝑀_𝑖=1^′_𝑗=1𝑢_𝑖𝜑₁𝑒_𝑗⋅𝑢_𝑖𝜑₁𝑒_𝑗∣𝜉_𝒜. (19) We similarly have 𝜉=^𝑁𝑁_𝑘=1^′_𝑙=1𝑣_𝑘𝜑₂𝑓_𝑙⋅𝑣_𝑘𝜑₂𝑓_𝑙∣𝜉_𝒜. (20)

We present some examples.

Examples. (1) Let 𝛼, 𝛽 be automorphisms of a unital 𝐶^∗-algebra 𝒜 satisfying 𝛼∘𝛽=𝛽∘𝛼. We set ℬ₁=ℬ₂=𝒜. Define right actions 𝜓_𝑖 of 𝒜 on ℬ_𝑖 by 𝑏₁𝜓₁(𝑎)=𝑏₁𝛼(𝑎),𝑏₂𝜓₂(𝑎)=𝑏₂𝛽(𝑎) (21) for 𝑏_𝑖∈ℬ_𝑖, 𝑎∈𝒜. We put ℋ_𝛼,𝛽=𝒜 and equip it with Hilbert 𝐶^∗-quad module structure over (𝒜;𝒜,𝒜) in the following way. For 𝜉=𝑥, 𝜉^′=𝑥^′∈ℋ_𝛼,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 𝜉⋅𝑎=𝑥𝑎,𝜉∣𝜉^′_𝒜=𝑥^∗𝑥^′. (22) Define the right actions 𝜑_𝑖 of ℬ_𝑖 with right ℬ_𝑖-valued inner products ⟨⋅∣⋅⟩_ℬ𝑖 and the left actions 𝜙_𝑖 of ℬ_𝑖 by setting 𝜉𝜑₁(𝑧)=𝑥𝛼(𝑧),𝜉𝜑₂(𝑤)=𝑥𝛽(𝑤),𝜉∣𝜉^′_ℬ1=𝛼⁻¹𝑥^∗𝑥^′,𝜉∣𝜉^′_ℬ2=𝛽⁻¹𝑥^∗𝑥^′,𝜙₁(𝑧)𝜉=𝛽(𝛼(𝑧))𝑥,𝜙₂(𝑤)𝜉=𝛼𝛽(𝑤)𝑥. (23) It is straightforward to see that ℋ_𝛼,𝛽 is a Hilbert 𝐶^∗-quad module over (𝒜;𝒜,𝒜) of strongly finite type.
(2) We fix natural numbers 1<𝑁, 𝑀∈ℕ. Consider finite-dimensional commutative 𝐶^∗-algebras 𝒜=ℂ,  ℬ₁=ℂ^𝑁, and ℬ₂=ℂ^𝑀. The right actions 𝜓_𝑖 of 𝒜 on ℬ_𝑖 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 ℋ_𝑀,𝑁 the tensor product ℂ^𝑀⊗ℂ^𝑁. Define the right actions 𝜑_𝑖 of ℬ_𝑖 with ℬ_𝑖-valued right inner products ⟨⋅∣⋅⟩_ℬ𝑖 and the left actions 𝜙_𝑖 of ℬ_𝑖 on ℋ_𝑀,𝑁=ℂ^𝑀⊗ℂ^𝑁 for 𝑖=1,2 by setting 𝜉⊗𝜂𝜑₁(𝑧)=𝜉⊗𝜂⋅𝑧,𝜉⊗𝜂𝜑₂(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 (24) for 𝑧∈ℬ₁, 𝜉,𝜉^′∈ℂ^𝑁, 𝑤∈ℬ₂, and 𝜂,𝜂^′∈ℂ^𝑀. Let 𝑒_𝑖, 𝑖=1,…,𝑀 and 𝑓_𝑘, 𝑘=1,…,𝑁 be the standard bases of ℂ^𝑀 and ℂ^𝑁, respectively. Put the finite bases 𝑢_𝑖=𝑒_𝑖⊗1∈ℋ_𝑀,𝑁𝑣,𝑖=1,…,𝑀,_𝑘=1⊗𝑓_𝑘∈ℋ_𝑀,𝑁,𝑖=1,…,𝑁. (25) It is straightforward to see that ℋ_𝑀,𝑁 is a Hilbert 𝐶^∗-quad module over (ℂ;ℂ^𝑁,ℂ^𝑀) of strongly finite type.
(3) Let (𝒜,𝜌,𝜂,Σ^𝜌,Σ^𝜂,𝜅) be a 𝐶^∗-textile dynamical system which means that for 𝑗∈Σ^𝜂, 𝑙∈Σ^𝜌 endomorphisms 𝜂_𝑗,𝜌_𝑙 of 𝒜 are given with commutation relations 𝜂_𝑗∘𝜌_𝑙=𝜌_𝑘∘𝜂_𝑖 if 𝜅(𝑙,𝑗)=(𝑖,𝑘). In [12], a Hilbert 𝐶^∗-quad module ℋ_𝜅^𝜌,𝜂 over (𝒜;ℬ₁,ℬ₂) from (𝒜,𝜌,𝜂,Σ^𝜌,Σ^𝜂,𝜅) is constructed (see [12] for its detail construction). The two triplets (𝒜,𝜌,Σ^𝜌) and (𝒜,𝜂,Σ^𝜂) are 𝐶^∗-symbolic dynamical systems [7], that yield 𝐶^∗-algebras 𝒪_𝜌 and 𝒪_𝜂, respectively. The 𝐶^∗-algebras ℬ₁ and ℬ₂ are defined as the 𝐶^∗-subalgebra of 𝒪_𝜂 generated by elements 𝑇_𝑗𝑦𝑇^∗_𝑗, 𝑗∈Σ^𝜂, 𝑦∈𝒜 and that of 𝒪_𝜌 generated by 𝑆_𝑘𝑦𝑆^∗_𝑘, 𝑘∈Σ^𝜌, 𝑦∈𝒜, respectively. Define the maps 𝜓_𝑖∶𝒜→ℬ_𝑖, 𝑖=1,2, by 𝜓₁(𝑎)=_{𝑗∈Σ𝜂}𝑇_𝑗𝑎𝑇^∗_𝑗,𝜓₂(𝑎)=_{𝑙∈Σ𝜌}𝑆_𝑙𝑎𝑆^∗_𝑙,𝑎∈𝒜, (26) which yield the right actions of 𝒜 on ℬ_𝑖,  𝑖=1,2. Define the maps 𝜆_𝑖∶ℬ_𝑖→𝒜,  𝑖=1,2 by 𝜆₁(𝑧)=_{𝑗∈Σ𝜂}𝑇^∗_𝑗𝑧𝑇_𝑗,𝜆₂(𝑤)=_{𝑙∈Σ𝜌}𝑆^∗_𝑙𝑤𝑆_𝑙,𝑧∈ℬ₁,𝑤∈ℬ₂. (27) Put 𝑒_𝑗=𝑇_𝑗𝑇^∗_𝑗∈ℬ₁, 𝑗∈Σ^𝜂. Let 𝑧=∑_𝑗∈Σ^𝜂𝑇_𝑗𝑧_𝑗𝑇^∗_𝑗 be an element of ℬ₁ for 𝑧_𝑗∈𝒜 with 𝑇^∗_𝑗𝑇_𝑗𝑧_𝑗𝑇^∗_𝑗𝑇_𝑗=𝑧_𝑗. As 𝜆₁(𝑒^∗_𝑗𝑧)=𝜆₁(𝑇_𝑗𝑧_𝑗𝑇^∗_𝑗)=𝑧_𝑗, one sees that 𝑧=_{𝑗∈Σ𝜂}𝑇_𝑗𝑇^∗_𝑗𝑇_𝑗𝑧_𝑗𝑇^∗_𝑗=_{𝑗∈Σ𝜂}𝑇_𝑗𝑇^∗_𝑗𝜓₁𝑧_𝑗=_{𝑗∈Σ𝜂}𝑒_𝑗𝜓₁𝜆₁𝑒^∗_𝑗𝑧. (28) We similarly have by putting 𝑓_𝑙=𝑆_𝑙𝑆^∗_𝑙∈ℬ₂, 𝑤=_{𝑙∈Σ𝜌}𝑓_𝑙𝜓₂𝜆₂𝑓^∗_𝑙𝑤for𝑤∈ℬ₂. (29) We see that ℋ_𝜅^𝜌,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 Hilbert 𝐶^∗-Quad Modules and Creation Operators

In this section, we will construct a 𝐶^∗-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 𝜙_𝑖 is a left action of ℬ_𝑖 on ℋ and 𝜑_𝑖 is a right action of ℬ_𝑖 on ℋ with ℬ_𝑖-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 ℋ⊗_ℬ1ℋ,ℋ⊗_ℬ2ℋ (31) as Hilbert 𝐶^∗-quad modules over (𝒜;ℬ₁,ℬ₂). The latter one should be written vertically as ℋ⊗_ℬ2ℋ (32) 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 𝜉⊗_ℬ1𝜁∣𝜉^′⊗_ℬ1𝜁^′_ℬ1∶=𝜁∣𝜙_ℬ1𝜉∣𝜉^′_ℬ1𝜁^′_ℬ1,𝜉⊗_ℬ1𝜁∣𝜉^′⊗_ℬ1𝜁^′_ℬ2∶=𝜁∣𝜙_ℬ1𝜉∣𝜉^′_ℬ1𝜁^′_ℬ2, (33) 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 𝜉⊗_ℬ1𝜁∣𝜉^′⊗_ℬ1𝜁^′_𝒜∶=𝜆₁𝜉⊗_ℬ1𝜁∣𝜉^′⊗_ℬ1𝜁^′_ℬ1=𝜆₂𝜉⊗_ℬ1𝜁∣𝜉^′⊗_ℬ1𝜁^′_ℬ2, (34) a right 𝒜-action id⊗𝑎 for 𝑎∈𝒜 and a left 𝒜-action 𝜙_𝒜⊗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. Let ℋ_𝑖=ℋ,  𝑖=1,2,3. The correspondences 𝜉₁⊗_ℬ1𝜉₂⊗_ℬ2𝜉₃∈ℋ₁⊗_ℬ1ℋ₂⊗_ℬ2ℋ₃⟶𝜉₁⊗_ℬ1𝜉₂⊗_ℬ2𝜉₃∈ℋ₁⊗_ℬ1ℋ₂⊗_ℬ2ℋ₃,𝜉₁⊗_ℬ2𝜉₂⊗_ℬ1𝜉₃∈ℋ₁⊗_ℬ2ℋ₂⊗_ℬ1ℋ₃⟶𝜉₁⊗_ℬ2𝜉₂⊗_ℬ1𝜉₃∈ℋ₁⊗_ℬ2ℋ₂⊗_ℬ1ℋ₃ (36) yield isomorphisms of Hilbert 𝐶^∗-quad modules, respectively.

We write the isomorphism class of the former Hilbert 𝐶^∗-quad modules as ℋ₁⊗_ℬ1ℋ₂⊗_ℬ2ℋ₃ and that of the latter ones as ℋ₁⊗_ℬ2ℋ₂⊗_ℬ1ℋ₃, respectively.

Note that the direct sum ℬ₁⊕ℬ₂ has a structure of a pre-Hilbert 𝐶^∗-right module over 𝒜 by the following operations. For 𝑏₁⊕𝑏₂, 𝑏^′₁⊕𝑏^′₂∈ℬ₁⊕ℬ₂, and 𝑎∈𝒜, set 𝑏₁⊕𝑏₂𝜓_𝒜(𝑎)∶=𝑏₁𝜓₁(𝑎)⊕𝑏₂𝜓₂(𝑎)∈ℬ₁⊕ℬ₂,𝑏₁⊕𝑏₂∣𝑏^′₁⊕𝑏^′_2𝒜∶=𝜆₁𝑏^∗₁𝑏^′₁+𝜆₂𝑏^∗₂𝑏^′₂∈𝒜. (37) By (8), the equality 𝑏₁⊕𝑏₂∣𝑏^′₁⊕𝑏^′₂𝜓_𝒜(𝑎)_𝒜=𝑏₁⊕𝑏₂∣𝑏^′₁⊕𝑏^′_2𝒜⋅𝑎 (38) 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 ℬ_𝑖-actions 𝜑_𝑖 and left ℬ_𝑖-action 𝜙_𝑖 by 𝑏₁⊕𝑏₂𝜑₁(𝑧)=𝑏₁𝑧⊕0,𝑏₁⊕𝑏₂𝜑₂(𝑤)=0⊕𝑏₂𝜙𝑤,₁(𝑧)𝑏₁⊕𝑏₂=𝑧𝑏₁⊕0,𝜙₂(𝑤)𝑏₁⊕𝑏₂=0⊕𝑤𝑏₂ (39) for 𝑏₁⊕𝑏₂∈ℬ₁⊕ℬ₂, 𝑧∈ℬ₁, and 𝑤∈ℬ₂.

We denote the relative tensor product ℋ⊗_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 Γ_𝑛={(𝑖₁,…,𝑖_𝑛))∣𝑖_𝑗=1,2}, 𝑛=1,2,…. We set 𝐹₁(ℋ)𝐹=ℋ,₂(ℋ)=ℋ⊗₁ℋ⊕ℋ⊗₂ℋ,𝐹₃(ℋ)=ℋ⊗₁ℋ⊗₁ℋ⊕ℋ⊗₁ℋ⊗₂ℋ⊕ℋ⊗₂ℋ⊗₁ℋ⊕ℋ⊗₂ℋ⊗₂ℋ⋮𝐹_𝑛(ℋ)=⊕_{(𝑖1,…,𝑖𝑛−1)∈Γ𝑛−1}ℋ⊗_𝑖1ℋ⊗_𝑖2⋯⊗_𝑖𝑛−1ℋ⋮ (40) as Hilbert 𝐶^∗-bimodules over 𝒜. We will define the Fock Hilbert 𝐶^∗-module 𝐹(ℋ) by setting 𝐹(ℋ)∶=⊕^∞_𝑛=0𝐹_𝑛(ℋ), (41) which is the completion of the algebraic direct sum ⊕^∞_𝑛=0𝐹_𝑛(ℋ) of the Hilbert 𝐶^∗-right module over 𝒜 under the norm ‖𝜉‖_𝒜 on ⊕^∞_𝑛=0𝐹_𝑛(ℋ) induced by the 𝒜-valued right inner product on ⊕^∞_𝑛=0𝐹_𝑛(ℋ). Then, 𝐹(ℋ) is a Hilbert 𝐶^∗-right module over 𝒜. It has a natural left ℬ_𝑖-action defined by 𝜙_𝑖 for 𝑖=1,2.

For 𝜉∈ℋ_𝜅, we define two operators 𝑠_𝜉∶𝐹_𝑛(ℋ)⟶𝐹_𝑛+1(ℋ)𝑡,𝑛=0,1,2,…,_𝜉∶𝐹_𝑛(ℋ)⟶𝐹_𝑛+1(ℋ),𝑛=0,1,2,… (42) by setting for 𝑛=0, 𝑠_𝜉𝑏₁⊕𝑏₂=𝜉𝜑₁𝑏₁,𝑏₁⊕𝑏₂∈ℬ₁⊕ℬ₂,𝑡_𝜉𝑏₁⊕𝑏₂=𝜉𝜑₂𝑏₂,𝑏₁⊕𝑏₂∈ℬ₁⊕ℬ₂, (43) and for 𝑛=1,2,…, 𝑠_𝜉𝜉₁⊗_𝜋1⋯⊗_𝜋𝑛−1𝜉_𝑛=𝜉⊗₁𝜉₁⊗_𝜋1⋯⊗_𝜋𝑛−1𝜉_𝑛,𝑡_𝜉𝜉₁⊗_𝜋1⋯⊗_𝜋𝑛−1𝜉_𝑛=𝜉⊗₂𝜉₁⊗_𝜋1⋯⊗_𝜋𝑛−1𝜉_𝑛 (44) for 𝜉₁⊗_𝜋1⋯⊗_𝜋𝑛−1𝜉_𝑛∈𝐹_𝑛(ℋ) with (𝜋₁,…,𝜋_𝑛−1)∈Γ_𝑛−1.

Lemma 7. For 𝜉∈ℋ, the two operators 𝑠_𝜉∶𝐹_𝑛(ℋ)⟶𝐹_𝑛+1(ℋ)𝑡,𝑛=0,1,2,…,_𝜉∶𝐹_𝑛(ℋ)⟶𝐹_𝑛+1(ℋ),𝑛=0,1,2,…, (45) are both right 𝒜-module maps.

Proof. We will show the assertion for 𝑠_𝜉. For 𝑛=0, we have for 𝑏₁⊕𝑏₂∈ℬ₁⊕ℬ₂ and 𝑎∈𝒜, 𝑠_𝜉𝑏₁⊕𝑏₂𝜓_𝒜(𝑎)=𝜉𝜑₁𝑏₁𝜓₁(𝑎)=𝜉𝜑₁𝑏₁𝑎=𝑠_𝜉𝑏₁⊕𝑏₂𝑎. (46) For 𝑛=1,2,…, one has 𝑠_𝜉𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛𝑎=𝑠_𝜉𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛𝑎=𝜉⊗₁𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛𝑎=𝜉⊗₁𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛𝑎=𝑠_𝜉𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛𝑎. (47)

It is clear that the two operators 𝑠_𝜉,𝑡_𝜉 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 𝑠_𝜉,𝑡_𝜉, respectively. The adjoints of 𝑠_𝜉,𝑡_𝜉∶𝐹(ℋ)→𝐹(ℋ) with respect to the 𝒜-valued right inner product on 𝐹(ℋ) map 𝐹_𝑛+1(ℋ) to 𝐹_𝑛(ℋ), 𝑛=0,1,2,….

Lemma 8. (i) For 𝜉,𝜉^′∈ℋ=𝐹₁(ℋ), one has 𝑠^∗_𝜉𝜉^′=𝜉∣𝜉^′_ℬ1⊕0∈ℬ₁⊕ℬ₂,𝑡^∗_𝜉𝜉^′=0⊕𝜉∣𝜉^′_ℬ2∈ℬ₁⊕ℬ₂. (48)
(ii) For 𝜉∈ℋ and 𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛𝜉_𝑛+1∈𝐹_𝑛+1(ℋ), 𝑛=1,2,…, we have 𝑠^∗_𝜉𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛𝜉_𝑛+1=𝜙₁⟨𝜉∣𝜉₁⟩_ℬ1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛𝜉_𝑛+1𝑖𝑓𝑖₁=1,0𝑖𝑓𝑖₁𝑡=2,_∗𝜉𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛𝜉_𝑛+1=0𝑖𝑓𝑖₁𝜙=1,₂𝜉∣𝜉_1ℬ2𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛𝜉_𝑛+1𝑖𝑓𝑖₁=2. (49)

Proof. We will show the assertions (i) and (ii) for 𝑠^∗_𝜉. (i) For 𝑏₁⊕𝑏₂∈ℬ₁⊕ℬ₂, we have 𝑏₁⊕𝑏₂∣𝑠^∗_𝜉𝜉^′_𝒜=𝜉𝜑₁𝑏₁∣𝜉^′_𝒜=𝜆₁𝑏^∗₁𝜉∣𝜉^′_ℬ1=𝑏₁⊕𝑏₂∣𝜉∣𝜉^′_ℬ1⊕0_𝒜 (50) so that 𝑠^∗_𝜉𝜉^′=⟨𝜉∣𝜉^′⟩_ℬ1⊕0.
(ii) For 𝜁₁⊗_𝑗1⋯⊗_𝑗𝑛−1𝜁_𝑛∈𝐹_𝑛(ℋ) with 𝑛=1,2,…, we have 𝜁₁⊗_𝑗1⋯⊗_𝑗𝑛−1𝜁_𝑛∣𝑠^∗_𝜉𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛𝜉_𝑛+1𝒜=𝜉⊗₁𝜁₁⊗_𝑗1⋯⊗_𝑗𝑛−1𝜁_𝑛∣𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛𝜉_𝑛+1𝒜=𝜆₁𝜉⊗₁𝜁₁⊗_𝑗1⋯⊗_𝑗𝑛−1𝜁_𝑛∣𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛𝜉_𝑛+1ℬ1=𝜁₁⊗_𝑗1⋯⊗_𝑗𝑛−1𝜁_𝑛∣𝜙₁𝜉∣𝜉_1ℬ1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛𝜉_𝑛+1𝒜if𝑖₁0=1,if𝑖₁=2. (51)

Denote by 𝜙_𝑖 the left actions of ℬ_𝑖,  𝑖=1,2 on 𝐹_𝑛(ℋ) and hence on 𝐹(ℋ), respectively. They satisfy the following equalities 𝜙₁(𝑧)𝑏₁⊕𝑏₂=𝑧𝑏₁⊕0,𝜙₂(𝑤)𝑏₁⊕𝑏₂=0⊕𝑤𝑏₂,𝜙₁(𝑧)𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝜙₁(𝑧)𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛,𝜙₂(𝑤)𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝜙₂(𝑤)𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛 (52) for 𝑧∈ℬ₁, 𝑤∈ℬ₂, 𝑏₁⊕𝑏₂∈ℬ₁⊕ℬ₂, and 𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛∈𝐹_𝑛(ℋ). More generally let us denote by ℒ_𝒜(ℋ) and ℒ_𝒜(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∈ℒ_𝒜(ℋ), define 𝐿∈ℒ_𝒜(𝐹(ℋ)) by 𝐿𝑏₁⊕𝑏₂=0for𝑏₁⊕𝑏₂∈ℬ₁⊕ℬ₂⊂𝐹₀(ℋ),𝐿𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝐿𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛 (53) for 𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛∈𝐹_𝑛(ℋ).

Lemma 9. Both the maps 𝜙_𝑖∶ℬ_𝑖→ℒ_𝒜(𝐹(ℋ)) for 𝑖=1,2 are faithful ∗-homomorphisms.

Proof. By assumption, the ∗-homomorphisms 𝜙_𝑖∶ℬ_𝑖→ℒ_𝒜(ℋ), 𝑖=1,2 are faithful, so that the ∗-homomorphisms 𝜙_𝑖∶ℬ_𝑖→ℒ_𝒜(𝐹(ℋ)), 𝑖=1,2 are both faithful.

Lemma 10. For 𝜉,𝜁∈ℋ, 𝑧∈ℬ₁,  𝑤∈ℬ₂, 𝐿∈ℒ_𝒜(ℋ), and 𝑐,𝑑∈ℂ, the following equalities hold on 𝐹(ℋ): 𝑠_{𝑐𝜉+𝑑𝜁}=𝑐𝑠_𝜉+𝑑𝑠_𝜁,𝑡_{𝑐𝜉+𝑑𝜁}=𝑐𝑡_𝜉+𝑑𝑡_𝜁,𝑠_{𝐿𝜉𝜑1(𝑧)}=𝐿𝑠_𝜉𝜙₁(𝑧),𝑡_{𝐿𝜉𝜑2(𝑤)}=𝐿𝑡_𝜉𝜙₂(𝑤),𝑠^∗_𝜁𝐿𝑠_𝜉=𝜙₁𝜁∣𝐿𝜉_ℬ1,𝑡^∗_𝜁𝐿𝑡_𝜉=𝜙₂𝜁∣𝐿𝜉_ℬ2. (54)(55)(56)

Proof. The equalities (54) are obvious. We will show the equalities (55) and (56) for 𝑠_𝜉. We have for 𝑏₁⊕𝑏₂∈ℬ₁⊕ℬ₂𝑠_{𝐿𝜉𝜑1(𝑧)}𝑏₁⊕𝑏₂=𝐿𝜉𝜑₁(𝑧)𝜑₁𝑏₁=𝐿𝜉𝜑₁𝑧𝑏₁=𝐿𝑠_𝜉𝑧𝑏₁=⊕0𝐿𝑠_𝜉𝜙₁(𝑧)𝑏₁⊕𝑏₂=𝐿𝑠_𝜉𝜙₁(𝑧)𝑏₁⊕𝑏₂. (57) For 𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛∈𝐹_𝑛(ℋ), 𝑛=1,2,…, we have 𝑠_{𝐿𝜉𝜑1(𝑧)}𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝐿𝜉𝜑₁(𝑧)⊗₁𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝐿𝜉𝜑₁(𝑧)⊗₁𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝐿𝜉⊗₁𝜙₁(𝑧)𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝐿𝑠_𝜉𝜙₁(𝑧)𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝐿𝑠_𝜉𝜙₁(𝑧)𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛 (58) so that 𝑠_{𝐿𝜉𝜑1(𝑧)}=𝐿𝑠_𝜉𝜙₁(𝑧) on 𝐹_𝑛(ℋ), 𝑛=0,1,…. Hence, the equalities (55) hold.
For 𝑏₁⊕𝑏₂∈ℬ₁⊕ℬ₂, we have 𝑠^∗_𝜁𝐿𝑠_𝜉𝑏₁⊕𝑏₂=𝑠^∗_𝜁𝐿𝜉𝜑₁𝑏₁=𝜁∣𝐿𝜉𝜑₁𝑏_1ℬ1=⊕0𝜁∣𝐿𝜉_ℬ1𝑏₁=⊕0𝜙₁𝜁∣𝐿𝜉_ℬ1𝑏₁⊕𝑏₂. (59) For 𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛∈𝐹_𝑛(ℋ), we have 𝑠^∗_𝜁𝐿𝑠_𝜉𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝑠^∗_𝜁𝐿𝜉⊗₁𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝜙₁𝜁∣𝐿𝜉_ℬ1𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝜙₁𝜁∣𝐿𝜉_ℬ1𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛 (60) so that 𝑠^∗_𝜁𝐿𝑠_𝜉=𝜙₁(⟨𝜁∣𝐿𝜉⟩_ℬ1) on 𝐹_𝑛(ℋ) for 𝑛=0,1,2,…. Hence, the equalities (56) hold.

The 𝐶^∗-subalgebra of ℒ_𝒜(𝐹(ℋ)) generated by the operators 𝑠_𝜉,𝑡_𝜉 for 𝜉∈ℋ is denoted by 𝒯_𝐹(ℋ) and is called the Toeplitz quad module algebra for ℋ.

Lemma 11. The 𝐶^∗-algebra 𝒯_𝐹(ℋ) contains the operators 𝜙₁(𝑧),𝜙₂(𝑤) for 𝑧∈ℬ₁,  𝑤∈ℬ₂.

Proof. By (56) in the preceding lemma, one sees that 𝑠^∗_𝜁𝑠_𝜉=𝜙₁𝜁∣𝜉_ℬ1,𝑡^∗_𝜁𝑡_𝜉=𝜙₂𝜁∣𝜉_ℬ2,𝜁,𝜉∈ℋ. (61) 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, 𝜙₁(ℬ₁), 𝜙₂(ℬ₂) are contained in 𝒯_𝐹(ℋ).

Lemma 12. There exists an action 𝛾 of ℝ/ℤ=𝕋 on 𝒯_𝐹(ℋ) such that 𝛾_𝑟𝑠_𝜉=𝑒^{√2𝜋−1𝑟}𝑠_𝜉,𝛾_𝑟𝑡_𝜉=𝑒^{√2𝜋−1𝑟}𝑡_𝜉𝛾,𝜉∈ℋ,_𝑟𝜙₁(𝑧)=𝜙₁(𝑧),𝑧∈ℬ₁,𝛾_𝑟𝜙₂(𝑤)=𝜙₂(𝑤),𝑤∈ℬ₂ (62) for 𝑟∈ℝ/ℤ=𝕋.

Proof. We will first define a one-parameter unitary group 𝑢_𝑟,  𝑟∈ℝ/ℤ=𝕋 on 𝐹(ℋ) with respect to the right 𝒜-valued inner product as in the following way.
For 𝑛=0∶𝑢_𝑟∶𝐹₀(ℋ)→𝐹₀(ℋ) is defined by 𝑢_𝑟𝑏₁⊕𝑏₂=𝑏₁⊕𝑏₂for𝑏₁⊕𝑏₂∈ℬ₁⊕ℬ₂. (63) For 𝑛=1,2,…∶𝑢_𝑟∶𝐹_𝑛(ℋ)→𝐹_𝑛(ℋ) is defined by 𝑢_𝑟𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝑒^{√2𝜋−1𝑛𝑟}𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛 (64) for 𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛∈𝐹_𝑛(ℋ). We therefore have a one-parameter unitary group 𝑢_𝑟 on 𝐹(ℋ). We then define an automorphism 𝛾_𝑟 on ℒ_𝒜(𝐹(ℋ)) for 𝑟∈ℝ/ℤ by 𝛾_𝑟(𝑇)=𝑢_𝑟𝑇𝑢^∗_𝑟 (65) for  𝑇∈ℒ_𝒜(𝐹(ℋ)),  𝑟∈ℝ/ℤ.
It then follows that for 𝑏₁⊕𝑏₂∈ℬ₁⊕ℬ₂𝛾_𝑟𝑠_𝜉𝑏₁⊕𝑏₂=𝑢_𝑟𝑠_𝜉𝑢^∗_𝑟𝑏₁⊕𝑏₂=𝑢_𝑟𝜉𝜑₁𝑏₁=𝑒^{√2𝜋−1𝑟}𝑠_𝜉𝑏₁⊕𝑏₂, (66) and for 𝜉₁⊗_𝑖1⋯⊗_𝑖𝑛−1𝜉_𝑛∈𝐹_𝑛(ℋ), 𝑛=1,2,…, 𝛾_𝑟𝑠_𝜉𝜉₁⊗_𝑖1⋯⊗_𝑖𝑛−1𝜉_𝑛=𝑒^{√−2𝜋−1𝑛𝑟}𝑢_𝑟𝜉⊗₁𝜉₁⊗_𝑖1⋯⊗_𝑖𝑛−1𝜉_𝑛=𝑒^{√2𝜋−1𝑟}𝑠_𝜉𝜉₁⊗_𝑖1⋯⊗_𝑖𝑛−1𝜉_𝑛. (67) Therefore, we conclude that 𝛾_𝑟(𝑠_𝜉)=𝑒^{√2𝜋−1𝑟}𝑠_𝜉 on 𝐹(ℋ) and similarly 𝛾_𝑟(𝑡_𝜉)=𝑒^{√2𝜋−1𝑟}𝑡_𝜉 on 𝐹(ℋ). It is direct to see that 𝛾_𝑟𝜙₁(𝑧)=𝜙₁(𝑧),𝛾_𝑟𝜙₂(𝑤)=𝜙₂(𝑤),for𝑧∈ℬ₁,𝑤∈ℬ₂. (68) It is also obvious that 𝛾_𝑟(𝒯_𝐹(ℋ))=𝒯_𝐹(ℋ) for 𝑟∈ℝ/ℤ.

Denote by 𝐽(ℋ) the 𝐶^∗-subalgebra of ℒ_𝒜(𝐹(ℋ)) generated by the elements ℒ_𝒜^finite_𝑛=0𝐹_𝑛(ℋ). (69) The algebra 𝐽(ℋ) is a closed two-sided ideal of ℒ_𝒜(𝐹(ℋ)).

Definition 13. The 𝐶^∗-algebra 𝒪_𝐹(ℋ) associated with the Hilbert 𝐶^∗-quad module ℋ of general type is defined by the quotient 𝐶^∗-algebra of 𝒯_𝐹(ℋ) by the ideal 𝒯_𝐹(ℋ)∩𝐽(ℋ).
We denote by [𝑥] the quotient image of an element 𝑥∈𝒯_𝐹(ℋ) under the ideal 𝒯_𝐹(ℋ)∩𝐽(ℋ). We set the elements of 𝒪_𝐹(ℋ)𝚂_𝜉=𝑠_𝜉,𝚃_𝜉=𝑡_𝜉,Φ₁(𝑧)=𝜙₁(𝑧),Φ₂(𝑤)=𝜙₂(𝑤) (70) for 𝜉∈ℋ and 𝑧∈ℬ₁, 𝑤∈ℬ₂. By the preceding lemmas, we have the following.

Proposition 14. The 𝐶^∗-algebra 𝒪_𝐹(ℋ) is generated by the family of operators 𝚂_𝜉, 𝚃_𝜉 for 𝜉∈ℋ. It contains the operators Φ₁(𝑧),Φ₂(𝑤) for 𝑧∈ℬ₁, 𝑤∈ℬ₂. They satisfy the following equalities, 𝚂_{𝑐𝜉+𝑑𝜁}=𝑐𝚂_𝜉+𝑑𝚂_𝜁,𝚃_{𝑐𝜉+𝑑𝜁}=𝑐𝚃_𝜉+𝑑𝚃_𝜁,𝚂_{𝜙1(𝑧′)𝜉𝜑1(𝑧)}=Φ₁𝑧^′𝚂_𝜉Φ₁(𝑧),𝚃_{𝜙1(𝑧′)𝜉𝜑2(𝑤)}=Φ₁𝑧^′𝚃_𝜉Φ₂(𝑤),𝚂_{𝜙2(𝑤′)𝜉𝜑1(𝑧)}=Φ₂𝑤^′𝚂_𝜉Φ₁(𝑧),𝚃_{𝜙2(𝑤′)𝜉𝜑2(𝑤)}=Φ₂𝑤^′𝚃_𝜉Φ₂(𝑤),𝚂^∗_𝜁𝚂_𝜉=Φ₁⟨𝜁∣𝜉⟩_ℬ1,𝚃^∗_𝜁𝚃_𝜉=Φ₂⟨𝜁∣𝜉⟩_ℬ2 (71)(72)(73)(74) for 𝜉,𝜁∈ℋ, 𝑐,𝑑∈𝐶 and 𝑧,𝑧^′∈ℬ₁,  𝑤,𝑤^′∈ℬ₂.

4. The 𝐶^∗-Algebras of Hilbert 𝐶^∗-Quad Modules of Finite Type

In what follows, we assume that a Hilbert 𝐶^∗-quad module ℋ is of finite type. In this section, we will study the 𝐶^∗-algebra 𝒪_𝐹(ℋ) for a Hilbert 𝐶^∗-quad module ℋ of finite type. Let {𝑢₁,…,𝑢_𝑀} be a finite basis of ℋ as a Hilbert 𝐶^∗-right module over ℬ₁ and {𝑣₁,…,𝑣_𝑁} a finite basis of ℋ as a Hilbert 𝐶^∗-right module over ℬ₂. Keep the notations as in the previous section. We set 𝑠_𝑖=𝑠_𝑢𝑖for𝑡𝑖=1,…,𝑀,_𝑘=𝑡_𝑣𝑘for𝑘=1,…,𝑁. (75) By (10) and Lemma 10, we have for 𝜉∈ℋ𝑠_𝜉=^𝑀_𝑖=1𝑠_𝑖𝜙₁𝑢_𝑖∣𝜉_ℬ1,𝑡_𝜉=^𝑁_𝑘=1𝑡_𝑘𝜙₂𝑣_𝑘∣𝜉_ℬ2. (76) Let 𝑃_𝑛 be the projection on 𝐹(ℋ) onto 𝐹_𝑛(ℋ) for 𝑛=0,1,… so that ∑^∞_𝑛=0𝑃_𝑛=1 on 𝐹(ℋ).

Lemma 15. For 𝜉,𝜁∈ℋ, one has(i)𝑠^∗_𝜉𝑡_𝜁𝑃_𝑛=0 for 𝑛=1,2,… and hence 𝑠^∗_𝜉𝑡_𝜁=𝑠^∗_𝜉𝑡_𝜁𝑃₀.(ii)𝑡^∗_𝜁𝑠_𝜉𝑃_𝑛=0 for 𝑛=1,2,… and hence 𝑡^∗_𝜁𝑠_𝜉=𝑡^∗_𝜁𝑠_𝜉𝑃₀.

Proof. (i) For 𝑛=1,2,…, we have 𝑠^∗_𝜉𝑡_𝜁𝜉₁⊗_𝑖1⋯⊗_𝑖𝑛−1𝜉_𝑛=𝑠^∗_𝜉𝜁⊗₂𝜉₁⊗_𝑖1⋯⊗_𝑖𝑛−1𝜉_𝑛=0. (77) (ii) Is similar to (i).

Define two projections on 𝐹(ℋ) by 𝑃_𝑠=Theprojectiononto_{∞𝑛=0𝑖1,…,𝑖𝑛∈Γ𝑛}ℋ⊗₁ℋ⊗_𝑖1ℋ⊗_𝑖2⋯⊗_𝑖𝑛𝑃ℋ,_𝑡=Theprojectiononto_{∞𝑛=0𝑖1,…,𝑖𝑛∈Γ𝑛}ℋ⊗₂ℋ⊗_𝑖1ℋ⊗_𝑖2⋯⊗_𝑖𝑛ℋ. (78)

Lemma 16. Keep the above notations. ^𝑀_𝑖=1𝑠_𝑖𝑠^∗_𝑖=𝑃₁+𝑃_𝑠,^𝑁_𝑘=1𝑡_𝑘𝑡^∗_𝑘=𝑃₁+𝑃_𝑡. (79) Hence, ^𝑀_𝑖=1𝑠_𝑖𝑠^∗_𝑖+^𝑁_𝑘=1𝑡_𝑘𝑡^∗_𝑘+𝑃₀=1_𝐹(ℋ)+𝑃₁. (80)

Proof. For 𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛∈𝐹_𝑛(ℋ) with 2≤𝑛∈𝑁, we have 𝑠_𝑖𝑠^∗_𝑖𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝑢_𝑖⊗₁𝜙₁𝑢_𝑖∣𝜉_1𝜂𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛if𝑖₁0=1,if𝑖₁=2. (81) As 𝑢_𝑖⊗₁𝜙₁(⟨𝑢_𝑖∣𝜉₁⟩_ℬ1)𝜉₂=𝑢_𝑖𝜑₁(⟨𝑢_𝑖∣𝜉₁⟩_ℬ1)⊗₁𝜉₂, and ∑^𝑀_𝑖=1𝑢_𝑖𝜑₁(⟨𝑢_𝑖∣𝜉₁⟩_ℬ1)=𝜉₁, we have ^𝑀_𝑖=1𝑠_𝑖𝑠^∗_𝑖𝜉₁⊗_𝑖1𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛=𝜉₁⊗₁𝜉₂⊗_𝑖2⋯⊗_𝑖𝑛−1𝜉_𝑛if𝑖₁0=1,if𝑖₁=2 (82) and hence ^𝑀_𝑖=1𝑠_𝑖𝑠^∗_{𝑖⊕∞𝑛=2𝐹𝑛(ℋ)}=𝑃_{𝑠⊕∞𝑛=2𝐹𝑛(ℋ)}. (83) For 𝜉∈𝐹₁(ℋ)=ℋ, we have 𝑠_𝑖𝑠^∗_𝑖𝜉=𝑠_𝑖(⟨𝑢_𝑖∣𝜉⟩_ℬ1⊕0)=𝑢_𝑖𝜑₁(⟨𝑢_𝑖∣𝜉⟩_ℬ1) so that ^𝑀_𝑖=1𝑠_𝑖𝑠^∗_𝑖𝜉=^𝑀_𝑖=1𝑢_𝑖𝜑₁𝑢_𝑖∣𝜉_ℬ1=𝜉 (84) and hence ^𝑀_𝑖=1𝑠_𝑖𝑠^∗_{𝑖𝐹1(ℋ)}=1_𝐹1(ℋ). (85) As 𝑠_𝑖𝑠^∗_𝑖(𝑏₁⊕𝑏₂)=0 for 𝑏₁⊕𝑏₂∈ℬ₁⊕ℬ₂, we have ^𝑀_𝑖=1𝑠_𝑖𝑠^∗_{𝑖𝐹0(ℋ)}=0. (86) Therefore, we conclude that ^𝑀_𝑖=1𝑠_𝑖𝑠^∗_𝑖=𝑃_𝑠+𝑃₁andsimilarly^𝑁_𝑘=1𝑡_𝑘𝑡^∗_𝑘=𝑃_𝑡+𝑃₁. (87) As 𝑃_𝑠+𝑃_𝑡+𝑃₀+𝑃₁=1_𝐹(ℋ), one obtains (80).

We set the operators 𝚂_𝑖=𝚂_𝑢𝑖=𝑠_𝑖for𝚃𝑖=1,…,𝑀,_𝑘=𝚃_𝑣𝑘=𝑡_𝑖for𝑘=1,…,𝑁 (88) in the 𝐶^∗-algebra 𝒪_𝐹(ℋ). As two operators ∑^𝑀_𝑖=1𝑠_𝑖𝑠^∗_𝑖 and ∑^𝑁_𝑘=1𝑡_𝑘𝑡^∗_𝑘 are projections by (79), so are ∑^𝑀_𝑖=1𝚂_𝑖𝚂^∗_𝑖 and ∑^𝑁_𝑘=1𝚃_𝑘𝚃^∗_𝑘. Since 𝑃₁−𝑃₀∈𝐽(ℋ), the identity (80) implies ^𝑀_𝑖=1𝚂_𝑖𝚂^∗_𝑖+^𝑁_𝑘=1𝚃_𝑘𝚃^∗_𝑘=1. (89) Therefore, we have the following.

Theorem 17. Let ℋ be a Hilbert 𝐶^∗-quad module over (𝒜;ℬ₁,ℬ₂) of finite type with finite basis {𝑢₁,…,𝑢_𝑀} as a right ℬ₁-module and {𝑣₁,…,𝑣_𝑁} as a right ℬ₂-module. Then, one has the following. (i)The 𝐶^∗-algebra 𝒪_𝐹(ℋ) is generated by the operators 𝚂₁,…,𝚂_𝑀,𝚃₁,…,𝚃_𝑁 and the elements Φ₁(𝑧),Φ₂(𝑤) for 𝑧∈ℬ₁,  𝑤∈ℬ₂.(ii)They satisfy the following operator relations: ^𝑀_𝑖=1𝚂_𝑖𝚂^∗_𝑖+^𝑁_𝑘=1𝚃_𝑘𝚃^∗_𝑘=1,𝚂^∗_𝑗𝚃_𝑙=0,𝚂^∗_𝑖𝚂_𝑗=Φ₁𝑢_𝑖∣𝑢_𝑗ℬ1,𝚃^∗_𝑘𝚃_𝑙=Φ₂𝑣_𝑘∣𝑣_𝑙ℬ2,Φ₁(𝑧)𝚂_𝑗=^𝑀_𝑖=1𝚂_𝑖Φ₁𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1,Φ₁(𝑧)𝚃_𝑙=^𝑁_𝑘=1𝚃_𝑘Φ₂𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2,Φ₂(𝑤)𝚂_𝑗=^𝑀_𝑖=1𝚂_𝑖Φ₁𝑢_𝑖∣𝜙₂(𝑤)𝑢_𝑗ℬ1,Φ₂(𝑤)𝚃_𝑙=^𝑁_𝑘=1𝚃_𝑘Φ₂𝑣_𝑘∣𝜙₂(𝑤)𝑣_𝑙ℬ2 (90)(91)(92)(93) for 𝑧∈ℬ₁,  𝑤∈ℬ₂,  𝑖,𝑗=1,…,𝑀,  𝑘,𝑙=1,…,𝑁.(iii)There exists an action 𝛾 of  ℝ/ℤ=𝕋 on 𝒪_𝐹(ℋ) such that 𝛾_𝑟𝚂_𝑖=𝑒^{√2𝜋−1𝑟}𝚂_𝑖,𝛾_𝑟𝚃_𝑘=𝑒^{√2𝜋−1𝑟}𝚃_𝑘,𝛾_𝑟Φ₁(𝑧)=Φ₁(𝑧),𝛾_𝑟Φ₂(𝑤)=Φ₂(𝑤) (94) for 𝑟∈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)𝑢_𝑗=∑^𝑀_𝑖=1𝑢_𝑖𝜑₁(⟨𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗⟩_ℬ1) so that 𝜙₁(𝑧)𝑠_𝑗=𝑠_{𝜙1(𝑧)𝑢𝑗}=^𝑀_𝑖=1𝑠_𝑢𝑖𝜙₁𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1 (95) which goes to the first equality of (92). The other equalities of (92) and (93) are similarly shown.
(iii) The assertion is direct from Lemma 12.

The action 𝛾 of 𝕋 on 𝒪_𝐹(ℋ) defined in the above theorem (iii) is called the gauge action.

5. The Universal 𝐶^∗-Algebras Associated with Hilbert 𝐶^∗-Quad Modules

In this section, we will prove that the 𝐶^∗-algebra 𝒪_𝐹(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,…,𝑢_𝑀} as a right Hilbert ℬ₁-module and {𝑣₁,…,𝑣_𝑁} as a right Hilbert ℬ₂-module as in the previous section.

Let 𝒫_ℋ be the universal ∗-algebra generated by operators 𝑆₁,…,𝑆_𝑀,  𝑇₁,…,𝑇_𝑁 and elements 𝑧∈ℬ₁, 𝑤∈ℬ₂ subject to the relations ^𝑀_𝑖=1𝑆_𝑖𝑆^∗_𝑖+^𝑁_𝑘=1𝑇_𝑘𝑇^∗_𝑘=1,𝑆^∗_𝑗𝑇_𝑙=0,𝑆^∗_𝑖𝑆_𝑗=𝑢_𝑖∣𝑢_𝑗ℬ1,𝑇^∗_𝑘𝑇_𝑙=𝑣_𝑘∣𝑣_𝑙ℬ2,𝑧𝑆_𝑗=^𝑀_𝑖=1𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1,𝑧𝑇_𝑙=^𝑁_𝑘=1𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2,𝑤𝑆_𝑗=^𝑀_𝑖=1𝑆_𝑖𝑢_𝑖∣𝜙₂(𝑤)𝑢_𝑗ℬ1,𝑤𝑇_𝑙=^𝑁_𝑘=1𝑇_𝑘𝑣_𝑘∣𝜙₂(𝑤)𝑣_𝑙ℬ2 (96)(97)(98)(99) 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,…,𝑆_𝑀,  𝑇₁,…,𝑇_𝑁 satisfying the relations (ℋ).

Lemma 18. The sums ∑^𝑀_𝑖=1𝑆_𝑖𝑆^∗_𝑖 and ∑^𝑁_𝑘=1𝑇_𝑘𝑇^∗_𝑘 are both projections.

Proof. Put 𝑃=∑^𝑀_𝑖=1𝑆_𝑖𝑆^∗_𝑖 and 𝑄=∑^𝑁_𝑘=1𝑇_𝑘𝑇^∗_𝑘. 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)For 𝑖,𝑗=1,…,𝑀 and 𝑧∈ℬ₁, 𝑤∈ℬ₂, one has 𝑆^∗_𝑖𝑧𝑆_𝑗=𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1,𝑆^∗_𝑖𝑤𝑆_𝑗=𝑢_𝑖∣𝜙₂(𝑤)𝑢_𝑗ℬ1. (100)(ii)For 𝑘,𝑙=1,…,𝑁 and 𝑧∈ℬ₁, 𝑤∈ℬ₂, one has 𝑇^∗_𝑘𝑧𝑇_𝑙=𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2,𝑇^∗_𝑘𝑤𝑇_𝑙=𝑣_𝑘∣𝜙₂(𝑤)𝑣_𝑙ℬ2. (101)

Proof. (i) By (98), we have 𝑆^∗_𝑖𝑧𝑆_𝑗=^𝑀_ℎ=1𝑆^∗_𝑖𝑆_ℎ𝑢_ℎ∣𝜙₁(𝑧)𝑢_𝑗ℬ1=^𝑀_ℎ=1𝑢_𝑖∣𝑢_ℎℬ1𝑢_ℎ∣𝜙₁(𝑧)𝑢_𝑗ℬ1=𝑢_𝑖∣^𝑀_ℎ=1𝑢_ℎ𝑢_ℎ∣𝜙₁(𝑧)𝑢_𝑗ℬ1ℬ1=𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1. (102)  The other equality 𝑆^∗_𝑖𝑤𝑆_𝑗=⟨𝑢_𝑖∣𝜙₂(𝑤)𝑢_𝑗⟩_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)For 𝑤∈ℬ₂, 𝑗=1,…𝑀, the element 𝑆^∗_𝑗𝑤𝑆_𝑗 belongs to 𝒜 and the formula holds: ^𝑀_𝑗=1𝑆^∗_𝑗𝜉∣𝜂_ℬ2𝑆_𝑗=𝜉∣𝜂_𝒜𝑓𝑜𝑟𝜉,𝜂∈ℋ. (103)(ii)For 𝑧∈ℬ₁, 𝑙=1,…𝑁, the element 𝑇^∗_𝑙𝑧𝑇_𝑙 belongs to 𝒜 and the formula holds: ^𝑁_𝑙=1𝑇^∗_𝑙𝜉∣𝜂_ℬ1𝑇_𝑙=𝜉∣𝜂_𝒜𝑓𝑜𝑟𝜉,𝜂∈ℋ. (104)

Lemma 21. The following equalities for 𝑧∈ℬ₁ and 𝑤∈ℬ₂ hold: (i)𝑧=^𝑀_𝑖,𝑗=1𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1𝑆^∗_𝑗+^𝑁_𝑘,𝑙=1𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2𝑇^∗_𝑙,𝑤=^𝑀_𝑖,𝑗=1𝑆_𝑖𝑢_𝑖∣𝜙₂(𝑤)𝑢_𝑗ℬ1𝑆^∗_𝑗+^𝑁_𝑘,𝑙=1𝑇_𝑘𝑣_𝑘∣𝜙₂(𝑤)𝑣_𝑙ℬ2𝑇^∗_𝑙. (105)(106)(ii)𝑧^∗=^𝑀_𝑖,𝑗=1𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)^∗𝑢_𝑗ℬ1𝑆^∗_𝑗+^𝑁_𝑘,𝑙=1𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)^∗𝑣_𝑙ℬ2𝑇^∗_𝑙,𝑤^∗=^𝑀_𝑖,𝑗=1𝑆_𝑖𝑢_𝑖∣𝜙₂(𝑤)^∗𝑢_𝑗ℬ1𝑆^∗_𝑗+^𝑁_𝑘,𝑙=1𝑇_𝑘𝑣_𝑘∣𝜙₂(𝑤)^∗𝑣_𝑙ℬ2𝑇^∗_𝑙. (107)(108)(iii)𝑧𝑤=^𝑀_𝑖,𝑗=1𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)𝜙₂(𝑤)𝑢_𝑗ℬ1𝑆^∗_𝑗+^𝑁_𝑘,𝑙=1𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)𝜙₂(𝑤)𝑣_𝑙ℬ2𝑇^∗_𝑙,𝑤𝑧=^𝑀_𝑖,𝑗=1𝑆_𝑖𝑢_𝑖∣𝜙₂(𝑤)𝜙₁(𝑧)𝑢_𝑗ℬ1𝑆^∗_𝑗+^𝑁_𝑘,𝑙=1𝑇_𝑘𝑣_𝑘∣𝜙₂(𝑤)𝜙₁(𝑧)𝑣_𝑙ℬ2𝑇^∗_𝑙. (109)

Proof. (i) By (98) and (99), we have 𝑧𝑆_𝑗𝑆^∗_𝑗=^𝑀_𝑖=1𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1𝑆^∗_𝑗,𝑧𝑇_𝑙𝑇^∗_𝑙=^𝑁_𝑘=1𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2𝑇^∗_𝑙 (110) so that by (96) 𝑧=^𝑀_𝑖,𝑗=1𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1𝑆^∗_𝑗+^𝑁_𝑘,𝑙=1𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2𝑇^∗_𝑙. (111) Similarly, we have (106).
(ii) All the adjoints of 𝜙₁(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 𝑧𝑤=^𝑀_𝑖,𝑗=1𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1𝑆^∗_𝑗+^𝑁_𝑘,𝑙=1𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2𝑇^∗_𝑙⋅^𝑀_𝑔,ℎ=1𝑆_𝑔𝑢_𝑔∣𝜙₂(𝑤)𝑢_ℎℬ1𝑆^∗_ℎ+^𝑁_𝑚,𝑛=1𝑇_𝑚𝑣_𝑚∣𝜙₂(𝑤)𝑣_𝑛ℬ2𝑇^∗_𝑛. (112) As 𝑆^∗_𝑗𝑇_𝑚=𝑇^∗_𝑙𝑆_𝑔=0 for any 𝑗,𝑔=1,…,𝑀,  𝑙,𝑚=1,…,𝑁, it follows that 𝑧𝑤=^𝑀_{𝑖,𝑗,𝑔,ℎ=1}𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1×𝑆^∗_𝑗𝑆_𝑔𝑢_𝑔∣𝜙₂(𝑤)𝑢_ℎℬ1𝑆^∗_ℎ+^𝑁_{𝑘,𝑙,𝑚,𝑛=1}𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2×𝑇^∗_𝑙𝑇_𝑚𝑣_𝑚∣𝜙₂(𝑤)𝑣_𝑛ℬ2𝑇^∗_𝑛=^𝑀_{𝑖,𝑗,𝑔,ℎ=1}𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1×𝑢_𝑗∣𝑢_𝑔ℬ1𝑢_𝑔∣𝜙₂(𝑤)𝑢_ℎℬ1𝑆^∗_ℎ+^𝑁_{𝑘,𝑙,𝑚,𝑛=1}𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2×𝑣_𝑙∣𝑣_𝑚ℬ2𝑣_𝑚∣𝜙₂(𝑤)𝑣_𝑛ℬ2𝑇^∗_𝑛=^𝑀_{𝑖,𝑔,ℎ=1}𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑔ℬ1𝑢_𝑔∣𝜙₂(𝑤)𝑢_ℎℬ1𝑆^∗_ℎ+^𝑁_{𝑘,𝑚,𝑛=1}𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)^𝑁_𝑙=1𝑣_𝑙𝑣_𝑙∣𝑣_𝑚ℬ2ℬ2×𝑣_𝑚∣𝜙₂(𝑤)𝑣_𝑛ℬ2𝑇^∗_𝑛=^𝑀_{𝑖,𝑔,ℎ=1}𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑔ℬ1𝑢_𝑔∣𝜙₂(𝑤)𝑢_ℎℬ1𝑆^∗_ℎ+^𝑁_{𝑘,𝑚,𝑛=1}𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑚ℬ2𝑣_𝑚∣𝜙₂(𝑤)𝑣_𝑛ℬ2𝑇^∗_𝑛=^𝑀_𝑖,ℎ=1𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)𝜙₂(𝑤)𝑢_ℎℬ1𝑆^∗_ℎ+^𝑁_𝑘,𝑛=1𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)𝜙₂(𝑤)𝑣_𝑛ℬ2𝑇^∗_𝑛. (113)

Lemma 22. Let 𝑝(𝑧,𝑤) be a polynomial of elements of ℬ₁ and ℬ₂. Then, one has (i)𝑝(𝑧,𝑤)𝑆_𝑗=∑^𝑀_𝑖=1𝑆_𝑖𝑧_𝑖 for some 𝑧_𝑖∈ℬ₁.(ii)𝑝(𝑧,𝑤)𝑇_𝑙=∑^𝑁_𝑘=1𝑇_𝑘𝑤_𝑘 for some 𝑤_𝑘∈ℬ₂.

Proof. For 𝑧∈ℬ₁, 𝑤∈ℬ₂, and 𝑖,𝑗=1,…,𝑀, by putting 𝑧_𝑖,𝑗=⟨𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗⟩_ℬ1∈ℬ₁ and 𝑤_𝑖,𝑗=⟨𝑢_𝑖∣𝜙₂(𝑤)𝑢_𝑗⟩_ℬ1∈ℬ₁, the relations (98), (99) imply 𝑧𝑆_𝑗=^𝑀_𝑖=1𝑆_𝑖𝑧_𝑖,𝑗,𝑤𝑆_𝑗=^𝑀_𝑖=1𝑆_𝑖𝑤_𝑖,𝑗 (114) so that the assertion of (i) holds. (ii) is similar to (i).

Lemma 23. Let 𝑝(𝑧,𝑤) be a polynomial of elements of ℬ₁ and ℬ₂. Then, one has (i)𝑆^∗_𝑖𝑝(𝑧,𝑤)𝑆_𝑗 belongs to ℬ₁ for all 𝑖,𝑗=1,…,𝑀.(ii)𝑇^∗_𝑘𝑝(𝑧,𝑤)𝑇_𝑙 belongs to ℬ₂ for all 𝑘,𝑙=1,…,𝑁.(iii)𝑆^∗_𝑖𝑝(𝑧,𝑤)𝑇_𝑙=0 for all 𝑖=1,…,𝑀, 𝑙=1,…,𝑁.(iv)𝑇^∗_𝑘𝑝(𝑧,𝑤)𝑆_𝑗=0 for all 𝑘=1,…,𝑁, 𝑗=1,…,𝑀.

Proof. (i) By the previous lemma, we know 𝑝(𝑧,𝑤)𝑆_𝑗=^𝑀_ℎ=1𝑆_ℎ𝑧_ℎforsome𝑧_ℎ∈ℬ₁ (115) so that 𝑆^∗_𝑖𝑝(𝑧,𝑤)𝑆_𝑗=^𝑀_ℎ=1𝑆^∗_𝑖𝑆_ℎ𝑧_ℎ=^𝑀_ℎ=1𝑢_𝑖∣𝑢_ℎℬ1𝑧_ℎ. (116) As ⟨𝑢_𝑖∣𝑢_ℎ⟩_ℬ1𝑧_ℎ belongs to ℬ₁, we see the assertion.
(ii) Is similar to (i).
(iii) As 𝑇^∗_𝑘𝑆_𝑖=0, we have 𝑇^∗_𝑘𝑝(𝑧,𝑤)𝑆_𝑗=^𝑀_𝑖=1𝑇^∗_𝑘𝑆_𝑖𝑧_𝑖=0. (117)
(iv) Is similar to (i).

We set 𝑆_1,𝑖∶=𝑆_𝑖𝑆,𝑖=1,…,𝑀,_2,𝑘∶=𝑇_𝑘,𝑘=1,…,𝑁. (118) Put Σ₁=)},Σ{(1,𝑖∣𝑖=1,…,𝑀₂={(2,𝑘)∣𝑘=1,…,𝑁}. (119)

Lemma 24. Every element of 𝒫_ℋ can be written as a linear combination of elements of the form 𝑆_𝑔1,𝑖1𝑆_𝑔2,𝑖2⋯𝑆_{𝑔𝑚,𝑖𝑚}𝑏𝑆^∗_{ℎ𝑛,𝑗𝑛}⋯𝑆^∗_ℎ2,𝑗2𝑆^∗_ℎ1,𝑗1 (120) for some (𝑔₁,𝑖₁),  (𝑔₂,𝑖₂),…,(𝑔_𝑚,𝑖_𝑚),  (ℎ₁,𝑗₁),  (ℎ₂,𝑗₂),…,(ℎ_𝑛,𝑗_𝑛)∈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 of ℬ₁ and ℬ₂ on a Hilbert space 𝐻 together with operators 𝑆_𝑖,  𝑖=1,…,𝑀, 𝑇_𝑘,  𝑘=1,…,𝑁 satisfying the relations (ℋ) extends to a representation of 𝒫_ℋ on 𝐵(𝐻). We will endow 𝒫_ℋ 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 𝒫_ℋ because of the inequalities ‖𝑆_𝑖‖,‖𝑇_𝑘‖≤1, which come from (96). The completion of the algebra 𝒫_ℋ under the norm becomes a 𝐶^∗-algebra denoted by 𝒪_ℋ, which is called the universal 𝐶^∗-algebra subject to the relations (ℋ).

Denote by 𝐶^∗(𝜙₁(ℬ₁),𝜙₂(ℬ₂)) the 𝐶^∗-subalgebra of ℒ_𝒜(ℋ) generated by 𝜙₁(ℬ₁) and 𝜙₂(ℬ₂).

Lemma 25. An element 𝐿 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 𝐿𝜉𝜑_𝑖𝑏_𝑖=𝐿𝜉𝜑_𝑖𝑏_𝑖𝑓𝑜𝑟𝜉∈ℋ,𝑏_𝑖∈ℬ_𝑖 (121) hold.

Proof. Since both the operators 𝜙₁(𝑧) for 𝑧∈ℬ₁ and 𝜙₂(𝑤) for 𝑤∈ℬ₂ are right ℬ_𝑖-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 by ℬ_∘ the 𝐶^∗-subalgebra of 𝒪_ℋ generated by ℬ₁ and ℬ₂.

Lemma 26. The correspondence 𝑧,𝑤∈ℬ_∘⟶𝜙₁(𝑧),𝜙₂(𝑤)∈𝐶^∗𝜙₁ℬ₁,𝜙₂ℬ₂ (122) 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 𝜙₁∶𝑧∈ℬ₁⟶𝜙₁(𝑧)∈ℒ_𝒜(ℋ),𝜙₂∶𝑤∈ℬ₂⟶𝜙₂(𝑤)∈ℒ_𝒜(ℋ) (123) 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)∈𝒪_ℋ for 𝐿∈𝑃(𝜙₁(ℬ₁),𝜙₂(ℬ₂)) by 𝜋(𝐿)=^𝑀_𝑖,𝑗=1𝑆_𝑖𝑢_𝑖∣𝐿𝑢_𝑗ℬ1𝑆^∗_𝑗+^𝑁_𝑘,𝑙=1𝑇_𝑘𝑣_𝑘∣𝐿𝑣_𝑙ℬ2𝑇^∗_𝑙. (124) Let 𝑃_∘ be the ∗-subalgebra of 𝒫_ℋ 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 𝜋∶𝑃𝜙₁ℬ₁,𝜙₂ℬ₂⟶𝑃_∘⊂ℬ_∘ (125) yields a ∗-homomorphism. As 𝑆^∗_𝑖𝑇_𝑘=0 for 𝑖=1,…,𝑀,  𝑘=1,…,𝑁, we have ‖𝜋(𝐿)‖=Max^𝑀_𝑖,𝑗=1𝑆_𝑖𝑢_𝑖∣𝐿𝑢_𝑗ℬ1𝑆^∗_𝑗,_{𝑁𝑘,𝑙=1}𝑇_𝑘𝑣_𝑘∣𝐿𝑣_𝑙ℬ2𝑇^∗_𝑙. (126) We then have ^𝑀_𝑖,𝑗=1𝑆_𝑖𝑢_𝑖∣𝐿𝑢_𝑗ℬ1𝑆^∗_𝑗≤^𝑀_𝑖,𝑗=1𝑢_𝑖∣𝐿𝑢_𝑗ℬ1≤^𝑀_𝑖,𝑗=1𝑢_𝑖ℬ1𝑢_𝑗ℬ1‖𝐿‖ (127) and similarly ^𝑁_𝑘,𝑙=1𝑇_𝑘𝑣_𝑘∣𝐿𝑣_𝑙ℬ2𝑇^∗_𝑙≤^𝑁_𝑘,𝑙=1𝑣_𝑘ℬ2𝑣_𝑙ℬ2‖𝐿‖. (128) By putting 𝐶=Max{∑^𝑀_𝑖,𝑗=1‖𝑢_𝑖‖_ℬ1‖𝑢_𝑗‖_ℬ1,∑^𝑁_𝑘,𝑙=1‖𝑣_𝑘‖_ℬ2‖𝑣_𝑙‖_ℬ2}, one has ‖𝜋(𝐿)‖≤𝐶‖𝐿‖,∀𝐿∈𝑃𝜙₁ℬ₁,𝜙₂ℬ₂. (129) 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 that 𝜋∶𝐶^∗(𝜙₁(ℬ₁),𝜙₂(ℬ₂))→ℬ_∘ is injective. By (124), we have for 𝐿∈𝐶^∗(𝜙₁(ℬ₁),𝜙₂(ℬ₂)) and ℎ,ℎ^′=1,…,𝑀, 𝑆^∗_ℎ𝜋(𝐿)𝑆_ℎ′=^𝑀_𝑖,𝑗=1𝑆^∗_ℎ𝑆_𝑖𝑢_𝑖∣𝐿𝑢_𝑗ℬ1𝑆^∗_𝑗𝑆_ℎ′=^𝑀_𝑖,𝑗=1𝑢_ℎ∣𝑢_𝑖ℬ1𝑢_𝑖∣𝐿𝑢_𝑗ℬ1𝑢_𝑗∣𝑢_ℎ′ℬ1=^𝑀_𝑗=1𝑢_ℎ∣^𝑀_𝑖=1𝑢_𝑖𝑢_𝑖∣𝐿𝑢_𝑗ℬ1ℬ1𝑢_𝑗∣𝑢_ℎ′ℬ1=^𝑀_𝑗=1𝑢_ℎ∣𝐿𝑢_𝑗ℬ1𝑢_𝑗∣𝑢_ℎ′ℬ1=𝑢_ℎ∣𝐿𝑢_ℎ′ℬ1. (130) Suppose that 𝜋(𝐿)=0 so that ⟨𝑢_ℎ∣𝐿𝑢_ℎ′⟩_ℬ1=0. Since 𝐿𝑢_ℎ′=^𝑀_ℎ=1𝑢_ℎ𝑢_ℎ∣𝐿𝑢_ℎ′ℬ1, (131) we see that 𝐿𝑢_ℎ′=0 so that 𝐿=0. We thus conclude that 𝜋 is injective and hence isomorphic.

Denote by 𝜙_∘∶ℬ_∘→𝐶^∗(𝜙₁(ℬ₁),𝜙₂(ℬ₂)) the inverse 𝜋⁻¹ of the ∗-isomorphism 𝜋 given in the proof of the above lemma which satisfies 𝜙_∘(𝑧)=𝜙₁(𝑧)for𝑧∈ℬ₁,𝜙_∘(𝑤)=𝜙₂(𝑤)for𝑤∈ℬ₂. (132) We put ℱ⁰_ℋ=ℬ_∘. For 𝑛∈ℕ, we denote by ℱ^𝑛_ℋ the closed linear span of elements of the form 𝑆_𝑔1,𝑖1𝑆_𝑔2,𝑖2⋯𝑆_{𝑔𝑛,𝑖𝑛}𝑏𝑆^∗_{ℎ𝑛,𝑗𝑛}⋯𝑆^∗_ℎ2,𝑗2𝑆^∗_ℎ1,𝑗1 (133) for some (𝑔₁,𝑖₁),(𝑔₂,𝑖₂),…,(𝑔_𝑛,𝑖_𝑛),(ℎ₁,𝑗₁),(ℎ₂,𝑗₂),…,(ℎ_𝑛,𝑗_𝑛)∈Σ₁∪Σ₂ and 𝑏∈ℬ_∘. Let us denote by ℱ_ℋ the 𝐶^∗-subalgebra of 𝒪_ℋ generated by ∪^∞_𝑛=0ℱ^𝑛_ℋ. By the relations (105) and (106), we see the following.

Lemma 27. For 𝑥∈ℬ_∘, the following identity holds: 𝑥=^𝑀_𝑖,𝑗=1𝑆_𝑖𝑢_𝑖∣𝜙_∘(𝑥)𝑢_𝑗ℬ1𝑆^∗_𝑗+^𝑁_𝑘,𝑙=1𝑇_𝑘𝑣_𝑘∣𝜙_∘(𝑥)𝑣_𝑙ℬ2𝑇^∗_𝑙. (134)

Hence, by putting for 𝑏∈ℬ_∘𝑏_1,𝑖𝑗=𝑢_𝑖∣𝜙_∘(𝑏)𝑢_𝑗ℬ1𝑏,𝑖,𝑗=1,…,𝑀,_2,𝑘𝑙=𝑣_𝑘∣𝜙_∘(𝑏)𝑣_𝑙ℬ2,𝑘,𝑙=1,…,𝑁, (135) we have the following.

Lemma 28. For 𝑏∈ℬ_∘, the identity 𝑆_𝑔1,𝑖1𝑆_𝑔2,𝑖2⋯𝑆_{𝑔𝑛,𝑖𝑛}𝑏𝑆^∗_{ℎ𝑛,𝑗𝑛}⋯𝑆^∗_ℎ2,𝑗2𝑆^∗_ℎ1,𝑗1=^𝑀_𝑖,𝑗=1𝑆_𝑔1,𝑖1𝑆_𝑔2,𝑖2⋯𝑆_{𝑔𝑛,𝑖𝑛}𝑆_1,𝑖𝑏_1,𝑖𝑗𝑆^∗_1,𝑗𝑆^∗_{ℎ𝑛,𝑗𝑛}⋯𝑆^∗_ℎ2,𝑗2𝑆^∗_ℎ1,𝑗1+^𝑁_𝑘,𝑙=1𝑆_𝑔1,𝑖1𝑆_𝑔2,𝑖2⋯𝑆_{𝑔𝑛,𝑖𝑛}𝑆_2,𝑘𝑏_2,𝑘𝑙𝑆^∗_2,𝑙𝑆^∗_{ℎ𝑛,𝑗𝑛}⋯𝑆^∗_ℎ2,𝑗2𝑆^∗_ℎ1,𝑗1 (136) holds and induces an embedding of ℱ^𝑛_ℋ↪ℱ_ℋ^𝑛+1 for 𝑛∈ℤ₊.

Lemma 29. The 𝐶^∗-algebra ℱ_ℋ is the inductive limit lim_𝑛→∞ℱ^𝑛_ℋ of the sequence of the inclusions ℱ⁰_ℋ↪ℱ¹_ℋ↪ℱ²_ℋ↪⋯↪ℱ^𝑛_ℋ↪ℱ_ℋ^𝑛+1↪⋯↪ℱ_ℋ. (137)

Let 𝑒^{√2𝜋−1𝑟}∈𝕋 be a complex number of modulus one for 𝑟∈ℝ/ℤ. The elements 𝑒^{√2𝜋−1𝑟}𝑆_𝑖,𝑖=1,…,𝑀,𝑒^{√2𝜋−1𝑟}𝑇_𝑘,𝑘=1,…,𝑁,𝑧∈ℬ₁,𝑤∈ℬ₂ (138) in 𝒪_ℋ instead of 𝑆_𝑖,𝑖=1,…,𝑀,𝑇_𝑘,𝑘=1,…,𝑁,𝑧∈ℬ₁,𝑤∈ℬ₂ (139) satisfy the relations (ℋ). This implies the existence of an action on 𝒫_ℋ by automorphisms of the one-dimensional torus 𝕋 that acts on the generators by ℎ_𝑟𝑆_𝑖=𝑒^{√2𝜋−1𝑟}𝑆_𝑖,ℎ_𝑟𝑇_𝑘=𝑒^{√2𝜋−1𝑟}𝑇_𝑘,ℎ_𝑟(𝑧)=𝑧,ℎ_𝑟(𝑤)=𝑤 (140) for 𝑖=1,…,𝑀,  𝑘=1,…,𝑁,  𝑧∈ℬ₁,  𝑤∈ℬ₂, and 𝑟∈ℝ/ℤ=𝕋. As the 𝐶^∗-algebra 𝒪_ℋ has the largest norm on 𝒫_ℋ, the action (ℎ_𝑟)_𝑟∈𝕋 on 𝒫_ℋ extends to an action of 𝕋 on 𝒪_ℋ, still denoted by ℎ. The formula 𝑎∈𝒪_ℋ⟶_𝑟∈𝕋ℎ_𝑟(𝑎)𝑑𝑟∈𝒪_ℋ, (141) where 𝑑𝑟 is the normalized Lebesgue measure on 𝕋 defines a faithful conditional expectation denoted by ℰ_ℋ from 𝒪_ℋ onto the fixed-point algebra (𝒪_ℋ)^ℎ. The following lemma is routine.

Lemma 30. (𝒪_ℋ)^ℎ=ℱ_ℋ.

The 𝐶^∗-algebra 𝒪_ℋ 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 𝑆₁,…,𝑆_𝑀,  𝑇₁,…,𝑇_𝑁 in 𝒟 satisfying the relations ^𝑀_𝑖=1𝑆_𝑖𝑆^∗_𝑖+^𝑁_𝑘=1𝑇_𝑘𝑇^∗_𝑘=1,𝑆^∗_𝑗𝑇_𝑙𝑆=0,_∗𝑖𝑆_𝑗=Φ₁𝑢_𝑖∣𝑢_𝑗ℬ1,𝑇^∗_𝑘𝑇_𝑙=Φ₂𝑣_𝑘∣𝑣_𝑙ℬ2,Φ₁(𝑧)𝑆_𝑗=^𝑀_𝑖=1𝑆_𝑖Φ₁𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1,Φ₁(𝑧)𝑇_𝑙=^𝑁_𝑘=1𝑇_𝑘Φ₂𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2,Φ₂(𝑤)𝑆_𝑗=^𝑀_𝑖=1𝑆_𝑖Φ₁𝑢_𝑖∣𝜙₂(𝑤)𝑢_𝑗ℬ1,Φ₂(𝑤)𝑇_𝑙=^𝑁_𝑘=1𝑇_𝑘Φ₂𝑣_𝑘∣𝜙₂(𝑤)𝑣_𝑙ℬ2, (142) for 𝑧∈ℬ₁,  𝑤∈ℬ₂,  𝑖,𝑗=1,…,𝑀,  𝑘,𝑙=1,…,𝑁; then, there exists a unique ∗-homomorphism Φ∶𝒪_ℋ→𝒟 such that Φ𝑆_𝑖=𝑆_𝑖,Φ𝑇_𝑘=𝑇_𝑘,Φ(𝑧)=Φ₁(𝑧),Φ(𝑤)=Φ₂(𝑤) (143) for 𝑖=1,…,𝑀, 𝑘=1,…,𝑁 and 𝑧∈ℬ₁, 𝑤∈ℬ₂. We further assume that both the homomorphisms Φ_𝑖∶ℬ_𝑖→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 Lemma 26𝜙_∘∶𝑧,𝑤,∈ℬ_∘⟶𝜙₁(𝑧),𝜙₂(𝑤)∈𝐶^∗𝜙₁ℬ₁,𝜙₂ℬ₂ (144) yields an isomorphism of 𝐶^∗-algebras, it suffices to prove that the correspondence 𝜙₁(𝑧),𝜙₂(𝑤)∈𝐶^∗𝜙₁ℬ₁,𝜙₂ℬ₂⟶Φ₁(𝑧),Φ₂(𝑤)∈𝒟 (145) 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 𝜋(𝐿)=^𝑀_𝑖,𝑗=1𝑆_𝑖Φ₁𝑢_𝑖∣𝐿𝑢_𝑗ℬ1𝑆^∗_𝑗+^𝑁_𝑘,𝑙=1𝑇_𝑘Φ₂𝑣_𝑘∣𝐿𝑣_𝑙ℬ2𝑇^∗_𝑙∈𝒟. (146) 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 𝜋𝜙₁(𝑧)=^𝑀_𝑖,𝑗=1𝑆_𝑖Φ₁𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1𝑆^∗_𝑗+^𝑁_𝑘,𝑙=1𝑇_𝑘Φ₂𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2𝑇^∗_𝑙=Φ₁(𝑧), (147) 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^∗_ℎ𝜋(𝐿)𝑆_ℎ′=Φ₁(⟨𝑢_ℎ∣𝐿𝑢_ℎ′⟩_ℬ1) for all ℎ,ℎ^′=1,…,𝑀. Hence, the condition 𝜋(𝐿)=0 implies that Φ₁(⟨𝑢_ℎ∣𝐿𝑢_ℎ′⟩_ℬ1)=0. Since Φ₁ is injective, we have ⟨𝑢_ℎ∣𝐿𝑢_ℎ′⟩_ℬ1=0 for all ℎ,ℎ^′=1,…,𝑀. As 𝐿 is a right ℬ₁-module map, we have for 𝜉∈ℋ, 𝐿𝜉=^𝑀_ℎ′=1𝐿𝑢_ℎ′𝑢_ℎ′∣𝜉_ℬ1=^𝑀_ℎ′=1𝐿𝑢_ℎ′𝑢_ℎ′∣𝜉_ℬ1=0 (148) so that 𝐿=0. Therefore, 𝜋∶𝐶^∗(𝜙₁(ℬ₁),𝜙₂(ℬ₂))→𝒟 is injective. Hence, the composition Φ_∘∶𝜋∘𝜙_∘∶ℬ⟶_∘^𝜙∘𝐶^∗𝜙₁ℬ₁,𝜙₂ℬ₂𝜋⟶𝒟 (149) is injective.

We set 𝑆_1,𝑖∶=𝑆_𝑖𝑆,𝑖=1,…,𝑀,_2,𝑘∶=𝑇_𝑘,𝑘=1,…,𝑁. (150) We put ℱ⁰_ℋ=ℬ_∘. For 𝑛∈ℕ, let ℱ^𝑛_ℋ be the closed linear span in the 𝐶^∗-algebra 𝒪_ℋ of elements of the form 𝑆_𝑔1,𝑖1𝑆_𝑔2,𝑖2⋯𝑆_{𝑔𝑛,𝑖𝑛}Φ_∘(𝑏)𝑆^∗_{ℎ𝑛,𝑗𝑛}⋯𝑆^∗_ℎ2,𝑗2𝑆^∗_ℎ1,𝑗1 (151) for (𝑔₁,𝑖₁),(𝑔₂,𝑖₂),…,(𝑔_𝑛,𝑖_𝑛),(ℎ₁,𝑗₁),(ℎ₂,𝑗₂),…,(ℎ_𝑛,𝑗_𝑛)∈Σ₁∪Σ₂ and 𝑏∈ℬ_∘. Similar to the subalgebras ℱ^𝑛_ℋ,𝑛∈ℤ₊, of 𝒪_ℋ, one knows that the closed linear span ℱ^𝑛_ℋ is a 𝐶^∗-algebra and naturally regarded as a subalgebra of ℱ_ℋ^𝑛+1 for each 𝑛∈ℤ₊. Let us denote by ℱ_ℋ the 𝐶^∗-subalgebra of 𝒪_ℋ generated by ∪^∞_𝑛=0ℱ^𝑛_ℋ. Then, the 𝐶^∗-algebra ℱ_ℋ is the inductive limit lim_𝑛→∞ℱ^𝑛_ℋ of the sequence of the inclusions ℱ⁰_ℋ↪ℱ¹_ℋ↪ℱ²_ℋ↪⋯↪ℱ^𝑛_ℋ↪ℱ_ℋ^𝑛+1↪⋯↪ℱ_ℋ. (152)

Lemma 32. Suppose that both the ∗-homomorphisms Φ_𝑖∶ℬ_𝑖→𝒪_ℋ, 𝑖=1,2 are injective. Then, the restriction of Φ to the subalgebra ℱ_ℋ yields a ∗-isomorphism Φ|_ℱℋ∶ℱ_ℋ→ℱ_ℋ.

Proof. By the universality of 𝒪_ℋ, the restriction of Φ to ℱ_ℋ yields a surjective ∗-homomorphism Φ|_ℱℋ∶ℱ_ℋ→ℱ_ℋ. It suffices to show that Φ|_ℱℋ is injective. Suppose that Ker(Φ|_ℱℋ)≠{0} and put 𝐼=Ker(Φ|_ℱℋ). Since Φ_ℱ(ℱ^𝑛_ℋ)=ℱ^𝑛_ℋ and ℱ_ℋ=lim_𝑛→∞ℱ^𝑛_ℋ, there exists 𝑛∈ℤ₊ such that 𝐼∩ℱ^𝑛_ℋ≠0. Let us denote by Σ^𝑛 the set of 𝑛-tuples of Σ₁∪Σ₂: Σ^𝑛=𝜇₁,…,𝜇_𝑛∣𝜇₁,…,𝜇_𝑛∈Σ₁∪Σ₂. (153) For 𝜇=(𝜇₁,…,𝜇_𝑛)∈Σ^𝑛, denote by 𝑆_𝜇 the operator 𝑆_𝜇=𝑆_𝜇1𝑆_𝜇2⋯𝑆_𝜇𝑛, (154) where 𝑆_𝜇𝑚=𝑆_1,𝑖=𝑆_𝑖if𝜇_𝑚=(1,𝑖)∈Σ₁,𝑆_2,𝑘=𝑇_𝑘if𝜇_𝑚=(2,𝑘)∈Σ₂,𝑚=1,…,𝑛. (155) Any element of ℱ^𝑛_ℋ is of the form _{𝜇,𝜈∈Σ𝑛}𝑆_𝜇𝑏_𝜇,𝜈𝑆^∗_𝜈forsome𝑏_𝜇,𝜈∈ℬ_∘. (156) Hence, one may find a nonzero element ∑_{𝜇,𝜈∈Σ}^𝑛𝑆_𝜇𝑏_𝜇,𝜈𝑆^∗_𝜈∈𝐼∩ℱ^𝑛_ℋ. Since ∑^𝑀_𝑖=1𝑆_𝑖𝑆^∗_𝑖+∑^𝑁_𝑘=1𝑇_𝑘𝑇^∗_𝑘=1, the equality ∑_𝜇∈Σ^𝑛𝑆_𝜇𝑆^∗_𝜇=1 holds. For some 𝜔,𝛾∈Σ^𝑛, one then sees 0≠𝑆^∗_{𝜔𝜇,𝜈∈Σ𝑛}𝑆_𝜇𝑏_𝜇,𝜈𝑆^∗_𝜈𝑆_𝛾∈𝐼∩ℱ^𝑛_ℋ. (157) As 𝑆^∗_𝑖𝑇_𝑘=0 and 𝑆^∗_𝑖𝑆_𝑗=⟨𝑢_𝑖∣𝑢_𝑗⟩_ℬ1, 𝑇^∗_𝑘𝑇_𝑙=⟨𝑣_𝑘∣𝑣_𝑙⟩_ℬ2 for 𝑖,𝑗=1,…,𝑀,  𝑘,𝑙=1,…,𝑁, the element 𝑆^∗_𝜔(∑_{𝜇,𝜈∈Σ}^𝑛𝑆_𝜇𝑏_𝜇,𝜈𝑆^∗_𝜈)𝑆_𝛾 belongs to 𝐼∩ℬ_∘. By the preceding lemma, the homomorphism Φ_∘∶ℬ_∘→𝒪_ℋ is injective, so that we have Φ_∘(𝑆^∗_𝜔(∑_{𝜇,𝜈∈Σ}^𝑛𝑆_𝜇𝑏_𝜇,𝜈𝑆^∗_𝜈)𝑆_𝛾)≠0 a contradiction. Therefore we conclude that Φ|_ℱℋ∶ℱ_ℋ→ℱ_ℋ is injective and hence isomorphic.

The following theorem is one of the main results of the paper.

Theorem 33. Let 𝒟 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 𝑆₁,…,𝑆_𝑀,  𝑇₁,…,𝑇_𝑁 in 𝒟 satisfying the relations ^𝑀_𝑖=1𝑆_𝑖𝑆^∗_𝑖+^𝑁_𝑘=1𝑇_𝑘𝑇^∗_𝑘=1,𝑆^∗_𝑗𝑇_𝑙𝑆=0,_∗𝑖𝑆_𝑗=Φ₁𝑢_𝑖∣𝑢_𝑗ℬ1,𝑇^∗_𝑘𝑇_𝑙=Φ₂𝑣_𝑘∣𝑣_𝑙ℬ2,Φ₁(𝑧)𝑆_𝑗=^𝑀_𝑖=1𝑆_𝑖Φ₁𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1,Φ₁(𝑧)𝑇_𝑙=^𝑁_𝑘=1𝑇_𝑘Φ₂𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2,Φ₂(𝑤)𝑆_𝑗=^𝑀_𝑖=1𝑆_𝑖Φ₁𝑢_𝑖∣𝜙₂(𝑤)𝑢_𝑗ℬ1,Φ₂(𝑤)𝑇_𝑙=^𝑁_𝑘=1𝑇_𝑘Φ₂𝑣_𝑘∣𝜙₂(𝑤)𝑣_𝑙ℬ2 (158) 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 Φ_𝑖∶ℬ_𝑖→𝒜, 𝑖=1,2 are injective, then there exists a ∗-isomorphism Φ∶𝒪_ℋ→𝒪_ℋ satisfying Φ𝑆_𝑖=𝑆_𝑖,Φ𝑇_𝑘=𝑇_𝑘,Φ(𝑧)=Φ₁(𝑧),Φ(𝑤)=Φ₂(𝑤) (159) 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 ℰ_ℋ∶𝒪_ℋ⟶ℱ_ℋ (160) defined by ℰ_ℋ(𝑥)=_𝑟∈𝕋ℎ_𝑟(𝑥)𝑑𝑟for𝑥∈𝒪_ℋ. (161) By the universality of the algebra 𝒪_ℋ, there exists a surjective ∗-homomorphism Φ from 𝒪_ℋ to 𝒪_ℋ such that Φ𝑆_𝑖=𝑆_𝑖,Φ𝑇_𝑘=𝑇_𝑘,Φ(𝑧)=Φ₁(𝑧),Φ(𝑤)=Φ₂(𝑤) (162) for 𝑖,𝑗=1,…,𝑀,  𝑘,𝑙=1,…,𝑁,  𝑧∈ℬ₁,  𝑤∈ℬ₂. Then, Φ(ℱ_ℋ)=ℱ_ℋ and the following diagram (163) is commutative. Denote by Φ_∘ the restriction of Φ to the 𝐶^∗-subalgebra ℬ_∘ of 𝒪_ℋ generated by 𝑧∈ℬ₁,  𝑤∈ℬ₂. By assumption, both the maps Φ_𝑖∶ℬ_𝑖→𝒪_ℋ, 𝑖=1,2 are injective, so that Φ_∘∶ℬ_∘→𝒪_ℋ is injective by Lemma 31. By the preceding lemma, Φ|_ℱℋ∶ℱ_ℋ→ℱ_ℋ is an isomorphism. Since the conditional expectation ℰ_ℋ∶𝒪_ℋ→ℱ_ℋ is faithful, a routine argument shows that Φ is injective and hence isomorphic.

Therefore, we have the following.

Theorem 34. For a 𝐶^∗-quad module ℋ of finite type, the 𝐶^∗-algebra 𝒪_𝐹(ℋ) generated by the quotients [𝑠_𝜉],[𝑡_𝜉] of the creation operators 𝑠_𝜉,𝑡_𝜉 for 𝜉∈ℋ on the Fock spaces 𝐹(ℋ) is canonically isomorphic to the universal 𝐶^∗-algebra 𝒪_ℋ generated by operators 𝑆₁,…,𝑆_𝑀,  𝑇₁,…,𝑇_𝑁 and elements 𝑧∈ℬ₁, 𝑤∈ℬ₂ subject to the relations ^𝑀_𝑖=1𝑆_𝑖𝑆^∗_𝑖+^𝑁_𝑘=1𝑇_𝑘𝑇^∗_𝑘=1,𝑆^∗_𝑗𝑇_𝑙𝑆=0,_∗𝑖𝑆_𝑗=𝑢_𝑖∣𝑢_𝑗ℬ1,𝑇^∗_𝑘𝑇_𝑙=𝑣_𝑘∣𝑣_𝑙ℬ2,𝑧𝑆_𝑗=^𝑀_𝑖=1𝑆_𝑖𝑢_𝑖∣𝜙₁(𝑧)𝑢_𝑗ℬ1,𝑧𝑇_𝑙=^𝑁_𝑘=1𝑇_𝑘𝑣_𝑘∣𝜙₁(𝑧)𝑣_𝑙ℬ2,𝑤𝑆_𝑗=^𝑀_𝑖=1𝑆_𝑖𝑢_𝑖∣𝜙₂(𝑤)𝑢_𝑗ℬ1,𝑤𝑇_𝑙=^𝑁_𝑘=1𝑇_𝑘𝑣_𝑘∣𝜙₂(𝑤)𝑣_𝑙ℬ2 (164) for 𝑖,𝑗=1,…,𝑀,  𝑘,𝑙=1,…,𝑁 and 𝑧∈ℬ₁,  𝑤∈ℬ₂.

Proof. Theorem 17 implies that the operators 𝚂₁,…,𝚂_𝑀,  𝚃₁,…,𝚃_𝑁 and the elements Φ₁(𝑧), Φ₂(𝑤) for 𝑧∈ℬ₁, 𝑤∈ℬ₂ in 𝒪_𝐹(ℋ) satisfy the eight relations of Theorem 33. By Theorem 33, we see that the correspondences 𝑆_𝑖⟶𝚂_𝑖,𝑇_𝑘⟶𝚃_𝑘,𝑧∈ℬ₁⟶Φ₁(𝑧),𝑤∈ℬ₂⟶Φ₂(𝑤) (165) for 𝑖=1,…,𝑀, 𝑘=1,…,𝑁, and 𝑧∈ℬ₁, 𝑤∈ℬ₂ give rise to an isomorphism from 𝒪_ℋ to 𝒪_𝐹(ℋ).

The eight relations of the operators above are called the relations (ℋ). The above generating operators 𝑆₁,…,𝑆_𝑀 and 𝑇₁,…,𝑇_𝑁 of the universal 𝐶^∗-algebra 𝒪_ℋ correspond to two finite bases {𝑢₁,…,𝑢_𝑀} and {𝑣₁,…,𝑣_𝑁} of the Hilbert 𝐶^∗-quad module ℋ, respectively. On the other hand, the other 𝐶^∗-algebra 𝒪_𝐹(ℋ) is generated by the quotients of the creation operators 𝑠_𝜉,𝑡_𝜉 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 a 𝐶^∗-quad module ℋ of finite type, the universal 𝐶^∗-algebra 𝒪_ℋ generated by operators 𝑆₁,…,𝑆_𝑀,  𝑇₁,…,𝑇_𝑁 and elements 𝑧∈ℬ₁,𝑤∈ℬ₂ subject to the relations (ℋ) does not depend on the choice of the finite bases {𝑢₁,…,𝑢_𝑀} and {𝑣₁,…,𝑣_𝑁}.

6. 𝐾-Theory Formulae

Let ℋ 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 𝒪_𝐹(ℋ). By the previous section, the 𝐶^∗-algebra 𝒪_𝐹(ℋ) is regarded as the universal 𝐶^∗-algebra 𝒪_ℋ generated by the operators 𝑆₁,…,𝑆_𝑀 and 𝑇₁,…,𝑇_𝑁 and the elements 𝑧∈ℬ₁ and 𝑤∈ℬ₂ subject to the relations (ℋ). Let us denote by ℬ_∘ the 𝐶^∗-subalgebra of 𝒪_ℋ generated by elements 𝑧∈ℬ₁ and 𝑤∈ℬ₂. By Lemma 26, the correspondence 𝑧,𝑤∈ℬ_∘⟶𝜙₁(𝑧),𝜙₂(𝑤)∈𝐶^∗𝜙₁ℬ₁,𝜙₂ℬ₂⊂ℒ_𝒜(ℋ) (166) 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,…,𝑆_𝑀 and 𝑇₁,…,𝑇_𝑁 are partial isometries, and(ii)𝑆₁𝑆^∗₁,…,𝑆_𝑀𝑆^∗_𝑀,  𝑇₁𝑇^∗₁,…,𝑇_𝑁𝑇^∗_𝑁 commute with all elements of ℬ_∘. If the bases {𝑢₁,…,𝑢_𝑀} and {𝑣₁,…,𝑣_𝑁} satisfy the conditions 𝑢_𝑖∣𝑢_𝑗ℬ1=0for𝑣𝑖≠𝑗,_𝑘∣𝑣_𝑙ℬ2=0for𝑘≠𝑙, (167) the condition (i) holds. Furthermore, if 𝜙₁(𝑧) acts diagonally on {𝑢₁,…,𝑢_𝑀} for 𝑧∈ℬ₁ and 𝜙₂(𝑤) acts diagonally on {𝑣₁,…,𝑣_𝑁} for 𝑤∈ℬ₂, the condition (ii) holds. Recall that the gauge action is denoted by ℎ which is an action of 𝕋 on 𝒪_ℋ such that the fixed-point algebra (𝒪_ℋ)^ℎ under ℎ is canonically isomorphic to the 𝐶^∗-algebra ℱ_ℋ. Denote by ℎ the dual action of ℎ which is an action of ℤ=𝕋 on the 𝐶^∗-crossed product 𝒪_ℋ×_ℎ𝕋 by the gauge action ℎ of 𝕋. As in the argument of [16], 𝒪_ℋ×_ℎ𝕋 is stably isomorphic to ℱ_ℋ. Hence, we have that 𝐾_∗(𝒪_ℋ×_ℎ𝕋) is isomorphic to 𝐾_∗(ℱ_ℋ). The dual action ℎ induces an automorphism on the group 𝐾_∗(𝒪_ℋ×_ℎ𝕋) and hence on 𝐾_∗(ℱ_ℋ), which is denoted by 𝜎_∗. Then, by [16] (cf. [10, 17], etc.), we have the following.

Proposition 36. The following six-term exact sequence of 𝐾-theory holds: (168)

We put for 𝑥∈ℬ_∘𝜆_1,𝑖(𝑥)=𝑆^∗_𝑖𝑥𝑆_𝑖𝜆,𝑖=1,…,𝑀,_2,𝑘(𝑥)=𝑇^∗_𝑘𝑥𝑇_𝑘,𝑘=1,…,𝑁. (169) Both the families 𝜆_1,𝑖, 𝜆_2,𝑘 yield endomorphisms on ℬ_∘ which give rise to endomorphisms on the 𝐾-groups: 𝜆_1,𝑖∗∶𝐾₀ℬ_∘⟶𝐾₀ℬ_∘𝜆,𝑖=1,…,𝑀,_2,𝑘∗∶𝐾₀ℬ_∘⟶𝐾₀ℬ_∘,𝑘=1,…,𝑁. (170) We put 𝜆_∘=∑^𝑀_𝑖=1𝜆_1,𝑖∗+∑^𝑁_𝑘=1𝜆_2,𝑘∗ which is an endomorphism on 𝐾₀(ℬ_∘). Now, we further assume that 𝐾₁(ℱ_ℋ)={0}. It is routine to show that the groups Coker(id−𝜎_∗) in 𝐾₀(ℱ_ℋ) and Ker(id−𝜎_∗) in 𝐾₀(ℱ_ℋ) 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 have

Proposition 37. The following formulae hold: 𝐾₀𝒪_ℋ=𝐶𝑜𝑘𝑒𝑟𝑖𝑑−𝜆_∘𝑖𝑛𝐾₀ℬ_∘,𝐾₁𝒪_ℋ=𝐾𝑒𝑟𝑖𝑑−𝜆_∘𝑖𝑛𝐾₀ℬ_∘. (171)

7. Examples

In this section, we will study the 𝐶^∗-algebras 𝒪_ℋ for the Hilbert 𝐶^∗-quad modules presented in Examples in Section 2.

(1) Let 𝛼,𝛽 be automorphisms of a unital 𝐶^∗-algebra 𝒜 satisfying 𝛼∘𝛽=𝛽∘𝛼. Let ℋ_𝛼,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. The 𝐶^∗-algebra 𝒪_ℋ𝛼,𝛽 associated with the Hilbert 𝐶^∗-quad module ℋ_𝛼,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: 𝑈𝑈^∗+𝑉𝑉^∗=1,𝑈𝑈^∗𝑥=𝑥𝑈𝑈^∗,𝑉𝑉^∗𝑥=𝑥𝑉𝑉^∗,𝛼(𝑥)=𝑈^∗𝑥𝑈,𝛽(𝑥)=𝑉^∗𝑥𝑉 (172) for 𝑥∈𝒜.

(2) We fix natural numbers 1<𝑁,  𝑀∈ℕ. Consider finite-dimensional commutative 𝐶^∗-algebras 𝒜=ℂ,  ℬ₁=ℂ^𝑁, and  ℬ₂=ℂ^𝑀. 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 ℋ_𝑀,𝑁 the Hilbert 𝐶^∗-quad module ℂ^𝑀⊗ℂ^𝑁 over (ℂ;ℂ^𝑁,ℂ^𝑀) defined in (2) in Section 2. Put the finite bases 𝑢_𝑖=𝑒_𝑖⊗1∈ℋ_𝑀,𝑁,𝑖=1,…,𝑀asarightℬ₁-module,𝑣_𝑘=1⊗𝑓_𝑘∈ℋ_𝑀,𝑁,𝑘=1,…,𝑁asarightℬ₂-module. (173) We set Σ^∘={(𝑖,𝑘)∣1≤𝑖≤𝑀,1≤𝑘≤𝑁} and put 𝑒_(𝑖,𝑘)=𝑒_𝑖⊗𝑓_𝑘,  (𝑖,𝑘)∈Σ^∘ the standard basis of ℋ_𝑀,𝑁. Then, the 𝐶^∗-algebra ℬ_∘ on ℋ_𝑀,𝑁 generated by ℬ₁ and ℬ₂ is regarded as ℂ^𝑀⊗ℂ^𝑁=ℬ₂⊗ℬ₁. Hence, ℬ_∘=_{(𝑖,𝑘)∈Σ∘}ℂ𝑒_(𝑖,𝑘). (174)

Lemma 39. The 𝐶^∗-algebra 𝒪_ℋ𝑀,𝑁 is generated by operators 𝑆_𝑖,𝑇_𝑘,𝑒_(𝑖,𝑘),  𝑖=1,…,𝑀,  𝑘=1,…,𝑁 satisfying ^𝑀_𝑖=1𝑆_𝑖𝑆^∗_𝑖+^𝑁_𝑘=1𝑇_𝑘𝑇^∗_𝑘=1,𝑆^∗_𝑖𝑆_𝑗=𝛿_𝑖,𝑗,𝑇^∗_𝑘𝑇_𝑙=𝛿_𝑘,𝑙,𝑒_(𝑖,𝑘)𝑆_𝑗=𝛿^𝑀_{𝑖,𝑗ℎ=1}𝑆_𝑗𝑒_(ℎ,𝑘),𝑒_(𝑖,𝑘)𝑇_𝑙=𝛿^𝑁_{𝑘,𝑙𝑚=1}𝑇_𝑙𝑒_(𝑖,𝑚) (175)(176)(177) for 𝑖,𝑗=1,…,𝑀,  𝑘,𝑙=1,…,𝑁.

Proof. It suffices to show the equalities (177). We have 𝑒_(𝑖,𝑘)𝑆_𝑗=𝑆_𝑗𝑢_𝑗∣𝜙_∘(𝑒_(𝑖,𝑘))𝑢_𝑗ℬ1=𝑆_𝑗𝑒_𝑗⊗1∣(𝑒_𝑖⊗𝑓_𝑘)(𝑒_𝑗⊗1)_ℬ1=𝛿_𝑖,𝑗𝑆_𝑗1⊗𝑓_𝑘=𝛿_𝑖,𝑗𝑆_𝑗^𝑀_ℎ=1𝑒_(ℎ,𝑘). (178) The other equality of (177) is similarly shown.

Put 𝑆_(𝑖,𝑘)=𝑒_(𝑖,𝑘)𝑆_𝑖,𝑇_(𝑖,𝑘)=𝑒_(𝑖,𝑘)𝑇_𝑘for(𝑖,𝑘)∈Σ^∘. (179) Then we have the following.

Lemma 40. The following equalities hold: 𝑒_(𝑖,𝑘)=𝑆_(𝑖,𝑘)𝑆^∗_(𝑖,𝑘)+𝑇_(𝑖,𝑘)𝑇^∗_(𝑖,𝑘),𝑆_𝑖=^𝑁_𝑘=1𝑆_(𝑖,𝑘),𝑇_𝑘=^𝑀_𝑖=1𝑇_(𝑖,𝑘),_{(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)=^𝑀_𝑗=1𝑆_(𝑗,𝑘)𝑆^∗_(𝑗,𝑘)+𝑇_(𝑗,𝑘)𝑇^∗_(𝑗,𝑘),𝑇^∗_(𝑖,𝑘)𝑇_(𝑖,𝑘)=^𝑁_𝑙=1𝑆_(𝑖,𝑙)𝑆^∗_()𝑖,𝑙+𝑇_(𝑖,𝑙)𝑇^∗_()𝑖,𝑙 (180)(181)(182)(183)(184) for 𝑖=1,…,𝑀,  𝑘=1,…,𝑁 and (𝑖,𝑘)∈Σ^∘.

Proof. Since 𝑒_(𝑖,𝑘)𝑆_𝑗=𝛿_𝑖,𝑗𝑒_(𝑖,𝑘)𝑆_𝑖, we have 𝑆_(𝑖,𝑘)𝑆^∗_(𝑖,𝑘)=𝑒_(𝑖,𝑘)𝑆_𝑖𝑆^∗_𝑖𝑒_(𝑖,𝑘)=𝑒_(𝑖,𝑘)^𝑀_𝑗=1𝑆_𝑗𝑆^∗_𝑗𝑒_(𝑖,𝑘) (185) and similarly 𝑇_(𝑖,𝑘)𝑇^∗_(𝑖,𝑘)=𝑒_(𝑖,𝑘)(∑^𝑁_𝑙=1𝑇_𝑙𝑇^∗_𝑙)𝑒_(𝑖,𝑘). Hence, we have 𝑒_(𝑖,𝑘)=𝑒_(𝑖,𝑘)^𝑀_𝑗=1𝑆_𝑗𝑆^∗_𝑗+^𝑁_𝑙=1𝑇_𝑙𝑇^∗_𝑙𝑒_(𝑖,𝑘)=𝑆_(𝑖,𝑘)𝑆^∗_(𝑖,𝑘)+𝑇_(𝑖,𝑘)𝑇^∗_(𝑖,𝑘), (186) so that (180) holds. As 1=∑_{(𝑗,𝑘)∈Σ}^∘𝑒_(𝑗,𝑘), the equality (182) holds. Since 𝑒_(𝑗,𝑘)𝑆_𝑖=0 for 𝑗≠𝑖, we have 𝑆_𝑖=_{𝑗,𝑘∈Σ∘}𝑒_(𝑗,𝑘)𝑆_𝑖=^𝑁_𝑘=1𝑆_(𝑖,𝑘) (187) and similarly 𝑇_𝑘=∑^𝑀_𝑖=1𝑇_(𝑖,𝑘), so that (181) holds. By (177), it follows that 𝑆^∗_()𝑖,𝑘𝑆_(𝑖,𝑘)=𝑆^∗_𝑖^𝑀_𝑗=1𝑆_𝑖𝑒_(𝑗,𝑘)=^𝑀_𝑗=1𝑒_(𝑗,𝑘)=^𝑀_𝑗=1𝑆_(𝑗,𝑘)𝑆^∗_(𝑗,𝑘)+𝑇_(𝑗,𝑘)𝑇^∗_(𝑗,𝑘), (188) and similarly, we have 𝑇^∗_(𝑖,𝑘)𝑇_(𝑖,𝑘)=^𝑁_𝑙=1𝑆_(𝑖,𝑙)𝑆^∗_(𝑖,𝑙)+𝑇_(𝑖,𝑙)𝑇^∗_(𝑖,𝑙). (189)

Theorem 41. The 𝐶^∗-algebra 𝒪_ℋ𝑀,𝑁 associated with the Hilbert 𝐶^∗-quad module ℋ_𝑀,𝑁=ℂ^𝑀⊗ℂ^𝑁 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 _{(𝑖,𝑘)∈Σ∘}𝑆_(𝑖,𝑘)𝑆^∗_(𝑖,𝑘)+_{(𝑖,𝑘)∈Σ∘}𝑇_(𝑖,𝑘)𝑇^∗_(𝑖,𝑘)𝑆=1,_{∗(𝑖,𝑘)}𝑆_(𝑖,𝑘)=^𝑀_𝑗=1𝑆_(𝑗,𝑘)𝑆^∗_(𝑗,𝑘)+𝑇_(𝑗,𝑘)𝑇^∗_(𝑗,𝑘),𝑇^∗_(𝑖,𝑘)𝑇_(𝑖,𝑘)=^𝑁_𝑙=1𝑆_(𝑖,𝑙)𝑆^∗_(𝑖,𝑙)+𝑇_(𝑖,𝑙)𝑇^∗_(𝑖,𝑙) (190) for (𝑖,𝑘)∈Σ^∘.

Proof. By the preceding lemma, one knows that 𝑒_(𝑖,𝑘),𝑆_𝑖,𝑇_𝑘 are generated by the operators 𝑆_(𝑖,𝑘), 𝑇_(𝑖,𝑘) so that the algebra 𝒪_ℋ𝑀,𝑁 is generated by the partial isometries 𝑆_(𝑖,𝑘), 𝑇_(𝑖,𝑘), (𝑖,𝑘)∈Σ^∘.

Let 𝐼_𝑛 be the 𝑛×𝑛 identity matrix and 𝐸_𝑛 the 𝑛×𝑛 matrix whose entries are all 1s. For an 𝑀×𝑀-matrix 𝐶=[𝑐_𝑖,𝑗]^𝑀_𝑖,𝑗=1 and an 𝑁×𝑁-matrix 𝐷=[𝑑_𝑘,𝑙]^𝑁_𝑘,𝑙=1, denote by 𝐶⊗𝐷 the 𝑀𝑁×𝑀𝑁 matrix 𝑐𝐶⊗𝐷=¹¹𝐷𝑐₁₂𝐷⋯𝑐_1𝑀𝐷𝑐₂₁𝐷𝑐₂₂𝐷⋯𝑐_2𝑀𝐷𝑐⋮⋮⋱⋮_𝑀1𝐷𝑐_𝑀2𝐷⋯𝑐_𝑀𝑀𝐷 (191) so that 𝐸_𝑀⊗𝐼_𝑁=𝐼_𝑁𝐼_𝑁⋯𝐼_𝑁𝐼_𝑁𝐼_𝑁⋯𝐼_𝑁𝐼⋮⋮⋱⋮_𝑁𝐼_𝑁⋯𝐼_𝑁,𝐼_𝑀⊗𝐸_𝑁=𝐸_𝑁0⋯00𝐸_𝑁⋱⋮⋮⋱⋱00⋯0𝐸_𝑁. (192) The index set {(𝑖,𝑘)∣𝑖=1,…,𝑀,𝑘=1,…,𝑁} of the standard basis of ℂ^𝑀⊗ℂ^𝑁 is ordered lexicographically from left as in the following way: (1,1),…,(1,𝑁),(2,1),…,(2,𝑁),…,(𝑀,1),…,(𝑀,𝑁). (193) Put the 𝑀𝑁×𝑀𝑁 matrices 𝐴_𝑀,𝑁=𝐸_𝑀⊗𝐼_𝑁,𝐵_𝑀,𝑁=𝐼_𝑀⊗𝐸_𝑁 (194) and the 2𝑀𝑁×2𝑀𝑁 matrix 𝐻_𝑀,𝑁=𝐴_𝑀,𝑁𝐴_𝑀,𝑁𝐵_𝑀,𝑁𝐵_𝑀,𝑁. (195) Then, we have the following.

Theorem 42. The 𝐶^∗-algebra 𝒪_ℋ𝑀,𝑁 is isomorphic to the Cuntz-Krieger algebra 𝒪_{𝐻𝑀,𝑁} for the matrix 𝐻_𝑀,𝑁. The algebra 𝒪_{𝐻𝑀,𝑁} is simple and purely infinite and is isomorphic to the Cuntz-Krieger algebra 𝒪_{𝐴𝑀,𝑁+𝐵𝑀,𝑁} for the matrix 𝐴_𝑀,𝑁+𝐵_𝑀,𝑁.

Proof. By the preceding proposition, the 𝐶^∗-algebra 𝒪_ℋ𝑀,𝑁 is isomorphic to the Cuntz-Krieger algebra 𝒪_{𝐻𝑀,𝑁} for the matrix 𝐻_𝑀,𝑁. Since the matrix 𝐻_𝑀,𝑁 is aperiodic, the algebra is simple and purely infinite. The 𝑛th column of the matrix 𝐻_𝑀,𝑁 coincides with the (𝑛+𝑁)th column for every 𝑛=1,…,𝑀. One sees that the matrix 𝐴_𝑀,𝑁+𝐵_𝑀,𝑁 is obtained from 𝐻_𝑀,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  ℤ^𝑀𝑁/(𝐴_𝑀,𝑁+𝐵_𝑀,𝑁−𝐼_𝑀𝑁)ℤ^𝑀𝑁,Ker(𝐴_𝑀,𝑁+𝐵_𝑀,𝑁−𝐼_𝑀𝑁)inℤ^𝑀𝑁 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(𝒪_ℋ2,𝑁)=ℤ/(𝑁²−1)ℤ,𝐾₁(𝒪_ℋ2,𝑁)=0 (see [15] for details).

(3) For a 𝐶^∗-textile dynamical system (𝒜,𝜌,𝜂,Σ^𝜌,Σ^𝜂,𝜅), let ℋ_𝜅^𝜌,𝜂 be the 𝐶^∗-quad module over (𝒜;ℬ₁,ℬ₂) as in (3) in Section 2. The 𝐶^∗-algebra 𝒪_ℋ𝜅^𝜌,𝜂 has been studied in [12].

8. Higher-Dimensional Analogue

In this final section, we will state a generalization of Hilbert 𝐶^∗-quad modules to Hilbert modules with multi actions of 𝐶^∗-algebras.

Let 𝒜 be a unital 𝐶^∗-algebra and let ℬ₁,…,ℬ_𝑛 be 𝑛-family of unital 𝐶^∗-algebras. Suppose that there exists a unital embedding 𝜄_𝑖∶𝒜↪ℬ_𝑖 (196) for each 𝑖=1,…,𝑛. Suppose that there exists a right action 𝜓_𝑖 of 𝒜 on ℬ_𝑖 such that 𝑏_𝑖𝜓_𝑖(𝑎)∈ℬ_𝑖for𝑏_𝑖∈ℬ_𝑖,𝑎∈𝒜,𝑖=1,…,𝑛. (197) Hence, ℬ_𝑖 is a right 𝒜-module through 𝜓_𝑖 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 𝜙_𝒜 from 𝒜 to ℒ_𝒜(ℋ). It is called a Hilbert 𝐶^∗-multimodule over (𝒜;ℬ_𝑖,𝑖=1,…,𝑛) if ℋ has a multistructure of Hilbert 𝐶^∗-bimodules over ℬ_𝑖 for 𝑖=1,…,𝑛 such that for each 𝑖=1,…,𝑛 there exist a right action 𝜑_𝑖 of ℬ_𝑖 on ℋ and a left action 𝜙_𝑖 of ℬ_𝑖 on ℋ and a ℬ_𝑖-valued right inner product ⟨⋅∣⋅⟩_ℬ𝑖 such that 𝜙_𝑖(𝑧_𝑖)∈ℒ_𝒜(ℋ) and 𝜙_𝑖𝑧_𝑖𝜉𝜑_𝑗𝑤_𝑗=𝜙_𝑖𝑧_𝑖𝜉𝜑_𝑗𝑤_𝑗,𝜉𝜑_𝑗𝑧_𝑗𝜓_𝑗(𝑎)=𝜉𝜑_𝑗𝑧_𝑗𝑎 (198) for 𝜉∈ℋ, 𝑧_𝑖∈ℬ_𝑖, 𝑤_𝑗∈ℬ_𝑗, 𝑎∈𝒜, 𝑖,𝑗=1,…,𝑛 and 𝜙_𝒜(𝑎)=𝜙_𝑖𝜄_𝑖(𝑎),𝑎∈𝒜,𝑖=1,…,𝑛. (199) The operator 𝜙_𝑖(𝑧_𝑖) on ℋ is adjointable with respect to the inner product ⟨⋅∣⋅⟩_ℬ𝑖 whose adjoint 𝜙_𝑖(𝑧_𝑖)^∗ coincides with the adjoint of 𝜙_𝑖(𝑧_𝑖) with respect to the inner product ⟨⋅∣⋅⟩_𝒜 so that 𝜙_𝑖(𝑧_𝑖)^∗=𝜙_𝑖(𝑧^∗_𝑖). We assume that the left actions 𝜙_𝑖 of ℬ_𝑖 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 ℬ_𝑖 through 𝜓_𝑖: 𝜉∣𝜂𝑎_ℬ𝑖=𝜉∣𝜂_ℬ𝑖𝜓_𝑖(𝑎),𝜉,𝜂∈ℋ,𝑎∈𝒜,𝑖=1,…,𝑛. (200) 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;ℬ_𝑖,𝑖=1,…,𝑛) is said to be of general type if there exists a faithful completely positive map 𝜆_𝑖∶ℬ_𝑖→𝒜 for 𝑖=1,…,𝑛 such that 𝜆_𝑖𝑏_𝑖𝜓_𝑖(𝑎)=𝜆_𝑖𝑏_𝑖𝑎,𝑏_𝑖∈ℬ_𝑖𝜆,𝑎∈𝒜,_𝑖𝜉∣𝜂_ℬ𝑖=𝜉∣𝜂_𝒜,𝜉,𝜂∈ℋ,𝑖=1,…,𝑛. (201) A Hilbert 𝐶^∗-multimodule ℋ over (𝒜;ℬ_𝑖,𝑖=1,…,𝑛) is said to be of finite type if there exists a family {𝑢₁^(𝑖),…,𝑢_𝑀^(𝑖)_(𝑖)}, 𝑖=1,…,𝑛 of finite bases of ℋ as a right Hilbert ℬ_𝑖-module for each 𝑖=1,…,𝑛 such that ^𝑀(𝑖)_𝑗=1𝑢_𝑗^(𝑖)𝜑_𝑖𝑢_𝑗^(𝑖)∣𝜉_ℬ𝑖𝑢=𝜉,𝜉∈ℋ,𝑖=1,…,𝑛_𝑗(𝑖)∣𝜙_𝑘(𝑤_𝑘)𝑢_ℎ^(𝑖)_ℬ𝑖∈𝒜,𝑤_𝑘∈ℬ_𝑘,𝑗,ℎ=1,…,𝑀^(𝑖),_{𝑀(𝑖)𝑗=1}𝑢_𝑗^(𝑖)∣𝜙_𝑘(𝜉∣𝜂_ℬ𝑘)𝑢_𝑗^(𝑖)_ℬ𝑖=𝜉∣𝜂_𝒜 (202) for all 𝜉,𝜂∈ℋ, 𝑖,𝑘=1,…,𝑛 with 𝑖≠𝑘.

By a generalizing argument to the preceding sections, we may construct a 𝐶^∗-algebra 𝒪_𝐹(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 𝑠_𝜉^(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. Let ℋ be a Hilbert 𝐶^∗-multimodule over (𝒜;ℬ_𝑖,𝑖=1,…,𝑛) of finite type with a finite basis {𝑢₁^(𝑖),…,𝑢_𝑀^(𝑖)_(𝑖)} of ℋ as a Hilbert 𝐶^∗-right module over ℬ_𝑖 for each 𝑖=1,…,𝑛. Then, the 𝐶^∗-algebra 𝒪_𝐹(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 𝒪_ℋ generated by the operators 𝑆₁^(𝑖),…,𝑆_𝑀^(𝑖)_(𝑖) and elements 𝑧_𝑖∈ℬ_𝑖 for 𝑖=1,…,𝑛 subject to the relations ^𝑛𝑀_𝑖=1^(𝑖)_𝑘=1𝑆_𝑘^(𝑖)𝑆_𝑘^(𝑖)∗=1,𝑆_𝑘^(𝑖)∗𝑆_𝑚^(𝑗)𝑆=0,𝑖≠𝑗,_{𝑘(𝑖)∗}𝑆_𝑙^(𝑖)=𝑢_𝑘^(𝑖)∣𝑢_𝑙^(𝑖)_ℬ𝑖,𝑧_𝑗𝑆_𝑘^(𝑖)=^𝑀(𝑖)_𝑙=1𝑆_𝑙^(𝑖)𝑢_𝑙^(𝑖)∣𝜙_𝑗(𝑧_𝑗)𝑢_𝑘^(𝑖)_ℬ𝑖 (203) for 𝑧_𝑗∈ℬ_𝑗, 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 Theorem 1.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by JSPS KAKENHI Grant Number 23540237.