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