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## Academic research paper on topic "-Algebras Associated with Hilbert -Quad Modules of Finite Type"

﻿Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2014, Article ID 952068, 21 pages http://dx.doi.org/10.1155/2014/952068

Research Article

C* -Algebras Associated with Hilbert C*-Quad Modules of Finite Type

Kengo Matsumoto

Department of Mathematics, Joetsu University of Education, Joetsu 943-8512, Japan Correspondence should be addressed to Kengo Matsumoto; kengo@juen.ac.jp 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.

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 A-graph system, and an automorphism of a unital C*-algebra. It is denoted by (A, ft, Z) and consists of a finite family |fta}agZ of endomorphisms of a unital C*-algebra A such that ^„(Zrf) c a e Z and ft,(1) > 1 where ZA denotes the center of A. It provides a subshift A p over Z and a Hilbert C*-bimodule HPA over A which gives rise to a C*-algebra @p 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, ft, Zp, Z^, k) consists of two C*-symbolic dynamical systems (A, ft, Zp) and (A, Z*) with common unital C*-algebra A and commutation relations k between the endomorphisms fta, a e Zp

and e Z*. 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, B, and B2 be unital C* -algebras. Assume that A has unital embeddings into both and B2. A Hilbert C*-quad module H over (A; B:, 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; with right actions <p; of B; and left actions of B; and Bj-valued inner products (• | -)B. 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 |«!,..., mm} of H as a Hilbert C*-right module over and

a finite basis {v1, ..., vn} of H as a Hilbert C*-right module

over B2 such that

{(ï\v).b2 )»>)B =I(vk \ti {(H\l)

k=l = (t \ n)>

for e 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^, t^ for % e H on F(H). The C*-algebra on F(H) generated by them is denoted by TF(H) and called the Toeplitz quad module algebra. We then define the C*-algebra @F(H) associated with the Hilbert C*-quad module H by the quotient C * -algebra of TF(H) by the ideal generated by the finite-rank operators. We will then prove that the C*-algebra @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,..uM} of H as a Hilbert C*-right module over B1 and a finite basis {v1, ..., vn} of H as a Hilbert C* -right module over B2. Then, the C*-algebra @F(H) generated by the quotients [s^j, [i^j of the creation operators s^, t^ for % e H on the Fock spaces F(H) is canonically isomorphic to the universal C*-algebra OH generated by operators S1,..., SM, T1,...,TN and elements z e B1, w e B2 subject to the relations

Ysts; + YTkTki = s]Ti =

i=l k=l

S'Sj = (ut \ Uj)Bi, Tft = (vk \ Vi)B2,

zSj = YSt(ut \4i (z)Uj) , i=i 1

zTi = TTk(vk \ <h (z) Vi)B2' k=l

WSJ = TSi(Ui \ $2 (w)uj)Bi ' wTl = YTk(vk \$2 (w) VÙ

for z e By, we B2, i,j = !,...,M,k, 1= 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 OHgenerated by operators

S1,..., SM, T1,...,TN and elements z e B1, w e B2 subject to the relations (H) does not depend on the choice of the finite bases {u1,..., uM} and {v1, ..., vn}.

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 ®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 ®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 @F(H) has the universal property stated as in Theorem 1. A strategy to prove Theorem 1 is to show that the C*-algebra @F(H) is regarded as a Cuntz-Pimsner algebra for a Hilbert C*-bimodule over the C*-algebra generated by (By) and <p2(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 OH. 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.

Throughout the paper, we fix three unital C* -algebras A, By, and B2 such that A c By, A c B2 with common units. We assume that there exists a right action of A on Bi so that

bifi (a) e B for bt e B¡, ae A, i = 1,2,

which satisfies

¡b^ (fl)|| < \\biW \\a\\, btfi (ad) = btfi (a) (a) (4)

for bt e B, a,d e A, i = 1,2. Hence, Bt is a right Amodule through 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 <pA from A to the algebra of all bounded adjointable right Amodule maps LA(H) satisfying the following.

(i) {■ I ■) a is linear in the second variable.

(ii) I no) a = I n)Aa for t»ne H, ae A.

(iii) «U)A =knI^)A for Ue H.

(iv) $I Oa * 0, and$ Ii) a = 0 if and only if $= 0. A Hilbert C* -bimodule H over A is called a Hilbert C*-quad module over (A; B1 , B2) if H has a further structure of a Hilbert C*-bimodule over B, for each i = 1,2 with right action <p, of B, and left action of B, and Brvalued right inner product (• | •}B, such that for z e B^ w e B2, (z),02 Me La I [01 (z) £] ^2 M = 01 (Z) [^2 M] ; [02 M £] <?1 (z) = 02 M [^1 (Z)] ; £<P1 (fl)) = [£<P1 (z)] fl, ^2 (^^2 (fl)) = [&2 M] A-(5) for £ e H, z e B1, w e B2, fl e A, and 0A (fl) = 01 (fl) = 02 (fl) for fl e A, where A is regarded as a subalgebra of B;. The left action of B; on H means that 0;(fc;) for e B; is a bounded adjointable operator with respect to the inner product (• | -}B. for each i = 1,2. The operator 0;(fc;) for e B; is also adjointable with respect to the inner product (• | -}A. We assume that the adjoint of 0;(fc;) with respect to the inner product (• | -}B, coincides with the adjoint of 0;(fc;) with respect to the inner product (• | -}A. Both of them coincide with 0j(fc*). We assume that the left actions of B; on H for i = 1,2 are faithful. We require the following compatibility conditions between the right A-module structure of H and the right A-module structure of B; through (£ | rçfl)B = (£ | ^ (fl) for £ rç e H, fl e A, i = 1,2. We further assume that H is a full Hilbert C*-bimodule with respect to the three inner products (• | -}A, (• | -}B , and (• | -}B for each. This means that the C*-algebras generated by elements {tf | ^ | U e H}, {« | ^ | e H} and {(^ | ^}B2 | e H} coincide with A, Bx, and B2, respectively. For a vector$ e H, denote by UflA IKIIb, , and ||£||b2

the norms | OaI|1/2, II« | Ob, I1/2, and | 0b2|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 A; : B; ^ A for i = 1,2 such that

A; (fcj^j (a)) = A; a for bt e B;, a e A, (8) ^ (<^>b,) = (?|^>A, for Ue H, ¿=1,2. (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 {m1 ,..., wM} of H as a right Hilbert B1-module and a finite basis {v1, ..., vn} of H as a right Hilbert B2-module; that is,

((", I Ob,) = X^ ((Vfc I ^)b2) = 6 ç e H

such that

(«; | 02 M "j)Bi e A, Î, j = 1,..., M,

(vfc | 01 (z) V;)b2 e A, fc,/=1,...,N for w e B2, z e B1 and

K", | 02 ((Ç|rç)B2)">)B, ^^

I<V* | 01 ((Ç|rç)Bi ) v^)b = (*U)A

for all ^ e H. Following [13], {m1, ..., wM} and {v1 ,..., vn} 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,..., eM/} of B1 as a right A-module through o A and a finite basis {/1,..., fN>} of B2 as a right A-module through o A2. This means that the following equalities hold:

z=Zei^1 (A1 W2^ ze B1, j=1

^=£/^2 (^2 C/T^ B2.

We note that for a Hilbert C*-quad module of general type, the conditions (9) imply

ll<m>A№(i)ll ll(m)|L, £ e h. (14)

< CJKHb,. Hence,

Put C; = ||A;(1)||1/2 >0 so that ||J the identity operators from the Banach spaces (H, || • ||B) to (H, | • ||a) are bounded linear maps. By the inverse mapping theorem, there exist constants C' such that ||£||B. < C'||£||A for £ e H. Therefore, the three norms || • ||A, || • ||B, and i = 1,2, induced by the three inner products (• | ^}A, (• | , and « = 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 {m1,...,mm} of H as a right Hilbert B1-module and {v1, ..., vn} of H as a right Hilbert B2-module as above. We put

¿1 (¿) = X(vfc | 01 (z) vfc)B2 e A, ze B1,

fc=1 M

^2 = | 02 ">)Bl e a, we

They give rise to faithful completely positive maps A; : B; ^ A, ( = 1,2. The equalities (12) imply that

^ «*U>b.) = (?|^>A, for H, ¿=1,2. (16)

It then follows that

A, f, (a)) = X, ((Ultia) )

= ($\n)Aa = Xi (($\n)B)a' ae

(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,...,eMi} of B1 as a right A-module through f1 ° X1 and a finite basis [f1,..., fNi} of B2 as a right A-module through f2 ° Let {u1,..uM} and {v1,..., vn} be finite bases of H satisfying (10). Then, two families {uif1(e j) | i = 1,...,M, j =

1,...,M'} and {vkcp2(fi) I k= 1,...,N, 1= 1,...,N'} of H form bases of H as right A-modules, respectively.

Proof. For % e H, by the equalities

¡=1 M

(u. Wb (Xi (e)(u> \t)B ))'

it follows that

S = Tui<Pi ( (X1 (e](u> \ )) i=i \j=i

= H»,<Pi (e)xi (e*j(ui \ Ob )

¡=i j=i M m'

= Hu.Vi (ej)'Xi {{u.Vi (ej)\^)B1

i=i j=i

= T Tui<pi (ej) ■ (ut<pi (ej) \ 0 i=i j=i

We similarly have

S=Hvk<p2 (fi) ■ (vk(p2 (Jl)\^)i k=i l=i

We present some examples.

(20) □

Examples. (1) Let a, ft be automorphisms of a unital C*-algebra A satisfying a ° p = p ° a. We set B1 = B2 = A. Define right actions fi of A on Bi by

hf1 (a) = h1a (a), hi Vi (a) = hp (a) (21) for bt e B, a e A. We put = A and equip it

with Hilbert C*-quad module structure over (A; A, A) in the following way. For £ = x, = x e = A, a e A,

z e B1 = A, and w e B2 = A, define the right A-module structure and the right A-valued inner product (■ I ■)A by

Z-a = xa, (ZIZ')A = x*x'. (22)

Define the right actions of B; with right Brvalued inner products (■ \ ■ )b. and the left actions fa of Bt by setting

(z) = xa (z), (S\S')ai = ai (x*x' ), fa (z)Ç = p(a(z))x,

ifp2 (w) = xp (w),

(t\ï).B2 =^-1 (*'*')>

fa (w)$= a(P(wj)x. It is straightforward to see that Hap is a Hilbert C*-quad module over (A; A, A) of strongly finite type. (2) We fix natural numbers 1 < N, Me N. Consider finite-dimensional commutative C*-algebras A = C, B1 = CN, and B2 = CM. The right actions of A on Bi 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 HM N the tensor product CM 8 CN. Define the right actions of Bi with Brvalued right inner products {■ | -)B, and the left actions fa of Bt on HM>N = CM 8 CN for i= 1,2 by setting 8 n) <P1 (z) =$8(q^z), 8 t]) (fi (w) = (^w)8t],

(^^i' 8^')Bi = ■n e B1,

{$8n|Z,' 8^')Bi =$* ■{.'(n^^v e Bi,

fa (z)(S8y) = S8(z^), fai (w)(S8y) = (W^)8t]

for z e B1, e CN, we B2, and e CM Let e{, i = 1,.. .,M and fk, k = 1,.. .,N be the standard bases of CM and CN, respectively. Put the finite bases

Ui = e{ »le HM>N, i = 1, ...,M, yk = 18fk e i=1,...,N.

It is straightforward to see that HMN is a Hilbert C*-quad module over (C; CN, CM) of strongly finite type.

(3) Let (A, p, q, If, IP, k) be a C*-textile dynamical system which means that for j e In, I e Ip endomorphisms ilj,pi of A are given with commutation relations ^ °pi = pk if K(l,j) = (i,k). In [12], a Hilbert C*-quad module H™ over (A; Bi, B2) from (A, p, q, Ip, In, k) is constructed (see [12] for its detail construction). The two triplets (A,p,Ip) and (A,q,I^) are C*-symbolic dynamical systems [7], that yield C*-algebras @p and respectively. The C*-algebras Bi and B2 are defined as the C*-subalgebra of generated by elements T^yT*, j e I, ye A and that of @p generated by SkyS*, k e Ip, ye A, respectively. Define the maps fi : A ^ Biy i = 1,2,by

yi (a) = T V2 (a) = T , a e A,

which yield the right actions of A on B;, i = 1,2. Define the maps A; : B; ^ A, i = 1,2 by

A1 (Z) = Z rJ z7j, A2 M = X

^ fezp (27)

z e B1, w e B2.

Put = T;T* e B1, j e I*. Let z = TjzjT;* be an element of B1 for z.- e A with T*T.z.T*T: = z.-. As

A1(e*z) = A1(TJzJT*) = Zj, one sees that

z=XT;T;T;Z;T; = ^r/r;^ (z;)

'j riZirj L^-rj

jeZ^ jeZ^

We similarly have by putting /; = S;S* e B2,

w = Z /^2 (A2 (//"w)) for w e B2-

We see that is a Hilbert C*-quad module of strongly finite type. In particular, two nonnegative commuting matrices A, B with a specification k coming from the equality Aß = 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;B^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 H, <pj) over and the Hilbert C*-bimodule (02, H, ^2) over B2, where is a left action of B; on H and <p; is a right action of B; on H with Brvalued right inner product (• | •}B, for each i = 1,2. This situation is written as in the figure

-> —B1

We will define two kinds of relative tensor products

b H, H®b2 H

as Hilbert C*-quad modules over (A; B:, B2). The latter one should be written vertically as

rather than horizontally H®B2 H. The first relative tensor product H®Bi H is defined as the relative tensor product as Hilbert C*-modules over Bj, where the left H is a right Bj -module through ^ and the right H is a left B:-module through .It has a right B: -valued inner product and a right B2-valued inner product defined by

<^Bim'®BiObi = (^ «?!?'>BiK)B ,

c I C>B2 := (<I*B, K)b2,

respectively. It has two right actions: id ® ^ from B: and id ® <p2 from B2. It also has two left actions: ^ ® id from B: and 02®id from B2. Bythese operations, H®B H is a Hilbert C*-bimodule over B: as well as a Hilbert C*-bimodule over B2. It also has a right A-valued inner product defined by

<^®BIC I *'®BIC>A

(<^®BI C|^'®Bi C>BI ) (34)

= A2 (<^®BIm'®BiC>B2),

a right A-action id ® a for a e A and a left A-action 0A ® id. By these structure H®B H is a Hilbert C* -quad module over (A; B1, B2):

^2 ® id

Bi ^ id ^^Bi

id ® f2 B2

We denote the above operations ^ ® id, 02 ® id, id ® ip^ and id ® <p2 still by ^, 02, ipj, and <p2, respectively. Similarly, we consider the other relative tensor product H®B by the relative tensor product as Hilbert C*-modules over B2, where the left H is a right B2-module through <p2 and the right H is a left B2-module through 02. By a

symmetric discussion to the above,

H is a Hilbert

C*-quad module over (A;B1, B2). The following lemma is routine.

Lemma 6. Let H; = H, i = 1,2, 3. The correspondences

(^»B6 (H1®B H2)®b2H3

^ 6 H1®B (H2®BH3) ,

(^1®B2^2)»B1 ^3 6 (H1®BH2)»B1 H3

S 1®B2 (^2»B1 ^ 6 H1®B2 (H2»B1 H3)

yield isomorphisms of Hilbert C* -quad modules, respectively.

We write the isomorphism class of the former Hilbert C*-quad modules as H1»B H2®B H3 and that of the latter ones as H1»B2 H2»B1 H3, 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 fc1 ® fc2, ® &2 6 B1 ® B2, and a 6 A, set

(&1 ® ^A («) := («) ® («) 6 B ® B2,

/ / / / (37)

(&1 ® &2 I ® &2>a := A1 (fc1*+ A2 (&2*fc2) 6 A By (8), the equality

(&1 ® &2 1 (&i ® &2) ^A («)>A = ® &2 1 ® ^2>a • A

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 Br actions <p; and left Braction fa by

(fc1 ® fc2) (z) = fc1z ® 0, (fc1 ® fc2) <p2 (w) = 0 ® fa (z) (fc1 ® fc2) = zfc1 ® 0, fa (w) (fc1 ® fc2) = 0 ®

for fc1 ® fc2 6 B1 ® B2, z 6 B1, and w 6 B2.

We denote the relative tensor product H®BH and elements ^ by H®;H and respectively, for « = 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 r0 = {0} and r„ = |(«1,..., «„)) | = 1,2}, « = 1,2,.... Weset

F1 (H) = H, F2 (H) = (H»1H) ® (H®2H), F3 (H) = (H»1H»1H) ® (H»1H»2H)

® (H»2H»1H)®(H»2H»2H)

(H) = ®(l1 >...>1„-1)6r„-1 H»l1 H®,2 • • • H

as Hilbert C*-bimodules over A. We will define the Fock Hilbert C* -module F(H) by setting

F(H) :=®™0F„ (H), (41)

which is the completion of the algebraic direct sum ®™0F„(H) of the Hilbert C*-right module over A under the norm ||£||A on ®™0F„(H) induced by the A-valued right inner product on ®™0F„(H). Then, F(H) is a Hilbert C*-right module over A. It has a natural left -action defined by fa for i = 1,2.

For £ 6 HK, we define two operators

s? :F„ (H) —> Fn+1 (H), «=0,1,2,...,

i? :F„ (H) —> Fn+1 (H), « = 0,1,2,... by setting for « = 0,

Sr (fc1 ® fc2) = ^ (fc1), fc1 ® fc2 6 B1 ® B2,

i? (61 ® 62) = tyi (62), ® fc2 6 B1 ® B2, and for « = 1,2,...,

i? • • • ^ = • • • for • • • 6 F„(H) with (^1,..., ^„-1) 6 r„-1.

Lemma 7. For £ 6 H, the two operators

s? :F„ (H) —> Fn+1 (H), «=0,1,2,...,

i? :F„ (H) —> Fn+1 (H), « = 0,1,2,..., are both right A-module maps.

Proof. We will show the assertion for . For « = 0, we have for fc1 ® fc2 6 B1 ® B2 and a 6 A,

((&1 ® &2) ^a («)) = («)) = (fc1)) «

= (s? (fc1 ®&2))«.

For « = 1,2,..., one has

((^1®i1 ^»¡2 •••»¡„-1 ^»)fl)

= s? (^1®i1 ^2®i2 •••®i„-1 = ?®1?1®i1 ^2®i2 •••®i„-1 (47)

= (^®1^1®;1 ^2®i2 •••®i„-1 = (^1®i1 ^2®i2 •••®»„1

It is clear that the two operators s^, 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^, i^, respectively. The adjoints of , : F(H) ^ F(H) with respect to the A-valued right inner product on F(H) map F„+1(H) to F„(H), « = 0,1,2,....

International Journal of Mathematics and Mathematical Sciences Lemma 8. (i) For £ e H = F1(H), one has

= <m')ai ® 0 6 Bi

f = 0®<m'L e B1

(¿¿J For Ç e H and Ç1®ii^ •••«^Ç„+1 e F„+1 1,2,..., we feave

«ç (?i®i1 •••«¡„4+0

01 )^2«Î2 •••«;„ ^»+1 f h = 1> 0 ¿f ¿1 =2,

0 ¿f ¿1 = 1, 02 ((^ 4>b2)4«>2 •••»;„£,+1 ¿f «1 =2.

Proof. We will show the assertions (i) and (ii) for . (i) For fc1 ® fc2 e B1 ® B2, we have

<61 K^A = <fo (Ml^A = ^1 (^O

= (fc1 ®fc2|<^|^'>Bi ®°)a

so that = ®°.

(ii) For C1®ji • • • «^ i e F„(H) with n = 1,2,.. .,we have

<<1®;i • • • «J„_i4 | Mi^2«,2 • • • «,„^»+1)>a = <?«1<1 • • • «j„-i C, | 4 ®fi • • • \ 4+1 >A = ¿1 ((Ç®1<1®À •••«j„-i4 | 4«^4«2 •••«;„4+1 >Bi)

<<1 «ii • • • ®Vi | | ^ >Bi ) 4«,2 • • • «>„4+1 > A

if <1 = 1,

0 if i1 = 2.

Denote by ^ the left actions of B;, i = 1,2 on F„(H) and hence on F(H), respectively. They satisfy the following equalities

(z) (fc1 ® fc2) = zfc1 ® 0, 02 (w) (fc1 ® fc2) = 0 ® wfc2,

01 (z)(^1«ii4«;2 •••«¡„-i4)

= (01 (z) O)«^ 4«;2 •••«¡„-i 4,

02 M(4 «ii 4«^ •••«¡„-i 4)

= (02 M4K 4«i2 •••«¡„-i 4

for z 6 B1, w 6 B2, fc1 ® fc2 6 B1 ® B2, and ^1®ii^2®j2 • • • ®in-ie F„(H). More generally let us denote by LA(H) and LA(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 e La(H), define I e LA(F(H)) by

L(fc1 ® fc2) = 0 for fc1 ® fc2 e B1 ® B2 c F0 (H),

ï(^1«ii4«^ • • • «>„_i4) = (i4 K4«^ • • • «>„_i4

for ^1«fi^2«i2 •••«in_i4 e F„(H).

Lemma 9. Both the maps : B; ^ LA(F(H))/or «'= 1,2 are faithful *-homomorphisms.

Proof. By assumption, the *-homomorphisms : B; ^ La(H), ( = 1,2 are faithful, so that the *-homomorphisms 0; : B, ^ La (F(H)), i=1,2 are both faithful. □

Lemma 10. For e H, z e B1, we B2, Le LA a«d c,d e C, thefollow¿ng equal¿t¿es hold on F(H):

ScÇ+di = + ^ ^cÇ+dC = ci? + díC, (54) = ïs?01 (z) , = ï^02 (w) , (55)

5ÇI5Ç = 01 (((HObi ), iC*ii? = 02 (<mOB2 )

Proof. The equalities (54) are obvious. We will show the equalities (55) and (56) for s^. We have for fc1 ® fc2 e B1 ® B2

(fc1 ® h) = (z)] <P1 (fc1)

= L[Çy1 (z&1)] =I[sç (z&1 ® 0)] = I[5ç [01 (z) (61 ®&2)]]

= [Isç01 (z)](&1 ®62). For ^1«fi^2«i2 • • • «in-1 ç„ e F„(H), « = 1,2,.. .,we have

(z))«1^1 «;i^2«i2 ••• «¡„_1 ?»

(z))«1^1 «fi^2«i2 •• •«¡„_1

ï[ç«1 (01 (z) ?1)«fi^2«i2 • ••«¡„_1 ?»]

X[S? ((01 (Z) ^1)«ii^2«i2 •• '•«.„_i ?»)]

Fs?01 (z) [^1«ii^2«i2 •••«>„ _i ?»]

so that Sl^z) = Liç01(z) on F„(H), n = 0,1,____Hence, the

equalities (55) hold.

For fcj ® fc2 e Bt ® B2, we have s^ ®fc2) = ic (L^i ))

= <i|L^i (&I))bi ®0

= (CHO^fci ® 0 = 01 (<C|L^)Bi ®^2).

For ^i®fi^ • • • ®i„-Ie F„(H), we have s^ ^^^ •••®in-I^

= (0i (<C|L^)Bi KK ^2®,2

= 0i (<CI^)BI )(?I®.I ^2®,2 •••^.„-I *»)

so that sc*Ls? = 01((C | L0Bi) on F„(H) for n = 0,1,2,.... Hence, the equalities (56) hold. □

The C*-subalgebra of LA(F(H)) generated by the operators s?, i? for £ e H is denoted by TF(H) and is called the Toeplitz quad module algebra for H.

Lemma 11. The C*-algebra TF(H) contains the operators 01(z),02(w)/or z e B1, w e B2.

Proof. By (56) in the preceding lemma, one sees that

V? = 01 «CI?>Bi ), = 02 ((CI^)b2 ),

C, ? e H.

Since H is a full C*-quad module, the inner products (£ | 0Bi, (C I 0B2 for e Hgenerate the C*-algebras Bt, B2,

respectively. Hence, 0i(Bi), 02(B2) are contained in TF(H).

Lemma 12. There exists a« action y of R/Z = T on TF(H) swcfo that

yr (s?) = e2W-% yr (i?) = e2^, £ e H,

yr (fa (z))=fa (z), ze Bi, (62)

yr (02 M) = 02 , ^ e B2 for r e R/Z = T.

Proof. We will first define a one-parameter unitary group wr, r e R/Z = T on F(H) with respect to the right A-valued inner product as in the following way.

For n = 0 : wr : F0(H) ^ F0(H) is defined by

wr (fci ® fc2) = fci ® fc2 for fci ® fc2 e Bi ® B2. (63)

For n = 1,2,... : wr : F„(H) ^ F„(H) is defined by Mr (^i®,i^2®,2 • • • ®,„-I?„) = e2*^!®,!^2®,2 • • • ®,„-I4

for ^1®ii£2®j2 e F„(H). We therefore have a one-

parameter unitary group wr on F(H). We then define an automorphism yr on (F(H)) for r e R/Z by

yr (T) = MrTw*

for T e LA(F(H)), r e

It then follows that for fci ® fc2 e Bi ® B2

yr (s?) (fci ® &2) = Mr* (fci ® &2) = (<Pi (fci)))

= e2W-% (fci ®62), and for ^i®ii • • • ®i„-ie F„(H), n = 1,2,...,

yr (s?)(^i®iI •••®,„-1

(^®i^I®,i •••®,„-I*„) (67)

Therefore, we conclude that yr(s?) = e - on F(H) and similarly yr(i?) = e27lV=Trf? on F(H). It is direct to see that

yr (0i(z)) = 0i(z), yr (02(w)) = 02M,

for z e Bt, w e B2.

It is also obvious that yr(TF(H)) = TF(H) for r e

Denote by /(H) the Cr-subalgebra of LA(F(H)) generated by the elements

finite

La (0F„ (H) ).

The algebra /(H) is a closed two-sided ideal of LA(F(H)).

Definition 13. The C*-algebra OF(H) associated with the Hilbert C*-quad module H of general type is defined by the quotient C*-algebra of TF(H) by the ideal TF(H) n /(H).

We denote by [x] the quotient image of an element x e TF(H) under the ideal TF(H) n /(H). We set the elements of

S? =[s?|, Tf =|if|,

i = ll?J

Oi (z)=[0i (z)], O2 (^)=[02 (^)j

for ^ e H and z e B1, w e B2. By the preceding lemmas, we have the following.

Proposition 14. The C*-algebra @F(H) is generated by the family of operators S^, T for Ç e H.It contains the operators <&1(z),<&2(w) for z e B1, w e B2. Theysatisfythefollowing equalities,

Sc$+di; = + Tcç+di; = cT^ + dl^ (71) =®1 {?) S^1 (z), T0! (z')fy2(w) =®1 {?) T^ M, M = ®2 ) S?^1 W, T</,2(w')^2(W) = ®2 (w') T^2 (W) , S**S? = ®1 ), T**T? = («\0b2 ) (74) for e H,c,d eC and z, z' e B1, w, w' e B2. 4. The C*-Algebras of Hilbert C*-Quad Modules of Finite Type In what follows, we assume that a Hilbert C*-quad module H is of finite type. In this section, we will study the C*-algebra @F(H) for a Hilbert C* -quad module H of finite type. Let {u1uM] be a finite basis of H as a Hilbert C*-right module over B1 and {v1,..., vn| a finite basis of H as a Hilbert C*-right module over B2. Keep the notations as in the previous section. We set S: = su, for i = 1,..., M, tk = t for k = 1,...,N. By (10) and Lemma 10, we have for Ç e H st = Xs^1 ((ui \*>)®1 )■■ h = Ih$2 «Vc \0«2).

Let Pn be the projection on F(H) onto Fn(H) for n= 0,1,... so that iZoPn =1 on F(H).

Lemma 15. For e H, one has

(i) spp = 0 for n= 1,2,... and hence s**t¡- = s**t^P0.

(ii) t**s^Pn = 0 for n= 1,2,... and hence t**s^ = t**s^P0. Proof. (i) For n= 1,2,.. .,we have

s*^ (Ï1®h • • • = s* (^1®,, • • • ®>_Jn) = 0. (77)

(ii) Is similar to (i). □

Define two projections on F(H) by Ps = The projection onto

n=0 (¡1,..Jn)cTn

1H®1 • ••®i H,

1 '1 '2 'n

Pt = The projection onto

n=0 (¡i,...,in)€Tn

•••®i H.

2 '1 '2 ln

Lemma 16. Keep the above notations.

Ists* =P1 + Ps, Itkt*k =P1 + Pt.

¡=1 k=1

Hence,

Isis* + Itktl +P0 = 1F(H) + P1.

i=1 k=1

Proof. For ^1®ii^2®ii •••®i e Fn(H) with 2 < n e N,we have

ss •••®i„_1%n)

i i S2 >2

u,®1$1 ((u, \$1)^2% •••%_iL if i1 = 1, (81) 0 if i1 = 2.

As ui»1^1({ui \ Ç^B)%2 = uicp1({ui \ Ç^B)»1%2, and

I^1uiP1({ui \ ) = 4,wehave

Zsis* (^1®i1^2®i2 •••^i„_1^n)

Ï1®1Ï2®Î2 •••\_1Ïn if i1 = 1, 0 if i1 = 2

and hence

For £ e F1(H) = H, we have s^Ç = st({u, \ © 0) = uP1 ({u \£,)b1 ) so that

Isis^^ = IuiP1 ((ui I Ob) = t i=1 i=1

and hence

F1(H).

Therefore, we conclude that

£sts* =PS + p1 and similarly £tkt*k = Pt + P1. (87)

i=1 k=l

As Ps + Pt + P0 +P1 = \F{H), one obtains (80). □

We set the operators

s, = su (=[s,]) for i=l,...,M,

Tk = TVi (=[ti]) for k=l,...,N

in the Ck-algebra OF(H). As two operators £M1 s,s* and ^N=1 tkt* are projections by (79), so are £MM1 S,S* and £N=1 tkt*. Since P1 -P0 e J(H), the identity (80) implies

£sis* + £Tkt* = I.

i=l k=l

Therefore, we have the following.

Theorem 17. Let H be a Hilbert C*-quad module over (A; B1, B2) of finite type with finite basis [u1,... ,uM} as a right B1-module and [vlt..., vn} as a right B2-module. Then, one has the following.

(i) The C*-algebra @F(H) is generated by the operators s1,..., SM, t1,..., tn and the elements ^ (z), 02(w) for z e B1, we B2.

(ii) They satisfy the following operator relations:

£sis* + £tkt* = 1, s * t = 0, (90)

i=1 k=1

1 {(ui IuJ)B1)' t*t = 02 ({Vk I ),

®1 (z) Sj = (("i I fa(z)Uj)Bi)

®1 (z) Tl = ^Tk^2 ({Vk Ifa(z)vt)B2 ), k=1 M

02 (w) Sj = YSi®1 ((ui 1 ),

02 M Tl = TTk02 ({Vk 1 M^VÙB, )

for z e B1, we B2, i,j = !,...,M, k,l = 1,...,N.

(iii) There exists an action y of R/Z = T on @F(H) such that

Yr (si) = e2^1rS,, yr (Tk) = lrTk, Yr (01 (Z)) = 01 (Z) , Yr (02 (W)) = 02 (w)

for r e R/Z = T, i = 1,...,M, k = 1,...,N, and z e B1, w e B2.

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

(ii) The first equality of (90) is (89). As the projection P0 belongs to J(H), Lemma 15 ensures us the second equality of (90). The equalities (91) come from (74). For z e B1 and

j = 1,... ,M,wehave <p1(z)uj = £M1 u^^ | fa(z)uj)B1) so that

$1 (z) Sj = S<h(z)uj = Xsu fa {(ui 1 fa(z)uj)B ) (95) i=11 which goes to the first equality of (92). The other equalities of (92) and (93) are similarly shown. (iii) The assertion is direct from Lemma 12. □ The action y of T on @f(h) defined in the above theorem (iii) is called the gauge action. 5. The Universal Ck-Algebras Associated with Hilbert Ck-Quad Modules In this section, we will prove that the Ck-algebra @F(H) associated with a Hilbert Ck -quad module of finite type is the universal Ck-algebra subject to the operator relations stated in Theorem 17 (ii). Throughout this section, we fix a Hilbert Ck-quad module H over (A; B1, B2) of finite type with finite basis {u1,... ,uM} as a right Hilbert B1-module and {v1,..., vn} as a right Hilbert B2-module as in the previous section. Let PH be the universal *-algebra generated by operators S1,..., SM, T1t...,TN and elements z e B1, w e B2 subject to the relations Ysts* + YTkT* = 1, S*Ti = 0, (96) i=1 k=1 S*SJ = (u> I ui)Bl' T*T1 = {Vk I VÙB2> (97) zSj = Zsi(uiI fa(z)u) zTl = YTk{Vk I fa(z)Vl)B2' k=1 WS1 = TSi(ui I ^2(w)ui)B > wTl = YTk{Vk I 02(w)Vl) k=1 for z e B1, w e B2, i,j = 1,...,M, k,l = 1,...,N. The above four relations (96), (97), (98), and (99) are called the relations (H). In what follows, we fix operators S1,..., SM, T1t...,TN satisfying the relations (H). Lemma 18. The sums SiS* and TkT* are both projections. Proof. PutP = XMi SiS* andQ = XN=i TkTk*.Bytherelations (96), one sees that 0 < P, Q<1, P + Q=1, and PQ = 0. It is easy to see that both P and Q are projections. □ Lemma 19. Keep the above notations. (i) For i, j = 1,.. ,,M and z e Bi, w e B2, one has S*zSj = (Ui |$i(z)Uj)B, S*wSj = (Ui | <p2(w)Uj)

(ii) For k,l = 1,.. .,N and z e Bi, w e B2, one has

TkzTi = (Vk 1 ^i(z)vi)B2 ' TkwTi = iVk 1 ^2(w)vi)

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

S*ZSJ = £S*iSh(uh 1 (z)uj)

= £("i 1 Uh).B, (uh 1 <h (z)uj)B h=i 1

= \ui 1 £uh(uh 1 <Pi (z)uJ)Bl

= (Ui Wi (z)Uj)Bi.

The other equality S*wSj = {ui | <p2(w)uf)B is similar to the above equalities.

(ii) Is similar to (i). □

By the equalities (12), we have the following. Lemma 20. Keep the above notations.

(i) For w e B2, j = 1,... M, the element S*wSj belongs to A and the formula holds:

Ys*(tlv)B2sj = (Sll)* for ive H. (103)

(ii) For z e Bi, I = 1, ...N, the element Tj*zTi belongs to A and the formula holds:

jTi,(^ln)B1 T1 =(Sln)* for ine H. (104)

Lemma 21. The following equalities for z e Bi and w e B2 hold:

z= £Si(Ui l^i(z)Uj)BiS*

£Tk(Vk l $i(z)Vi)B2Tl W= £si(ui l ^2(w)uj)B S ij=i + £Tk(Vk l ^2(W)Vi)B2 T*. kl=i z* = £Si(ui l<Pi(Zyuj)Bis* + £Tk(vk l$i(z)*Vi)B2Ti

ki=i M

W* = £Si(Ui l $2(w)*Uj)B^ S* + £Tk(Vk l ^2(W)*Vi)B2 T* ki=i ZW= £ Si(Ui l ^i{z)^2(w)Uj)B S* ij=i £ Tk(Vk l ^i(z)^2(w)vi)B2Ti* wz= £Si(Ui l$2(w)$i(z)Uj)B,S* + £ Tk(Vk l ^2(w)^i(z)vi)B2Ti Proof. (i) By (98) and (99), we have zSjS*j = lSi(ui l$i(z)uj)B,Sj'

zTiTl = £Tk(Vk l 0i(z)vi) k=i

so that by (96)

z = YSt{u> I^1(z)uj)Bl SJ

+ TTk(Vk I ^1(z)vl)B2Tl-k,l=\

Similarly, we have (106).

(ii) All the adjoints of ^1(z), $2(w) for z e B1, we B2 by the three inner products {■ | ■)B , {■ I ■)B , and {■ | ■) A on H coincide with <p1(z*), <p2(w*), respectively. Hence, the assertions are clear. (iii) By (i), we have ZW= I YS'{U' 1 h (z)uj )B Sj + T Tk(Vk 1 ^1(z)vl)B2Tl k,l=l lsg(ug I^2 (w) uh)BiS, + T Tm{vm 1 ^2(w)v„)b2T* ) . m,n=l J As S*Tm = T*Sg = 0 for any j,g=l,...,M, l,m=l,...,N, it follows that zw= T si(ui Ifa(Z)Uj) i,j,g,h=1 xS)Sg{ug 1$2 (w)Uh)B,Sh

+ T Tk{vk (z) Vi)b2

I 02(w)vn)

= T Si(Ui I fa(z)Uj)

i,j,g,h=1

X{Uj IU9)B1 (U0 I$2(™)Uh)B, S> + T Tk(Vk I ^i(z)Vi) k,l,m,n=1 X {Vl I Vm)B, {Vm I 02(W)vn)B2 K = T Si(ui I$1 (z)ug)Bi (ug I $2(w)uh)BiS1 i,g,h=1 k,m,n=1 + T Tk (vk I^1(z)TVl(vl I Vm) B2 X {Vm I = T Si(ui I$1(z)ug)Bi {ug I $2(w)uh)BiS, ifgfh=1 + T Tk{vk I 01(z)vm)B2 {vm I 02(™)vn)B2 k,m,n=1 = TSi{ui I fa(z)$2(w)uh )b,

+ T Tk{vk I fa(Z)\$2(W)vn)B2C

(113) □

Lemma 22. Let p(z,w) be a polynomial of elements of B1 and B2. Then, one has

(i) p(z,w)Sj = JM1 S,z, for some zi e B1.

(ii) p(z,w)Tl = Tkwk for some wk e B2.

Proof. For z e B1, w e B2, and i, j = 1,..., M, by putting = (ui I ^i(z)ui>Bl e Bi and w,j = (u, | ^M«,-^ e B1, the relations (98), (99) imply

zSj = IS, Zij, wSj = Y,Siwi,j (114)

i=1 i=1

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

Lemma 23. Let p(z, w) be a polynomial of elements ofB1 and B2. Then, one has

(i) S* p(z, w)Sj belongs to B1 for all i, j = 1,..., M.

(ii) Tkp(z, w)Ti belongs to B2 for all k,l = 1,...,N.

(iii) S*p{z, w)Ti = 0 for all i=1,...,M, l=1,...,N.

(iv) Tkp(z, w)Sj = 0 for all k=1,...,N, j=1,...,M.

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

p(z,w)Sj = ^Shzh for some zh e B1 (115)

S*p(z, w) Sj = TS*ShZh =T{Ui I Uh)Blzh. (116) h=1 h=1

As {ui I Uh)B, Zh belongs to B1, we see the assertion.

(ii) Is similar to (i).

(iii) As T*Si = 0, we have

T^p(z,w)Sj = TT*S,Zi = 0.

(iv) Is similar to (i).

We set

S1>; Sj,

Z -{(I,i)\i-1,...,M], Z2 - {(2, k) \k-1,...,N].

Lemma 24. Every element of PH can be written as a linear combination ofelements oftheform

S S ■

U02 'i2

■Sa i bSu :

0 m'im nri 'In

for some (gv ii), (g2, i2),..., (gm, im), (huji), J2), ■ ■■, (hn, j n) e Z1 where b is a polynomial of elements of B1 and

Proof. The assertion follows from the preceding lemmas. □

By construction, every representation of B1 and B2 on a Hilbert space H together with operators Si, i - 1,..., M, Tk, k - 1,...,N satisfying the relations (H) extends to a representation of PH on B(H). We will endow PH with the norm obtained by taking the supremum of the norms in B(H) over all such representations. Note that this supremum is finite for every element of PH because of the inequalities II^II, ||Tkl| < 1, which come from (96). The completion of the algebra PH under the norm becomes a C*-algebra denoted by OH, which is called the universal C*-algebra subject to the relations (H).

Denote by C* (fa(B1),fa(B2)) the C*-subalgebra of LA(H) generated by fa1(B1) and fa(B2).

Lemma 25. An element L of the C*-algebra C*(fa1(B1), fa(B2)) is both a right B1-module map and a right B2-module map. This means that the equalities

№ cp, (k) - L (b,)] for H,b, e Bi (121)

Proof. Since both the operators fa (z) for z e B1 and fa2(w) for w e B2 are right B,-module maps for i - 1,2, any element of the *-algebra algebraically generated by fa1(B1) and fa2(B2) is both a right B1-module map and a right B2-module map. Hence, it is easy to see that any element L of the C*-algebra C* (fa(B1),fa(B2)) is both a right B1-module map and a right B2-module map. □

Denote by B0 the C*-subalgebra of OH generated by B1 and B2.

Lemma 26. The correspondence

z,we Ba 1 (z), ^ (w) e C* (fa (Bx), fa (B2))

gives rise to an isomorphism from B0 onto C*(fa1(B1), fa(B2)) as C*-algebras.

Proof. We note that by hypothesis both the maps

fa : z e B1 fa (z) e La (H), fa :we B2 ~^fa (w) e La (H)

are injective. Denote by P(fa1(B1 ),fa2(B2)) the *-algebra on H algebraically generated by fa1(z), fa2(w) for z e B1, w e B2. Define an operator n(L) e OH for L e P(fa (B1 ),fa(B2)) by

n(L) - fsi(ui \ Luj)b S* + !Tk(vk \ Lv1)BiT*.

j 1 ~"U B"J i,j=1 k'l=1

Let P0 be the *-subalgebra of PH algebraically generated by B1 and B2.Since n(fa(z)) - z for z e B1 and n(fa(w)) - w for w e B2 and by Lemma 21, the map

n:P(fa B),fa2 (B2))-^P. c B„ (125)

yields a *-homomorphism. As S**Tk - 0 for i - 1,..., M, k -1,..., N,we have

IIn(L)H - Max

lSi(ui \luj)b1 S. i'j=1

Z TkiVk \ Lvi)b2T* k'l=1

We then have

Is, (u, \Luj)b1Sj i'j=1

< z \\(u, \ luj) i'}=1

and similarly

Z Tk(Vk \ lv^bt*

,i'i=1

\UiI^Bl imb,

<( IMb2IIviIB )nm. (128)

,k'l=1

k,l=1 By putting

C - Max{lM=1 MYb,IIuj\IBi, Tk,i=1 IIvkIIB2MB2], one has

IIn(L)II<CHm, VLeP(fa (B1), fa (B2)). (129)

Hence, n extends to the C* -algebra C* (fa1(B1),fa2(B2)) such that n(C*(fa (B1), fa2(B2))) - B„. The equality (124) holds for L e C*(fa1(B1),fa2(B2)).

We will next show that n : C*(fa1(B1),fa2(B2)) ^ B„ is injective. By (124), we have for L e C*(fa1(B1),fa2(B2)) and h,h' = 1,...,M,

S'hn(L)Sh, = Ys'hsi(ui I Luj)B s*jsh' i,j=i 1

I k^h K^, {Ui ILuj)B, (Uj Iuh' )B ,

ij=l M M

= I\Uh Ilui(ui ILuj)B) (Uj Iuh' ) j=l\ i=l

= Y(uh 1 Luj>Bl (ui 1 uh'>B = 1 Luh'!• j=i 1 1

Suppose that n(L) = 0 so that {uh | Luhi)B 1 = 0. Since

Luh> = Yuh (uh 1 Luh' ,' (131)

we see that Luy = 0 so that L = 0. We thus conclude that n is injective and hence isomorphic. □

Denote by fa„ : B„ ^ C*(fa1(B1),fa2(B2)) the inverse n-1 of the *-isomorphism n given in the proof of the above lemma which satisfies

fa (z) = fa (z) for z e B1, fa (w) = fa2 (w) for w e B2.

We put FH = B„. For ne N, we denote by FnH the closed linear span of elements of the form

C1 7 oh Oh Oh

for some (g1, i1), (g2, i2),..., (gn, in), (h1,j1), (h^, )2),..., (hn, jn) e ^ U 12 and b e B0. Let us denote by FH the Cj-subalgebra of OH generated by U^FH^. Bythe relations (105) and (106), we see the following.

Lemma 27. For x e B0, thefollowing identity holds:

x=I Si(ui I fao(x)Uj)B Sj + 1 Tk{vk I fa(x)vl)B2 Tj. i,j=l 1 k,l=l

Hence, by putting for b e B0

hij ={ui Ifa°(b)uj)Bl> i,j = h...,M, hki = (*k Ifa(b)vi)a2, k,l=\,...,N, we have the following.

Lemma 28. For b e B0, the identity

c1 c1 C1 7 o* ri* ri h

^2 ■ ■ ■ b9„,i„ °\,j„ ■ ■ ■ \,j2

ZOO 0 0 7 o* ri* ri * ri *

^02,i2 ■ ■ ■ b9„,i„ ¿l,iDl,ij^l,j\,j„ ■ ■ ■ \2,j2\,h

Zc1 c1 0 0 7 O * O * o* o *

' '' Un^n 2,k 2,kl 2,1 hn,jn • • • \2,j2bhl,jl

holds and induces an embedding of FnH ^ FH+1 for n e Z+.

Lemma 29. The C*-algebra is the inductive limit

limn^mFnn of the sequence of the inclusions

F°x w Fnr '

Let e n ~ 1 e T be a complex number of modulus one for r e R/Z. The elements

C à:,

z e Bu we B2

e2nV-IrTk, k=l,

St, i=l,...,M, Tk, k=l,

ze Bu we B2

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

hr (Si) = e2^rS„ hr (Tk) = emrTk, hr (z) = z, hr (w) = w

for i = \,...,M, k = \,...,N, z e Bv w e B2, and r e R/Z = T. As the C*-algebra has the largest norm on the action (hr)reT on extends to an action of T on still denoted by h. The formula

e On \ hr (a) dr e

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

Lemma 30. (6H)h = F;

The C*-algebra satisfies the following universal property. Let D be a unital C*-algebra and ^ : B1 ^ D,

D be *-homomorphisms such that

01(a) = O2(a) for a e A. Assume that there exist elements S1;..., SM, T1,...,TN in D satisfying the relations

+ = 1, S*T; = 0,

>=l fc=l

S*S, = (<M, | Mj>Bi), T^ = O2 ((vfc | V;)b2)

Oi (z)S, = £s,Oi (<«, |fa(z)Mj>B ), >=l 1

01 (z)T; = £rfcO2 ((vfc |0i(z)vj)b2),

02 = £?,Oi (<«, IfaM"i>B ),

O2 (w)T; = £rfcO2 ((Vfc |02(^)V^)b2 ),

for z e we B2, (, j = 1,..., M, M = 1,..., N; then, there exists a unique *-homomorphism O : OH ^ D such that

O(S,) = S,; O(Tfc) = Tfc, O (z) = Ol (z), O (w) = O2 (w)

for i = 1,..., M, fc = 1,..., N and z e we B2. We further assume that both the homomorphisms O; : B; ^ D, « = 1,2 are injective. We denote by O0 : B0 ^ D the restriction of O to the subalgebra B0. Let us denote by ©H the C*-subalgebra of D generated by Tk, i = 1,..., M, fc = 1,... and O1(z), O2(w) for z e w e B2.

Lemma 31. Keep the above situation. The *-homomorphism O0 : B0 ^ D is injective.

Proof. Since the correspondence in Lemma 26

fa : z, w, e Bo fa 1 (z), 02 M e C* (fa (Bi), fa (B2))

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

fa (z),fa (w) eC* (fa (Bi),fa (B2)) —1 (z),O2 (w) e D

yields an isomorphism. Let B0 be the C*-subalgebra of ©H generated by elements O1(z), O2(w) e A for z e B1,

w e B2. Define an element 7r(L) of D for L e C*(0l(Bl),02(B2)) by setting

(L)= £s,Oi (<M,|LMj>B i ;>j=l

+ ((vfc | ¿v;)b2 K e D.

fc,i=i

As in the proof of Lemma 26, one sees that n gives rise to a *-homomorphism from C*(01(B1), fa(B2)) into D. Since

Tî(fa (z))= Ol (<«, |fa(z)Mj>Bi)s

+ ((vfc |0i(z)v;)b2)T;

W=i = Oi (z),

and similarly fr(fa(w)) = O2(w), it is enough to show that 7T is injective. Suppose that fr(L) = 0 for some L e C*(0l(Bl), 02(B2)). By following the proof of Lemma 26, one sees that S*7r(L)Sh/ = Ol((wh | Lw«/}Bi) for all fo, fo' = 1,...,M. Hence, the condition 7r(L) = 0 implies that Ol((wh | Lw«/}Bi) = 0. Since Ol is injective, we have (wh | }B i = 0 for all fo, fo' = 1,..., M.As L is a right Bl-module map, we have for £ e H,

^ = | Ob i ) = Z ) H' | ^ = 0

h'=l h'=l

so that L = 0. Therefore, tt : C*(0l(Bl),02(B2)) ^ D is injective. Hence, the composition

O0 : tt o fa : B, C* (fa (Bi) ,fa (B2)) D (149) is injective. □

We set

Sl>j :=S;, ¿=1,...,M, ?2>fc :=îfc, fc=1,...,N.

We put FH = B0. For n e N, let FH be the closed linear span in the C*-algebra of elements of the form

sa i sa i -"S. i o0(fc)s*; . •••si* . s*;

for (^l, Í1), (^2, <2), . . . , ), (^1, J1X (^2, i2), . . . , jj e Z1UZ2 and fc e B0. Similar to the subalgebras FH, n e Z+, of

OH, one knows that the closed linear span FH is a C* -algebra

and naturally regarded as a subalgebra of FH+1 for each n e

Z+. Let us denote by FH the C*-subalgebra of QH generated

by U^=°FH• Then, the C*-algebra FH is the inductive limit such that 01(a) = O2(a) for a e A and there exist elements limn ^ œFH of the sequence of the inclusions S1

FH — FH — FH — ••• — — FH

Lemma 32. Suppose that both the *-homomorphisms O; : B> — @H, t = 1,2 are injective. Then, the restriction of O to the subalgebra FH yields a *-isomorphism

O|FH : FH — FH.

Proof. By the universality of OH, the restriction of O to FH yields a surjective *-homomorphism O|Fh : FH — FH. It suffices to show that O|f is injective. Suppose that Ker(O|FH) = {0} and put I = Ker^O^)• Since Of(F^) = FH and FH = limn^œFH, there exists ne Z+ such that I n FH = 0. Let us denote by Z" the set of n-tuples of Z1 U Z2 :

Z" = eZj U Z2}. (153)

For ^ = ,..., e Z", denote by S^ the operator

if Fm = (1,') e Z1,

s2,fc = rfc if Fm = (2, fc) e Z2, m = 1,..., n.

Any element of FH is of the form

X S* for some e B0.

Hence, one may find a nonzero element ^ ]gz„ S^fc^S* e

I n FH. *Since S* + TfcT* = l, the equality Z^sz" ^= 1 holds. For some w,ye Z", one then sees

0 = s; ( X S^A* )sr em FH-

As s;rfc = 0 and s;s, = (M; 1 «,)Bi, =

(vfc | vl)B2 for i, j = 1,..., M, fc, / = 1,..., N, the element sAA*)sy belongs to I n Bo. By the preceding

lemma, the homomorphism O0 : B0 — is injective, so that we have O0(S* S^.A^y) = 0 a contradiction.

Therefore we conclude that 0|Fh : FH — FH is injective and hence isomorphic. □

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

Theorem 33. Let D beaunital C*-algebra. Suppose that there exist *-homomorphisms O1 : B1 — D, O2 : B2 — D

S*Tl = 0, j l

, SM, T1,...,TN in D satisfying the relations

Xs,s; + X^r; = 1, >=1 ¿=1

= O1 (<M,|M;>BI ),

rfc*% = O2 «vfc | Vi>B2),

01 (z)S, = X?,O1 (<m, I01(z)«;->B)

01 (Z)Ti = XrfcO2 ((Vfc |01(z)Vi>B2),

02 (^)S, = X?,O1 (<M, IfaM",>B ),

02 (w)Tl = X% ((Vfc |faMv*>B2)

for z e B1, w e B2, (, j = 1,...,M, M = 1,...,N. Let us denote by the C*-subalgebra of D generated by S;, Tfc, « = 1,...,M, fc = 1,...,NandO^z), O2(w),forz e B1, w e B2. One further assumes that the algebra admits a gauge action. If both the *-homomorphisms O; : B; ^ A, ( = 1,2 are injective, then there exists a *-isomorphism O : OH ^ satisfying

O(S,) = S„ O(Tfc) = Tfc, O (z) = O1 (z), O (w) = O2 (w)

for î = 1,..., M, fc = 1,..., N and z e B1, w e B2.

Proof. By assumption, admits a gauge action, which we denote by k Let us denote by (0H)'i the fixed-point algebra of under the gauge action h and by FH the C* -subalgebra of defined by the inductive limit (152). Then, it is routine to

check that ( ©H)'i is canonically *-isomorphic to FH. There exists a conditional expectation

defined by

(*) = f 5

/ir (x) dr for x e

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

O(S,) = S„ O(Tfc) = Tfc, O (z) = O1 (z), O (w) = O2 (w)

for i,j= 1,...,M, k,l = 1,...,N, z e B1, we B2. Then, Q(FH) = FH and the following diagram

is commutative. Denote by O0 the restriction of O to the C*-subalgebra B0 of OH generated by z e B1, w e B2. By assumption, both the maps O; : ^ 0H, i = 1,2 are injective, so that O0 : B0 ^ OH is injective by Lemma 31. By the preceding lemma, 0\Fh : FH ^ FH is an isomorphism. Since the conditional expectation EH : OH ^ FH is faithful, a routine argument shows that O is injective and hence isomorphic. □

Therefore, we have the following.

Theorem 34. For a C*-quad module H of finite type, the C*-algebra OF(H) generated by the quotients [s^j, [t^] of the creation operators s^, t^ for % e H on the Fock spaces F(H) is canonically isomorphic to the universal C*-algebra OH generated by operators S1,...,SM, T1,...,TN and elements z e B1, w e B2 subject to the relations

S*T, = 0,

is, s; + ïtkt; = 1,

i=1 k=l

S;Sj = (ui | Uj)b, = (vk I v1)b2

ZSj = ^si(ui I^i(Z)Uj)bi, ¡=1 1

zTl = tTk(vk 1 fa1(z)vl).B2 , k=l

WSJ = tSi(Ui 1 fa2(w)uj)B , ¡=1 1

wTl = y,Tk(vk 1 fa2(w)vl).B2

for i,j= 1,...,M, k,l= 1,...,N and z e B1, w e B2.

Proof. Theorem 17 implies that the operators sj,..., sm, tj,..., tn and the elements <^1(z), <&2(w) for z e B1, w e B2 in 0F(H) satisfy the eight relations of Theorem 33. By Theorem 33, we see that the correspondences

S, si,

z e B1 —> 01 (z),

k ~ w e

O2 (w)

for i = 1,...,M, k = 1,...,N, and z e B1, we B2 give rise to an isomorphism from OH to 0F(H). □

The eight relations of the operators above are called the relations (H). The above generating operators S1,..., SM and T1,...,TN of the universal C*-algebra OH correspond to two finite bases {u1,..., uM} and {v1,.. ., vn} of the Hilbert C* -quad module H, respectively. On the other hand, the other C* -algebra OF(H) is generated by the quotients of the creation operators for % e H on the Fock spaces F(H),whichdo not depend on the choice of the two finite bases. Hence, we have the following.

Corollary 35. For a C*-quad module H of finite type, the universal C*-algebra OH generated by operators S1,...,SM, T1,...,TN and elements z e B1,w e B2 subject to the relations (H) does not depend on the choice of the finite bases {u1,..., uM} and {v1,.. ., vn}.

6. K-Theory Formulae

Let H be a Hilbert C*-quad module over (A; B1, B2) of finite type as in the preceding section. In this section, we will state K-theory formulae for the C*-algebra OF(H). By the previous section, the C*-algebra OF(H) is regarded as the universal C*-algebra OH generated by the operators S1 ,...,SM and T1,...,TN and the elements z e B1 and w e B2 subject to the relations (H). Let us denote by B0 the C*-subalgebra of OH generated by elements z e B1 and w e B2. By Lemma 26, the correspondence

z,w e B0 —> 01 (z), fa (w) e C* (fa (B), fa (B)) c La (H)

gives rise to a *-isomorphism from B0 onto C* (fa1(B1), fa(B2)) as C*-algebras, which is denoted by fa. We will restrict our interest to the case when

(i) S1,..., S M and T1,...,TN are partial isometries, and

(ii) SlS*,... ,SмSM, T1T*,...,TnTN commute with aU

elements of B0.

If the bases [u1,..., uM} and {v1,..., vn} satisfy the conditions

(u, IUj)B =0 for i = j,

(Vk I vi)b =0 for k = l

the condition (i) holds. Furthermore, if fa1 (z) acts diagonally on {u1,... ,uM} for z e B1 and fa2(w) acts diagonally on {v1,..., vn} for w e B2, the condition (ii) holds. Recall that the gauge action is denoted by h which is an action of T on OH such that the fixed-point algebra (@H)h under h is canonically isomorphic to the C*-algebra FH. Denote by h the dual action of h which is an action of Z = T on the C* -crossed product OHXhT by the gauge action h of T. As in the argument of [16], OHXhT is stably isomorphic to FH. Hence, we have that K*(@HXhT) is isomorphic to K* (FH). The dual action h induces an automorphism on the group K*(@HXhT ) and hence on K*(FH), which is denoted by a*. Then, by [16] (cf. [10,17], etc.), we have the following.

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

I (168)

We put for Bo

Xu (x) = S**xSi, i=1,...,M, xxk(х) = т;хтк, k = i,...,N.

Both the families X1t, X2k yield endomorphisms on B° which give rise to endomorphisms on the K-groups:

^2,k* : Ко

k=1,...,N.

We put X0 = + ZZi which is an endomor-

phism on K0(Bo). Now, we further assume that K1(FH) = {0}. It is routine to show that the groups Coker(id - a*) in K0(FH) and Ker(id - a*) in K0(FH) are isomorphic to the groups Coker(id-X0) in K0(BJ, and Ker(id-X0) in K0(Bo), respectively by an argument of [17]. Therefore, we have

Proposition 37. The following formulae hold:

K0 (OH) = Coker (id - X0) in K01 K1 (OH) = Ker (id - X„) in K0 (

7. Examples

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

(1) Let a, p be automorphisms of a unital C*-algebra A satisfying a ° p = p ° a. Let Hap be the associated Hilbert C*-quad module of finite type as in (1) in Section 2. It is easy to see the following proposition.

Proposition 38. The C*-algebra OH associated with the Hilbert C*-quad module Hap coming from commuting automorphisms a, p of a unital C*-algebra A is isomorphic to the universal C*-algebra generated by two isometries U,V and elements x of A subject to the following relations:

UU* + W* = 1,

UU*x = xUU*, VV*x = xVV*, (172)

a(x) = U*xU, ¡3(x) = V*xV

for x e A.

(2) We fix natural numbers 1 < N, Me N. Consider finite-dimensional commutative C*-algebras A = C,

B1 = CN, and B2 = CM. 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 HM^N the Hilbert C* -quad module CM 0 CN over (C; CN, CM) defined in (2) in Section 2. Put the finite bases

ut = et 0 1 e HMN, i = I,..., M

as a right B1 -module, vk =l0fk e HMt№ k=h...,N

as a right B2-module.

We set = {(«, k) \ 1<i< M, 1 < k < N} and put e(iJi) = ei ® fk, (i, k) e the standard basis of HM N. Then, the C* -algebra B° on HM N generated by B1 and B2 is regarded as

CM ® CN = B2 0 B1. Hence,

Bo = Y Ce{t,ky

(i,k)еГ

Lemma 39. The C*-algebra 0Hmn is generated by operators si, Tk> en,k), i=1,...,M, k=1,...,N satisfying

!sts; + YTkTk =

i=1 k=1

S* О ^ гт-т* гр Л

i si = s>,p Tk Tl = skp

emSj = 8Uj^SjeiKky e{hk)Tl = Ski £ Tle{ijn) (177)

h=1 m=1

for i,j =1,...,M, k,l=1,...,N.

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

e(i,k)Sj = sj(uj \ fa(e{iM))Uj)B

= Sj{ej ®1\(ei ®fk)(ej 0 1))B

= Si,jSj (l0fk) = Si,jSj Ye(h,k).

The other equality of (177) is similarly shown.

sd,k) = ed,k)sf Td,k) = en,k)Tk for (i,k) e (179) Then we have the following.

Lemma 40. The following equalities hold:

em - s(i,k)s(t,k) + TrnT(.i,ky (180)

Si - XS(>,k), Tk - Jrm, (181)

k=1 i=1

Z S(i,k)S{i,k) + Z T(ikk)TW - 1 (182)

(i,k)ez° (i,k)tT

S(i,k)s(i,k) - Z {S(j,k)S*w + TwT(jk)), (183)

T(*i,k)T(i,k) - X (S(i,l)S(iJ) + T(i,l)T(i,l)) (184)

for i = 1,...,M, k = 1,...,N and (i, k) e Proof. Since emSj = 8UjemSt, we have

S(i,k)S(i,k) - e(i,k)SiS*em - em (ZSjSj j em (185)

and similarly TmT(Uk) - em(£N=i T{T*)ß(i,k). Hence, we have

e(i,k) - e(i,k) I TSjS* + ZTiTI ) e(ik

" j ' l~(ijc)

J=1 1=1 / (186)

- S(i,k)S(i,k) + TmTm,

so that (180) holds. As 1 = X(jk)eL° e(j,k), the equality (182) holds. Since £(jk)Si = 0 for j = i, we have

si-( Z

(jk) ) Si - ZS(i,k) / k=1

and similarly Tk = Xi=i ^(¡,k), so that (181) holds. By (177), it follows that

S(i,k)S(i,k) - S* Yßie(jkk) - Ze(j,k)

j=1 j=1

- Z (S(j,k)S*j,k) + T(j,k)T(j,k)) , j=1

and similarly, we have

T(*i,k)T(i,k) - Z (SaA + T(i,i) T*m) . (189)

Theorem 41. The C*-algebra associated with the

d H M.N

partial isometries S^ijc),T^ijc) for (i,k) e = {(i,k) | i = 1,...,M, k = 1, ...,N] satisfying the relations

Z S(i,k)S(*i,k) + Z T0.k)T*,k) = 1

(i,k)€V (ik)ez'

n^ n \ / n n^ m rr-r* \

\ikrm = L \(j,k)^(j,k) + 1 (jk)1 (jkk)), (190)

qri* qri \ ' / ft q* qn qri* \

1(ik)1(i,k) - Z (^(i,l)^(i,l) + 1(i,l)1(i,l)J

for (i, k) e

Proof. By the preceding lemma, one knows that e^), Sit Tk are generated by the operators S^ k), T(ik) so that the algebra 0Hmn is generated by the partial isometries S^ijc), T(i,k), (i,k)eT. □

Let In be the nxn identity matrix and En the nxn matrix whose entries are all 1s. For an M x M-matrix C = [ci j]ij=1

and an N x N-matrix D = [dkj]kl=1, denote by C 8 D the MN x MN matrix

cnD cnD ■■■ C1mD

C21D c22D

CM1D cm2D ■■■ cmmD

\in in ■ ■ in

in in ■ ■ in

.in in ■ '■ in

en 0 ■ ■■ 0

The index set {(i,k) | i = 1,...,M, k = 1,...,N} of the standard basis of CM ® CN is ordered lexicographically from left as in the following way:

(1,1),...,(1,N),(2,1),...,(2,N),...,(M,1),...,(M,N).

Put the MN x MN matrices

^m,n = ® ^n> and the 2MN x 2MN matrix U

bm,n - im ® en

M,N Amn

bM,N bMN

Hilbert C*-quad module HMN = CM 8 CN is generated by Then, we have the following.

Theorem 42. The C*-algebra OH is isomorphic to the Cuntz-Krieger algebra OH for the matrix HM N. The algebra OH is simple and purely infinite and is isomorphic to the Cuntz-Krieger algebra @amn+bmn for the matrix AM N+BM N.

Proof. By the preceding proposition, the C*-algebra OH is isomorphic to the Cuntz-Krieger algebra OH for the matrix HM>N. Since the matrix HM N is aperiodic, the algebra is simple and purely infinite. The nth column of the matrix HMN coincides with the (n + N)th column for every n = 1,...,M. One sees that the matrix Am,n + ^m,n is obtained from HMN by amalgamating them. The procedure is called the column amalgamation and induces an isomorphism on their Cuntz-Krieger algebras (see [15]). □

The operator fa(z; ) on H is adjointable with respect to the inner product (• | whose adjoint fa(z;)* coincides with the adjoint of fa(z;) with respect to the inner product (• | so that fa(z;)* = fa(z*). We assume that the left actions fa of B; on H for i = 1,2 are faithful. We require the following compatibility conditions between the right Amodule structure of H and the right A-module structure of B; through

të I ^fl>Bf = I ^>BfYt (fl) > ^ e H, fl e A, i = 1,...,n.

In [15], the abelian groups ZMN/(A

M,N + ^M,N

We further assume that H is a full Hilbert C*-bimodule with respect to the inner product (• | (• | for each.

M,N + ^M,N

imn)Zmn, Ker(A computed by using Euclidean algorithms. For the case M = 2, they are Z/(N2 - 1)Z, {0}, respectively, so that we see Ko(Oh2>n) = Z/(N2 - 1)Z,^(Oh2,n) = 0 (see [15] for details).

(3) For a C*-textile dynamical system (A, Z^, k),

let be the C*-quad module over (A; B1, B2) as in (3) in Section 2. The C*-algebra Ohm has been studied in [12].

8. Higher-Dimensional Analogue

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

Let A be a unital C*-algebra and let B1,..., Bn be n-family of unital C*-algebras. Suppose that there exists a unital embedding

) in jmn have been A Hilbert C*-multimodule H over (A;B,, i = 1,...,n) is

l: A ^ B:

for each i = 1,..., n. Suppose that there exists a right action y, of A on B, such that

fc,w (fl) e B, for fc, e B,, a e A, i = 1,..., n. (197)

Hence, B, is a right A-module through for i = 1,..., n. Let H be a Hilbert C*-bimodule over A with a right action of A, an A-valued right inner product (• | -}A, and a *-homomorphism 0A from A to LA(H). It is called a Hilbert C*-multimodule over (A;B,, i = 1,...,n) if H has a multistructure of Hilbert C*-bimodules over B, for i = 1,..., n such that for each i = 1,..., n there exist a right action <p, of B, on H and a left action fa of B, on H and a B,-valued right inner product (• | •}B such that fa(z,) e LA(H) and

[0, (z,)^- (^.) = fa (z,) [^ (^.)]

tyj (Zjfj (fl)) = [£<Pj (z;)] fl for £ e H, z, e B,, e Bj, fl e A, i, j = 1,..., n and

0A (fl) = (', (fl)), fl e A, i=1,...,n. (199)

said to be of general type if there exists a faithful completely positive map A; : B; ^ A for i = 1,..., n such that

A, (fc, fi (fl)) = A; (fc;)fl, fc, e B ((^>b,) = (*U>*, Ue H

fl e A,

i = 1,..., n.

A Hilbert C*-multimodule H over (A; B;, ( = 1,..., n) is said to be of finite type if there exists a family j«^',..., }, « = 1,..., n of finite bases of H as a right Hilbert Brmodule for each i = 1,..., n such that

X^V, «Mf |Ç>B) = Ç, £ e H, ;=!,...,«

<«j° | faH)«j,0>B. e A, e Bfc, j,h = 1,..., M(,),

j i TK*~K>--n IB M'"

for all ^ e H, (', fc = 1,..., n with i = fc.

By a generalizing argument to the preceding sections, we may construct a C* -algebra OF(H) associated with the Hilbert C*-multimodule H by a similar manner to the preceding sections; that is, the C*-algebra is generated by the quotients of the n-kinds of creation operators , £ e H, i = 1,..., n on the generalized Fock space F(H) by the ideal generated by the finite-rank operators. One may show the following generalization.

Proposition 43. Let H be a Hilbert C*-multimodule over (A; B;, i = 1,...,n) of finite type with a finite basis {m(,), ..., «M(i)} of H as a Hilbert C*-right module over B; for each i = 1,..., n. Then, the C*-algebra OF(H) generated by the quotients of the n-kinds of creation operators on the generalized Fock spaces F(H) is canonically isomorphic to the universal

C*-algebra generated by the operators S(l),..., t) and elements zi e Bt for i = 1 ,...,n subject to the relations

>=1 k= 1

i x«* = i, =o, i=j,

sfs« = (4° | M(')>B1, (203)

forZj e Bj, i,j=1,...,n, k,l= 1,...,M(i\m = 1,...,M{J).

The proof of the above proposition is similar to the proof of Theorem 1.

Conflict of Interests

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

Acknowledgment

This work was supported by JSPS KAKENHI Grant Number 23540237.

[11] K. Matsumoto, "C*-algebras associated with textile dynamical systems," , http://arxiv.org/pdf/1201.1056.pdf.

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