International Journal of Mathematics and Mathematical Sciences

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

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

Research Article

-Algebras Associated with Hilbert -Quad Modules of Finite Type

Kengo Matsumoto

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

By (10) and Lemma 10, we have for

Let be the projection on onto for so that on .

Lemma 15. For , one has

(i) for and hence .(ii) for and hence .

Proof. (i) For , we have

(ii) Is similar to (i).

Define two projections on by

Lemma 16. Keep the above notations.

Hence,

Proof. For with , we have

As , and , we have

and hence

For , we have so that

and hence

As for , we have

Therefore, we conclude that

As , one obtains (80).

We set the operators

in the -algebra . As two operators and are projections by (79), so are and . Since , the identity (80) implies

Therefore, we have the following.

Theorem 17. Let be a Hilbert -quad module over of finite type with finite basis as a right -module and as a right -module. Then, one has the following.

(i)The -algebra is generated by the operators and the elements for ,  .(ii)They satisfy the following operator relations:

 for ,  ,  ,  .(iii)There exists an action of   on such that

 for , ,  , and ,  .

Proof. (i) The assertion comes from the equalities (76).

(ii) The first equality of (90) is (89). As the projection belongs to , Lemma 15 ensures us the second equality of (90). The equalities (91) come from (74). For and , we have so that

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 of on defined in the above theorem (iii) is called the gauge action.

5. The Universal -Algebras Associated with Hilbert -Quad Modules

In this section, we will prove that the -algebra associated with a Hilbert -quad module of finite type is the universal -algebra subject to the operator relations stated in Theorem 17 (ii). Throughout this section, we fix a Hilbert -quad module over of finite type with finite basis as a right Hilbert -module and as a right Hilbert -module as in the previous section.

Let be the universal -algebra generated by operators ,   and elements , subject to the relations

for , , , . The above four relations (96), (97), (98), and (99) are called the relations . In what follows, we fix operators ,   satisfying the relations .

Lemma 18. The sums and are both projections.

Proof. Put and . By the relations (96), one sees that , , , and . It is easy to see that both and are projections.

Lemma 19. Keep the above notations.

(i)For and , , one has

(ii)For and , , one has

Proof. (i) By (98), we have

 The other equality 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 , , the element belongs to and the formula holds:

(ii)For , , the element belongs to and the formula holds:

Lemma 21. The following equalities for and hold:

(i)

(ii)

(iii)

Proof. (i) By (98) and (99), we have

so that by (96)

Similarly, we have (106).

(ii) All the adjoints of , for , by the three inner products , , and on coincide with , , respectively. Hence, the assertions are clear.

(iii) By (i), we have

As for any ,  , it follows that

Lemma 22. Let be a polynomial of elements of and . Then, one has

(i) for some .(ii) for some .

Proof. For , , and , by putting and , the relations (98), (99) imply

so that the assertion of (i) holds. (ii) is similar to (i).

Lemma 23. Let be a polynomial of elements of and . Then, one has

(i) belongs to for all .(ii) belongs to for all .(iii) for all , .(iv) for all , .

Proof. (i) By the previous lemma, we know

so that

As belongs to , we see the assertion.

(ii) Is similar to (i).

(iii) As , we have

(iv) Is similar to (i).

We set

Put

Lemma 24. Every element of can be written as a linear combination of elements of the form

for some ,  ,  ,   where is a polynomial of elements of and .

Proof. The assertion follows from the preceding lemmas.

By construction, every representation of and on a Hilbert space together with operators ,  , ,   satisfying the relations extends to a representation of on . We will endow with the norm obtained by taking the supremum of the norms in over all such representations. Note that this supremum is finite for every element of because of the inequalities , which come from (96). The completion of the algebra under the norm becomes a -algebra denoted by , which is called the universal -algebra subject to the relations .

Denote by the -subalgebra of generated by and .

Lemma 25. An element of the -algebra is both a right -module map and a right -module map. This means that the equalities

hold.

Proof. Since both the operators for and for are right -module maps for , any element of the -algebra algebraically generated by and is both a right -module map and a right -module map. Hence, it is easy to see that any element of the -algebra is both a right -module map and a right -module map.

Denote by the -subalgebra of generated by and .

Lemma 26. The correspondence

gives rise to an isomorphism from onto as -algebras.

Proof. We note that by hypothesis both the maps

are injective. Denote by the -algebra on algebraically generated by , for , . Define an operator for by

Let be the -subalgebra of algebraically generated by and . Since for and for and by Lemma 21, the map

yields a -homomorphism. As for ,  , we have

We then have

and similarly

By putting

 ,

one has

Hence, extends to the -algebra such that . The equality (124) holds for .

We will next show that is injective. By (124), we have for and ,

Suppose that so that . Since

we see that so that . We thus conclude that is injective and hence isomorphic.

Denote by the inverse of the -isomorphism given in the proof of the above lemma which satisfies

We put . For , we denote by the closed linear span of elements of the form

for some and . Let us denote by the -subalgebra of generated by . By the relations (105) and (106), we see the following.

Lemma 27. For , the following identity holds:

Hence, by putting for

we have the following.

Lemma 28. For , the identity

holds and induces an embedding of for .

Lemma 29. The -algebra is the inductive limit of the sequence of the inclusions

Let be a complex number of modulus one for . The elements

in instead of

satisfy the relations . This implies the existence of an action on by automorphisms of the one-dimensional torus that acts on the generators by

for ,  ,  ,  , and . As the -algebra has the largest norm on , the action on extends to an action of on , still denoted by . The formula

where is the normalized Lebesgue measure on defines a faithful conditional expectation denoted by from onto the fixed-point algebra . The following lemma is routine.

Lemma 30. .

The -algebra satisfies the following universal property. Let be a unital -algebra and , be -homomorphisms such that for . Assume that there exist elements ,   in satisfying the relations

for ,  ,  ,  ; then, there exists a unique -homomorphism such that

for , and , . We further assume that both the homomorphisms , are injective. We denote by the restriction of to the subalgebra . Let us denote by the -subalgebra of generated by , , and , for , .

Lemma 31. Keep the above situation. The -homomorphism is injective.

Proof. Since the correspondence in Lemma 26

yields an isomorphism of -algebras, it suffices to prove that the correspondence

yields an isomorphism. Let be the -subalgebra of generated by elements for , . Define an element of for by setting

As in the proof of Lemma 26, one sees that gives rise to a -homomorphism from into . Since

and similarly , it is enough to show that is injective. Suppose that for some . By following the proof of Lemma 26, one sees that for all . Hence, the condition implies that . Since is injective, we have for all . As is a right -module map, we have for ,

so that . Therefore, is injective. Hence, the composition

is injective.

We set

We put . For , let be the closed linear span in the -algebra of elements of the form

for and . Similar to the subalgebras , of , one knows that the closed linear span is a -algebra and naturally regarded as a subalgebra of for each . Let us denote by the -subalgebra of generated by . Then, the -algebra is the inductive limit of the sequence of the inclusions

Lemma 32. Suppose that both the -homomorphisms , are injective. Then, the restriction of to the subalgebra yields a -isomorphism .

Proof. By the universality of , the restriction of to yields a surjective -homomorphism . It suffices to show that is injective. Suppose that and put . Since and , there exists such that . Let us denote by the set of -tuples of :

For , denote by the operator

where

Any element of is of the form

Hence, one may find a nonzero element . Since , the equality holds. For some , one then sees

As and , for ,  , the element belongs to . By the preceding lemma, the homomorphism is injective, so that we have a contradiction. Therefore we conclude that is injective and hence isomorphic.

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

Theorem 33. Let be a unital -algebra. Suppose that there exist -homomorphisms , such that for and there exist elements ,   in satisfying the relations

for , , , . Let us denote by the -subalgebra of generated by , , and , for ,  . One further assumes that the algebra admits a gauge action. If both the -homomorphisms , are injective, then there exists a -isomorphism satisfying

for , and , .

Proof. By assumption, admits a gauge action, which we denote by . Let us denote by the fixed-point algebra of under the gauge action and by the -subalgebra of defined by the inductive limit (152). Then, it is routine to check that is canonically -isomorphic to . There exists a conditional expectation

defined by

By the universality of the algebra , there exists a surjective -homomorphism from to such that

for ,  ,  ,  . Then, and the following diagram

(163)

is commutative. Denote by the restriction of to the -subalgebra of generated by ,  . By assumption, both the maps , are injective, so that is injective by Lemma 31. By the preceding lemma, is an isomorphism. Since the conditional expectation is faithful, a routine argument shows that is injective and hence isomorphic.

Therefore, we have the following.

Theorem 34. For a -quad module of finite type, 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 ,   and ,  .

Proof. Theorem 17 implies that the operators ,   and the elements , for , in satisfy the eight relations of Theorem 33. By Theorem 33, we see that the correspondences

for , , and , give rise to an isomorphism from to .

The eight relations of the operators above are called the relations . The above generating operators and of the universal -algebra correspond to two finite bases and of the Hilbert -quad module , respectively. On the other hand, the other -algebra is generated by the quotients of the creation operators for on the Fock spaces , which do not depend on the choice of the two finite bases. Hence, we have the following.

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 .

6. -Theory Formulae

Let be a Hilbert -quad module over of finite type as in the preceding section. In this section, we will state -theory formulae for the -algebra . By the previous section, the -algebra is regarded as the universal -algebra generated by the operators and and the elements and subject to the relations . Let us denote by the -subalgebra of generated by elements and . By Lemma 26, the correspondence

gives rise to a -isomorphism from onto as -algebras, which is denoted by . We will restrict our interest to the case when

(i) and are partial isometries, and(ii),   commute with all elements of .

If the bases and satisfy the conditions

the condition (i) holds. Furthermore, if acts diagonally on for and acts diagonally on for , the condition (ii) holds. Recall that the gauge action is denoted by which is an action of on such that the fixed-point algebra under is canonically isomorphic to the -algebra . Denote by the dual action of which is an action of on the -crossed product by the gauge action of . As in the argument of [16], is stably isomorphic to . Hence, we have that is isomorphic to . The dual action induces an automorphism on the group and hence on , which is denoted by . Then, by [16] (cf. [10, 17], etc.), we have the following.

Proposition 36. The following six-term exact sequence of -theory holds:

(168)

We put for

Both the families , yield endomorphisms on which give rise to endomorphisms on the -groups:

We put which is an endomorphism on . Now, we further assume that . It is routine to show that the groups in and in are isomorphic to the groups in , and in , respectively, by an argument of [17]. Therefore, we have

Proposition 37. The following formulae hold:

7. Examples

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

(1) Let be automorphisms of a unital -algebra satisfying . Let be the associated Hilbert -quad module of finite type as in (1) in Section 2. It is easy to see the following proposition.

Proposition 38. The -algebra associated with the Hilbert -quad module coming from commuting automorphisms of a unital -algebra is isomorphic to the universal -algebra generated by two isometries and elements of subject to the following relations:

for .

(2) We fix natural numbers ,  . Consider finite-dimensional commutative -algebras ,  , and  . The algebras have the ordinary product structure and the inner product structure which we denote by and , respectively. Let us denote by the Hilbert -quad module over defined in (2) in Section 2. Put the finite bases

We set and put ,   the standard basis of . Then, the -algebra on generated by and is regarded as . Hence,

Lemma 39. The -algebra is generated by operators ,  ,   satisfying

for ,  .

Proof. It suffices to show the equalities (177). We have

The other equality of (177) is similarly shown.

Put

Then we have the following.

Lemma 40. The following equalities hold:

for ,   and .

Proof. Since , we have

and similarly . Hence, we have

so that (180) holds. As , the equality (182) holds. Since for , we have

and similarly , so that (181) holds. By (177), it follows that

and similarly, we have

Theorem 41. The -algebra associated with the Hilbert -quad module is generated by partial isometries for satisfying the relations

for .

Proof. By the preceding lemma, one knows that are generated by the operators , so that the algebra is generated by the partial isometries , , .

Let be the identity matrix and the matrix whose entries are all 1s. For an -matrix and an -matrix , denote by the matrix

so that

The index set of the standard basis of is ordered lexicographically from left as in the following way:

Put the matrices

and the matrix

Then, we have the following.

Theorem 42. The -algebra is isomorphic to the Cuntz-Krieger algebra for the matrix . The algebra is simple and purely infinite and is isomorphic to the Cuntz-Krieger algebra for the matrix .

Proof. By the preceding proposition, the -algebra is isomorphic to the Cuntz-Krieger algebra for the matrix . Since the matrix is aperiodic, the algebra is simple and purely infinite. The th column of the matrix coincides with the th column for every . One sees that the matrix is obtained from 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   have been computed by using Euclidean algorithms. For the case , they are , respectively, so that we see (see [15] for details).

(3) For a -textile dynamical system , let be the -quad module over as in (3) in Section 2. The -algebra has been studied in [12].

8. Higher-Dimensional Analogue

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

Let be a unital -algebra and let be -family of unital -algebras. Suppose that there exists a unital embedding

for each . Suppose that there exists a right action of on such that

Hence, is a right -module through for . Let be a Hilbert -bimodule over with a right action of , an -valued right inner product , and a -homomorphism from to . It is called a Hilbert -multimodule over if has a multistructure of Hilbert -bimodules over for such that for each there exist a right action of on and a left action of on and a -valued right inner product such that and

for , , , , and

The operator on is adjointable with respect to the inner product whose adjoint coincides with the adjoint of with respect to the inner product so that . 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 inner product for each. A Hilbert -multimodule over is said to be of general type if there exists a faithful completely positive map for such that

A Hilbert -multimodule over is said to be of finite type if there exists a family , of finite bases of as a right Hilbert -module for each such that

for all , with .

By a generalizing argument to the preceding sections, we may construct a -algebra associated with the Hilbert -multimodule by a similar manner to the preceding sections; that is, the -algebra is generated by the quotients of the -kinds of creation operators ,, on the generalized Fock space by the ideal generated by the finite-rank operators. One may show the following generalization.

Proposition 43. Let be a Hilbert -multimodule over of finite type with a finite basis of as a Hilbert -right module over for each . Then, the -algebra generated by the quotients of the -kinds of creation operators on the generalized Fock spaces is canonically isomorphic to the universal -algebra generated by the operators and elements for subject to the relations

for , ,  ,  .

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.

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