International Journal of Mathematics and Mathematical Sciences

Volume 2014 (2014), Article ID 952068, 21 pages

http://dx.doi.org/10.1155/2014/952068

## -Algebras Associated with Hilbert -Quad Modules of Finite Type

Department of Mathematics, Joetsu University of Education, Joetsu 943-8512, Japan

Received 30 September 2013; Accepted 2 December 2013; Published 30 January 2014

Academic Editor: A. Zayed

Copyright © 2014 Kengo Matsumoto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 The algebra is a closed two-sided ideal of .

*Definition 13. *The -algebra associated with the Hilbert -quad module of general type is defined by the quotient -algebra of by the ideal .

We denote by the quotient image of an element under the ideal . We set the elements of
for and , . By the preceding lemmas, we have the following.

Proposition 14. *The -algebra is generated by the family of operators , for . It contains the operators for , . They satisfy the following equalities,
**
for , and , .*

#### 4. The -Algebras of Hilbert -Quad Modules of Finite Type

In what follows, we assume that a Hilbert -quad module is of finite type. In this section, we will study the -algebra for a Hilbert -quad module of finite type. Let be a finite basis of as a Hilbert -right module over and a finite basis of as a Hilbert -right module over . Keep the notations as in the previous section. We set