#### Abstract

A Hilbert -quad module of finite type has a multistructure of Hilbert -bimodules with two finite bases. We will construct a -algebra from a Hilbert -quad module of finite type and prove its universality subject to certain relations among generators. Some examples of the -algebras from Hilbert -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 -algebras. Since then, there have been many studies on these -algebras by many authors (see, e.g., [1–6], etc.).

In [7], the author has introduced a notion of -symbolic dynamical system, which is a generalization of a finite labeled graph, a -graph system, and an automorphism of a unital -algebra. It is denoted by and consists of a finite family of endomorphisms of a unital -algebra such that , and where denotes the center of . It provides a subshift over and a Hilbert -bimodule over which gives rise to a -algebra as a Cuntz-Pimsner algebra ([7] cf. [8–10]). In [11, 12], the author has extended the notion of -symbolic dynamical system to -textile dynamical system which is a higher-dimensional analogue of -symbolic dynamical system. A -textile dynamical system consists of two -symbolic dynamical systems and with common unital -algebra and commutation relations between the endomorphisms and . A -textile dynamical system provides a two-dimensional subshift and a multistructure of Hilbert -bimodules that has multi right actions and multi left actions and multi inner products. Such a multi structure of Hilbert -bimodule is called a Hilbert -quad module. In [12], the author has introduced a -algebra associated with the Hilbert -quad module of -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 -algebra has been proved to have a universal property subject to certain operator relations of generators encoded by structure of -textile dynamical system [12].

In this paper, we will generalize the construction of the -algebras of Hilbert -quad modules of -textile dynamical systems. Let , , and be unital -algebras. Assume that has unital embeddings into both and . A Hilbert -quad module over is a Hilbert -bimodule over with -valued right inner product which has a multi structure of Hilbert -bimodules over with right actions of and left actions of and -valued inner products for satisfying certain compatibility conditions. A Hilbert -quad module is said to be of finite type if there exists a finite basis of as a Hilbert -right module over and a finite basis of as a Hilbert -right module over such that for (see [13] for the original definition of finite basis of Hilbert module). For a Hilbert -quad module, we will construct a Fock space from , which is a 2-dimensional analogue to the ordinary Fock space of Hilbert -bimodules (cf. [10, 14]). We will then define two kinds of creation operators , for on . The -algebra on generated by them is denoted by and called the Toeplitz quad module algebra. We then define the -algebra associated with the Hilbert -quad module by the quotient -algebra of by the ideal generated by the finite-rank operators. We will then prove that the -algebra for a -quad module of finite type has a universal property in the following way.

Theorem 1 (Theorem 34). *Let be a Hilbert -quad module over of finite type with a finite basis of as a Hilbert -right module over and a finite basis of as a Hilbert -right module over . Then, the -algebra generated by the quotients of the creation operators for on the Fock spaces is canonically isomorphic to the universal -algebra generated by operators , and elements , subject to the relations
**
for , , , .*

The eight relations of the operators above are called the relations . As a corollary, we have the following.

Corollary 2 (Corollary 35). *For a -quad module of finite type, the universal -algebra generated by operators , and elements , subject to the relations does not depend on the choice of the finite bases and .*

The paper is organized in the following way. In Section 2, we will define Hilbert -quad module and present some basic properties. In Section 3, we will define a -algebra from Hilbert -quad module of general type by using creation operators on Fock Hilbert -quad module. In Section 4, we will study algebraic structure of the -algebra for a Hilbert -quad module of finite type. In Section 5, we will prove, as a main result of the paper, that the -algebra has the universal property stated as in Theorem 1. A strategy to prove Theorem 1 is to show that the -algebra is regarded as a Cuntz-Pimsner algebra for a Hilbert -bimodule over the -algebra generated by and . We will then prove the gauge invariant universality of the -algebra (Theorem 33). In Section 6, we will present K-theory formulae for the -algebra . 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 the set of nonnegative integers and by the set of positive integers.

#### 2. Hilbert -Quad Modules

Throughout the paper, we fix three unital -algebras , , and such that , with common units. We assume that there exists a right action of on so that
which satisfies
for , , . Hence, is a right -module through for . Suppose that is a Hilbert -bimodule over , which has a right action of , an -valued right inner product , and a -homomorphism from to the algebra of all bounded adjointable right -module maps satisfying the following.(i) is linear in the second variable.(ii) for , .(iii) for .(iv), and if and only if .A Hilbert -bimodule over is called a Hilbert -*quad module over * if has a further structure of a Hilbert -bimodule over for each with right action of and left action of and -valued right inner product such that for , ,
for , , , , and
where is regarded as a subalgebra of . The left action of on means that for is a bounded adjointable operator with respect to the inner product for each . The operator for is also adjointable with respect to the inner product . We assume that the adjoint of with respect to the inner product coincides with the adjoint of with respect to the inner product . Both of them coincide with . We assume that the left actions of on for are faithful. We require the following compatibility conditions between the right -module structure of and the right -module structure of through :

We further assume that is a full Hilbert -bimodule with respect to the three inner products , , and for each. This means that the -algebras generated by elements , and coincide with , , and , respectively.

For a vector , denote by , , and the norms , , and induced by the right inner products, respectively. By definition, is complete under the above three norms for each.

*Definition 3. *(i) A Hilbert -quad module over is said to be *of general type* if there exists a faithful completely positive map for such that

(ii) A Hilbert -quad module over is said to be *of finite type* if there exist a finite basis of as a right Hilbert -module and a finite basis of as a right Hilbert -module; that is,
such that
for , and
for all . Following [13], and are called finite bases of , respectively.

(iii) A Hilbert -quad module over is said to be *of strongly finite type* if it is of finite type and there exist a finite basis of as a right -module through and a finite basis of as a right -module through . This means that the following equalities hold:

We note that for a Hilbert -quad module of general type, the conditions (9) imply
Put so that . Hence, the identity operators from the Banach spaces to are bounded linear maps. By the inverse mapping theorem, there exist constants such that for . Therefore, the three norms , , and , induced by the three inner products , and on , are equivalent to each other.

Lemma 4. *Let be a Hilbert -quad module over . If is of finite type, then it is of general type.*

*Proof. *Suppose that is of finite type with finite bases of as a right Hilbert -module and of as a right Hilbert -module as above. We put
They give rise to faithful completely positive maps , . The equalities (12) imply that
It then follows that
Since is full, the equalities (8) hold.

Lemma 5. *Suppose that a Hilbert -quad module of finite type is of strongly finite type with a finite basis of as a right -module through and a finite basis of as a right -module through . Let and be finite bases of satisfying (10). Then, two families , and , of form bases of as right -modules, respectively.*

*Proof. *For , by the equalities
it follows that
We similarly have

We present some examples.

*Examples. *(1) Let , be automorphisms of a unital -algebra satisfying . We set . Define right actions of on by
for , . We put and equip it with Hilbert -quad module structure over in the following way. For , , , , and , define the right -module structure and the right -valued inner product by
Define the right actions of with right -valued inner products and the left actions of by setting
It is straightforward to see that is a Hilbert -quad module over of strongly finite type.

(2) We fix natural numbers , . Consider finite-dimensional commutative -algebras , , and . The right actions of on are naturally defined as right multiplications of . The algebras have the ordinary product structure and the inner product structure which we denote by and , respectively. Let us denote by the tensor product . Define the right actions of with -valued right inner products and the left actions of on for by setting
for , , , and . Let , and , be the standard bases of and , respectively. Put the finite bases
It is straightforward to see that is a Hilbert -quad module over of strongly finite type.

(3) Let be a -textile dynamical system which means that for , endomorphisms of are given with commutation relations if . In [12], a Hilbert -quad module over from is constructed (see [12] for its detail construction). The two triplets and are -symbolic dynamical systems [7], that yield -algebras and , respectively. The -algebras and are defined as the -subalgebra of generated by elements , , and that of generated by , , , respectively. Define the maps , , by
which yield the right actions of on , . Define the maps , by
Put , . Let be an element of for with . As , one sees that
We similarly have by putting ,
We see that is a Hilbert -quad module of strongly finite type. In particular, two nonnegative commuting matrices with a specification coming from the equality yield a -textile dynamical system and hence a Hilbert -quad module of strongly finite type, which are studied in [15].

#### 3. Fock Hilbert -Quad Modules and Creation Operators

In this section, we will construct a -algebra from a Hilbert -quad module of general type by using two kinds of creation operators on Fock space of Hilbert -quad module. We first consider relative tensor products of Hilbert -quad modules and then introduce Fock space of Hilbert -quad modules which is a two-dimensional analogue of Fock space of Hilbert -bimodules. We fix a Hilbert -quad module over of general type as in the preceding section. The Hilbert -quad module is originally a Hilbert -right module over with -valued inner product . It has two other structures of Hilbert -bimodules: the Hilbert -bimodule over and the Hilbert -bimodule over , where is a left action of on and is a right action of on with -valued right inner product for each . This situation is written as in the figure (30) We will define two kinds of relative tensor products as Hilbert -quad modules over . The latter one should be written vertically as rather than horizontally . The first relative tensor product is defined as the relative tensor product as Hilbert -modules over , where the left is a right -module through and the right is a left -module through . It has a right -valued inner product and a right -valued inner product defined by respectively. It has two right actions: from and from . It also has two left actions: from and from . By these operations, is a Hilbert -bimodule over as well as a Hilbert -bimodule over . It also has a right -valued inner product defined by a right -action for and a left -action . By these structure is a Hilbert -quad module over : (35) We denote the above operations , , , and still by , , , and , respectively. Similarly, we consider the other relative tensor product defined by the relative tensor product as Hilbert -modules over , where the left is a right -module through and the right is a left -module through . By a symmetric discussion to the above, is a Hilbert -quad module over . The following lemma is routine.

Lemma 6. *Let , . The correspondences
**
yield isomorphisms of Hilbert -quad modules, respectively.*

We write the isomorphism class of the former Hilbert -quad modules as and that of the latter ones as , respectively.

Note that the direct sum has a structure of a pre-Hilbert -right module over by the following operations. For , , and , set By (8), the equality holds so that is a pre-Hilbert -right module over . We denote by the completion of by the norm induced by the inner product . It has right -actions and left -action by for , , and .

We denote the relative tensor product and elements by and , respectively, for . Let us define the Fock Hilbert -quad module as a two-dimensional analogue of the Fock space of Hilbert -bimodules. Put and , . We set as Hilbert -bimodules over . We will define the Fock Hilbert -module by setting which is the completion of the algebraic direct sum of the Hilbert -right module over under the norm on induced by the -valued right inner product on . Then, is a Hilbert -right module over . It has a natural left -action defined by for .

For , we define two operators by setting for , and for , for with .

Lemma 7. *For , the two operators
**
are both right -module maps.*

*Proof. *We will show the assertion for . For , we have for and ,
For , one has

It is clear that the two operators yield bounded right -module maps on having its adjoints with respect to the -valued right inner product on . The operators are still denoted by , respectively. The adjoints of with respect to the -valued right inner product on map to , .

Lemma 8. *(i) For , one has
**(ii) For and , , we have
*

*Proof. *We will show the assertions (i) and (ii) for . (i) For , we have
so that .

(ii) For with , we have

Denote by the left actions of , on and hence on , respectively. They satisfy the following equalities for , , , and . More generally let us denote by and the -algebras of all bounded adjointable right -module maps on and on with respect to their right -valued inner products, respectively. For , define by for .

Lemma 9. *Both the maps for are faithful -homomorphisms.*

*Proof. *By assumption, the -homomorphisms , are faithful, so that the -homomorphisms , are both faithful.

Lemma 10. *For , , , , and , the following equalities hold on :
*

*Proof. *The equalities (54) are obvious. We will show the equalities (55) and (56) for . We have for
For , , we have
so that on , . Hence, the equalities (55) hold.

For , we have
For , we have
so that on for . Hence, the equalities (56) hold.

The -subalgebra of generated by the operators for is denoted by and is called the Toeplitz quad module algebra for .

Lemma 11. *The -algebra contains the operators for , .*

*Proof. *By (56) in the preceding lemma, one sees that
Since is a full -quad module, the inner products for generate the -algebras , , respectively. Hence, , are contained in .

Lemma 12. *There exists an action of on such that
**
for .*

*Proof. *We will first define a one-parameter unitary group , on with respect to the right -valued inner product as in the following way.

For is defined by
For is defined by
for . We therefore have a one-parameter unitary group on . We then define an automorphism on for by
for , .

It then follows that for
and for , ,
Therefore, we conclude that on and similarly on . It is direct to see that
It is also obvious that for .

Denote by the -subalgebra of generated by the elements