数理解析研究所講究録 1291 巻 2002 年 84-94

84
CORING STRUCTURES AND HILBERT C MODULES*

大阪女子大学理学部-

大内 本夫 (MOTO O’UCHI)OSAKA WOMEN’S UNIVERSITY

1. INTRODUCTION

S. Baaj and G. Skandalis [1] introduced the notion of multiplicative unitaries andthey studied Hopf C’-algebras associated with them. J. M. Vallin introduced thenotion of pseud0-multiplicative unitaries and algebraic structures associated withthem ([11], [12]). M. Enock and Vallin [2] studied pseud0-multiplictive unitaries
and quantum groupoids associated with inclusions of von Neumann algebras. The
author introduced anotion of multiplicative unitary operators (MUO) on Hilbert$\mathrm{C}$’-modules([8], see also [6] and [7]). It is interesting to study natural algebraic

structures associated with MUO’s. In this note, we will show the relation btween

MUO’s and coring structures on Hilbert C*-modules. Coring structures were introduced by M. Sweedler [10] in the purely algebraic framework. Y. Watatani [13]showed that incusions of C’-algebras give natural coring structures in the framework of his index theory. In this note, we introduce notions of coring structures on

Hilbert C’-modules and study coring structures associated with MUO’s. In Sections
2and 3, we study coring structures associated with MUO’s arising from groupoids

and inclusions of C’-algebras of inedx finite type in the sense of Watatani. In thecase of groupoids, the base algebras are commutative. In the case of inclusions ofDate: May 29 2002.

2000 Mathematics Subject Classification.

$46\mathrm{L}08$.

Key words and phrases. Multiplicative unitary oprator, coproduct, inclusion ofgroupoid, compact operator$C^{*}$-algebras,

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C’-algebras of index finite type, we do not know any concrete examples of MUO’sfinite-dimensional Hilbert C’-modules. Therefore it is interesting to study con-oncrete examples of MUO’s and the associated coring structures such that the basealgebras are not commutative and the Hilbert C’-modules are

finite-dimensional.

In the last section, we study an MUO and the associatd coring structures on theHilbert$\mathrm{C}$’-module of compact operators. In this case, the Hilbert C’-module isinfinite-dimensional and the base algebra is the$\mathrm{C}$’-algebra of compact operators.

2. PRELIMINARIES

2.1. Multiplicative operators on Hilbert C’-modules. Let A be aC’-algebra,
let

$E$

be aHilbert

We assume thatWe defineand define$\mathrm{a}*$$\phi$$A$-module and letand$\psi$

and$\psi$$\mathrm{b}\mathrm{e}*$-homomorphisms of$A$

commute, that is, $\phi(a)\psi(b)=\psi(b)\phi(a)$ for all-homomorphism$\mathrm{a}*$$\phi$-homomorphism$\iota\otimes_{\phi}\psi$$\iota\otimes_{\psi}\phi$

Let $W$ be an operator in Ca(E$\otimes_{\psi}$to$\mathcal{L}_{A}(E\otimes_{\phi}E)$by$(\iota\otimes_{\phi}\psi)(a)=I\otimes_{\phi}\psi(a)$

.

of

$A$

to

$\mathcal{L}_{A}(E\otimes_{\psi}E)$by$(\iota\otimes_{\psi}\phi)(a)=I\otimes_{\psi}\phi(a)$.

$E,$ $E\otimes_{\phi}E$). We assume that $W$ satisfies the,$W(\iota\otimes_{\psi}\phi)(a)=(\phi\otimes_{\phi}\iota)(a)W$(2.2)$W(\psi\otimes_{\psi}\iota)(a)=$( Op ) $(a)W$,(2.3)$W(\phi\otimes_{\psi}\iota)(a)=$( &p ) $(a)W$$\psi$$\iota$$\psi$$\iota$for all $a\in A$ . Then we can define following operators;$E\otimes_{\psi}E$

C&p $E$ ,$E\otimes_{\phi}E\otimes_{\psi}E$),$I\otimes_{\phi\otimes\iota}W\in \mathrm{C}\mathrm{A}(\mathrm{E}\otimes_{\phi}E\otimes_{\psi}E, E\otimes_{\psi}E\otimes_{\phi}E)$$W\otimes_{\phi}I\in \mathrm{C}\mathrm{A}(\mathrm{E}\otimes_{\psi}E\otimes_{\phi}E, E\otimes_{\phi}E\otimes_{\phi}E)$$I\otimes_{\psi\otimes\iota}W\in \mathcal{L}_{A}$(, $b\in A$ .

$A$

(2.1)

(

$a$

of

following equations;$W\otimes_{\psi}I\in \mathcal{L}_{A}$to Ca(E).

$E\otimes_{\psi}E\otimes_{\psi}E$,$E\otimes_{\iota\otimes\psi}$,,( $E$ &p $E$ )),$I\otimes_{\iota\otimes\phi}W\in \mathcal{L}_{A}(E\otimes_{\iota\otimes\phi}(E\otimes_{\psi}E), E\otimes_{\phi}E\otimes_{\phi}E)$

.

86
Since

$\phi$

and$\psi$

commute, there exists an isomorphism

as Hilbert$E\otimes_{\iota\otimes\phi}(E\otimes_{\psi}E)$$A$$\Sigma_{12}$

of

onto

-modules such that, for $x_{i}\in E(i=1,2,3)$ ,$\Sigma_{12}(x_{1}\otimes(x_{2}\otimes x_{3}))=x_{2}\otimes(x_{1}\otimes x_{3})$Definition 2.1 ([8]). Let $W$ be an element of$W$$E\otimes_{\iota\otimes\psi}(E\otimes_{\phi}E)$.

$\mathcal{L}_{A}(E\otimes_{\psi}E, E\otimes_{\phi}E)$. Assume thatsatisfies the equations (2.1), (2.2) and (2.3). An operator $W$ is said to be multi-

plicative if it satisfies the pentagonal equation(2.4)$(W\otimes_{\phi}I)(I\otimes_{\phi\otimes\iota}W)(W\otimes_{\psi}I)=(I\otimes_{\iota\otimes\phi}W)\Sigma_{12}(I\otimes_{\psi\otimes\iota}W)$

Example 2.2. Suppose thatCc(E)$=\mathrm{C}(\mathrm{H})$$A=\mathrm{C}$. Then $E=H$ is ausual Hilbert space andis the C’-albebra of bounded linear operators on $H$ . Let $\phi=\psi=id$ ,. Then $E\otimes_{id}E$ is the usual tensor product $H\otimes H$ .

where$\mathrm{i}\mathrm{d}(\mathrm{A})=\lambda I_{H}$

for A

Let I

$\in \mathcal{L}(H\otimes H)$

be the flip, that is,$\mathcal{L}(H\otimes H)$$\in \mathrm{C}$$\Sigma(\xi\otimes\eta)=\eta\otimes\xi$. Let $W$ be an element of. Then the pentagonal equation (2.4) has the following form:(2.5)$(W\otimes I)(I\otimes W)(W\otimes I)=(I\otimes W)$

Defin an operator
if and only if$\overline{W}$$\overline{W}$by$\overline{W}=W\Sigma$(I$\otimes I$) $(I\otimes W)$ .

. Then $W$ satisfies the pentagonal equation (2.5)

satisfies the usual pentagonal equation ;

.

(2.6)

$\overline{W}_{12}\overline{W}_{13}\overline{W}_{23}=\overline{W}_{23}\overline{W}_{13}$2.2. Coproducts on Hilbert C’-modules. Let E be aHilbertbe$\mathrm{a}*$-homomorphism ofDefinition 2.3. Let
uct of $(E, \phi)$ if
(2.7)

(2.8)

.

$\delta$$\delta$$A$

to$\mathcal{L}_{A}(E)$$A$-module and$\phi$.

be an operator in$\mathcal{L}_{A}(E, E\otimes_{\phi}E)$. We say thatsatisfies the following equations;$\delta\phi(a)=(\phi\otimes\iota)(a)\delta$$(\delta\otimes I_{E})\delta=(I_{E}\otimes\delta)\delta$.

for all $a\in A$ ,$\delta$is acoprod-

87
Suppose that 6is acoproduct forby$\xi\eta=\delta^{*}(\xi\otimes_{\phi}\eta)$$E$. For , $\eta\in E$ , we define aproduct 47 in $E$$\xi$. It follows from (2.8) that this product is associative. Then $E$ is

an algebra over C. Note that we have$||\xi\eta||\leq||\delta||||\xi||||\eta||$.

2.3. Coproducts associated with MUO’s. Let E be aHilbertlet$\phi$

and$\psi$

be -homomorphisms of$*$$W\in \mathcal{L}_{A}(E\otimes_{\psi}E, E\otimes_{\phi}E)$

For an element$\xi_{0}$to$A$$\mathcal{L}_{A}(E)$such that$\phi$

and$A$-module andcommute. Let$\psi$

be amultiplicative unitary operator (MUO).

of $E$ , we say thathas Property El if it satisfies the following$\xi_{0}$conditions;(i)$W(\xi_{0}\otimes_{\psi}\xi_{0})=\xi_{0}\otimes_{\phi}\xi_{0}$.

(ii) For every $\xi\in E$ , there exists an element$<\eta$,Fix an elementby,$\pi_{\xi 0}(\xi)\zeta>=<W(\xi_{0}\otimes_{\psi}\eta)$$\xi_{0}$$\xi\otimes_{\phi}\zeta>$of$\mathcal{L}_{A}(E)$pentagonal equation,$\xi_{0}$$\delta$of. Then we havesuch thatfor every 7,

with Property El. Define an operator$\delta(\eta)=W(\xi_{0}\otimes_{\psi}\eta)$

For an element$\pi_{\xi_{0}}(\xi)$$\delta=\delta_{\xi 0}$$\delta^{*}(\xi\otimes\eta)=\pi_{\xi 0}(\xi)\eta$.

$($$\in E$in$\mathcal{L}_{A}(E, E\otimes_{\phi}E)$. Since $W$ satisfies theis acoproduct of $(E, \phi)$ .
$E$, we say thathas Property E2 if it satisfies the following$\xi_{0}$conditions;(i)$W(\xi_{0}\otimes_{\psi}\xi_{0})=\xi_{0}\otimes_{\phi}\xi_{0}$.

(ii) For every $\xi\in E$ , there exists an element$<\eta,\hat{\pi}_{\xi 0}(\xi)\zeta>=<W’(\xi_{0}\otimes_{\phi}\eta)$

Fix an elementby$\xi_{0}$,$\hat{\pi}_{\xi 0}(\xi)$$\xi\otimes_{\psi}\zeta>$of$\mathcal{L}_{A}(E)$for every 77,

with Property E2. Define an operator$\hat{\delta}(\eta)=W’(\xi_{0}\otimes_{\phi}\eta)$. Since$W$such that$\delta$$=\hat{\delta}_{\xi_{0}}$$\zeta\in E$

insatisfies the pentagonal equation,.

$\mathcal{L}_{A}(E, E\otimes_{\psi}E)$$\hat{\delta}$

is acoproductof $(E, \psi)$ .
3. CORING STRUCTURES ON HILBERT
Let
that

$E$

$A$

be aHilbert -module and let$A$itself is aHilbert$A$$\phi$

be$\mathrm{a}*$-module with the$\mathrm{C}’$-MODULES-homomorphism of

$A$

$A$

to

$\mathcal{L}_{A}(E)$. Note-valued inner product $<a$ , $b>=a^{*}b$ .

88
We denote by$i$the -homomorphism of$*$

there exists aunitary operator

in

$t$

to

$A$

$\mathcal{L}_{A}(A)$defined by $t(\xi\otimes_{i}a)=\xi a$ . If$\mathcal{L}_{A}(E\otimes_{i}A, E)$

non-degenerate, then there exists aunitary operator$t’(a\otimes_{\phi}\xi)=\phi(a)\xi$

$Q$

$t’$

in

$\mathcal{L}_{A}(A\otimes_{\phi}E, E)$$\phi$

issuch that.

Definition 3.1. Suppose thatand letdefined by $i(a)b=ab$ . Thenbe an element of(1) We say that$\phi$

is non-degenerate. Let$\mathcal{L}_{A}(E, A)$$(E, \phi, \delta, Q)$$\delta$be acoproduct of $(E, \phi)$, such that $Q\phi(a)=aQ$ for $a\in A$ .

is aright counital$A$-coring if it satisfies the following

equation;$t(I_{E}\otimes_{\phi}Q)\delta=I_{E}$

Then

$Q$

.

is called aright counit.

(2) We say that$(E, \phi, \delta, Q)$

is aleft counital$A$-coring if it satisfies the following

equation;$t’(Q\otimes_{\phi}I_{E})\delta=I_{E}$

Then

$Q$

is called aleft counit.

(3) We say that

Then

$Q$

.

$(E, \phi, \delta, Q)$

is acounital$A$-coring if$Q$is aright and left counit.

is called acounit.

For $n\geq 2$ , we set$E^{\otimes_{\phi}n}=E\otimes_{\phi}\cdots\otimes_{\phi}E$

Let $(E, \phi, \delta, Q)$ be aleft or right counital$\mathcal{L}_{A}(E^{\otimes_{\phi}4}, E^{\otimes_{\phi}2})$$E\otimes_{\phi}E$

by

$n$

-coring.

We defin an element$\omega$ofby$\omega$

Then we have$A$

( times ).

$=\{t(I_{E}\otimes_{\phi}Q)\otimes_{\phi}I_{E}\}(I_{E}\otimes_{\phi\otimes\iota}\delta’\otimes_{\phi}I_{E})$$\omega(\omega\otimes_{\phi\otimes\iota}I)=\omega(I\otimes_{\phi\otimes\iota}\omega)$$xy=\omega(x\otimes_{\phi\otimes\iota}y)$. Then$E\otimes_{\phi}E$.

. Therefore we can difine aproduct on
is an algebra over C. Note that we have$(\xi_{1}\otimes_{\phi}\xi_{2})(\eta_{1}\otimes_{\phi}\eta_{2})=(\xi_{1}Q(\xi_{2}\eta_{1}))\otimes_{\phi}\eta_{2}$.

89Definition 3.2. We say that and$Q$$\delta$$\delta(\xi\eta)=\delta(\xi)\delta(\eta)$

for every ,$\xi$

Example 3.3 ([13]). Let$A_{1}arrow A_{0}$$\eta\in E$

are compatible if the following equation holds;.

$1\in A_{0}\subset A_{1}$be an inclusion of C’-algebras and let$P_{1}$:

be afaithful positive conditional expectation of index finite type. Let$\{\mathrm{u}\mathrm{j}, u_{i}’;i=1, \cdots, N\}$be aquasi-basis of$P_{1}$. Let $E_{1}=A_{1}$ be aHilbert-valued inner product defined by $<a$ , $b>=P_{1}(a^{*}b)$ . Letwith the$A_{0}$$\mathcal{L}_{A_{0}}(E_{1})$be a -homomorphism defined by$*$

restriction of$\phi_{1}$The product onto$E_{1}$$A_{0}$. Define

induced by$\phi_{1}$-module:$\phi_{1}$(a)b=a&. We denote byby$\delta\in \mathcal{L}_{A_{0}}(E_{1}, E_{1}\otimes_{\phi_{0}}E_{1})$$\delta$$A_{0}$$A_{1}arrow$$\phi_{0}$the$\delta(\xi)=\sum_{i=1}^{N}(\xi u:)\otimes_{\phi_{0}}u_{i}^{*}$

agrees with the product on$A_{1}$. Then.

$(E_{1}, \phi_{0}, \delta, P_{1})$is acompatible counital A-coring.
Example 3.4. Let$E$

is aright$\xi\in E$

$A$

$G$

be afinite groupoid. Set-module with the right$A$, $a\in A$ and $x\in G$ . We define an$<\xi$,$A=\mathrm{C}(G^{(0)})$and $E=\mathrm{C}(G)$ . Then-action defined by $(\xi a)(x)=\xi(x)a(s(x))$ for$A$-valued inner product of $E$ by$\eta>(u)=\sum_{g\in G_{u}}\overline{\xi(g)}\eta(g)$for , $\eta\in E$ and $u\in G^{(0)}$ , where $G_{u}=s^{-1}(u)$ for $u\in G^{(0)}$ . Then $E$ is aHilbert
$\xi$

$A$

-module. Define -homomorphisms$*$$\phi$

andthat we have $E\otimes_{\psi}E=\mathrm{C}(G^{2}(ss))$ and$(0, h)\in G^{2};\mathrm{s}(\mathrm{p})=\mathrm{s}(\mathrm{p})$$(W\xi)(g, h)=\xi(h, gh)$

an element$\xi_{0}\in E$$Q_{\eta 0}$

tively. Then

induced by
uct onto$\mathcal{L}_{A}(E)$. Let$E\otimes_{\phi}E=\mathrm{C}(G^{(2)})$$W\in \mathrm{C}\mathrm{A}\{\mathrm{E}\otimes_{\psi}E,$,

by $\xi_{0}(g)=a_{0}(s(g))$ . Then

: $Earrow A$ by $Q_{\eta 0}(\xi)=<\eta_{0}$ ,$(E, \phi, \delta_{\xi_{0}}, Q_{\eta 0})$$\delta_{\xi_{0}}$$\mathrm{C}(G)$$Q_{\xi_{0}}$$E\otimes_{\phi}E$. Define an element $a_{0}\in A$ by$\xi_{0}$Note that we have $||\xi_{0}||=1$ . Define an element
operators$A$

by $(\phi(a)\xi)(x)=$and $\psi(a)=\xi a$ respectively for $a\in A$ , $\xi\in E$ and $x\in G$ . Note$a(r(x))\xi(x)${of$\psi$, where $G^{2}(ss)=$) be the MUO defined by$\mathrm{a}\mathrm{o}(\mathrm{w})=|G_{u}|^{-1/2}$

and definesatisfies Properties El and E2.

$\eta_{0}\in E$by$\eta_{0}=\chi_{G^{(0)}}a_{0}^{-1}$$\xi>\mathrm{a}\mathrm{n}\mathrm{d}Q_{\xi_{0}}(\xi)=<\xi_{0}$is acompatible counital

is of the form $\xi\eta=(\xi*\eta)a_{0}$ , where$A$,. Define$\xi>\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}-$-coring. The product on $E$$\xi*\eta$. We also have two compatible right counitalis the convolution prod$A$-corings$(E, \psi, \delta_{\xi_{0}}, Q_{\xi_{0}})$

90
and
$A$

. Two products on$(E, \psi, \delta_{\xi_{0}}, Q_{\eta 0})$associated with above right counital$E\otimes_{\psi}E$-corings are different.
4. CORING STRUCTURES ASSOCIATED WITH INCLUSIONS OF
Let 16be an inclusion of C’-algebras and let$A_{0}\subset A_{1}$$P_{1}$:$\mathrm{C}^{*}$-ALGEBRAS$A_{1}arrow A_{0}$be afaithful

positive conditional expectation of index-finite type with aquasi-basis
Let$E_{1}$,$\phi_{1}$andhomomorphism$A=\mathcal{L}_{A_{0}}(E_{1}, \phi_{1})$$\phi_{0}$be as in Example 3.3. Set:$\phi_{2}$$A_{1}arrow \mathcal{L}_{A_{0}}(E_{2})$and aHilbert

$A$

by

-module$a\in A$

. The

$A$

homomorphisms$\phi$

and$\psi$

of

$A$

to

,

$(E_{2}, \phi_{2}))$$x\in \mathcal{L}_{A_{0}}(E_{1}, E_{2})$-valued inner product on$\mathcal{L}_{A}(E)$$E$$V’\overline{V}=W\otimes_{i}I_{E_{1}}$, where$V$:$\mathrm{C}$$*-$’-algebraA by,such that $x\phi_{1}(a)=\phi_{2}(a)x$ for allis defined by $<x$ , $y>=x’ y$ . We$\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}*-$by $\phi(a)x=(a\otimes_{\phi_{0}}I)x$ and $\psi(a)x=(I\otimes_{\phi_{0}}a)x$respectively. We suppose that there exists an MUO

that. Define a

.

by

$E$

$E=\mathcal{L}_{A_{0}}$ $((E_{1}, \phi_{1})$that is, $E$ is the set of elements

and define a$E_{2}=E_{1}\otimes_{\phi_{0}}E_{1}$$\phi_{2}=\phi_{1}\otimes\iota$$\{u_{i}, u_{i}’\}_{i=1}^{N}$$E\otimes_{\phi}E$

(&

$i$

$W\in \mathrm{C}\mathrm{a}\{\mathrm{E}\otimes_{\psi}E,$$E_{1}arrow E_{3}$and$\tilde{V}$:$E\otimes_{\phi}E$) such$E\otimes_{\psi}E\otimes_{i}E_{1}arrow E_{3}$are operators defined in [8]. As for sufficient conditions for $W$ to exist, see [7] and[8]. Define an element $x_{0}\in E$ by$x_{0}(\xi)=\xi\otimes_{\phi_{0}}1$. Thenand E2. Note that we have $||x_{0}||=1$ . Define an element$\overline{y_{0}}(\xi)=\sum_{i=1}^{N}(\xi u_{i})\otimes_{\phi_{0}}u_{i}’$Note that we have$Q_{x0}$,$\tilde{y_{0}}’(\xi\otimes_{\phi 0}\eta)=\xi\eta$$Q_{\tilde{y0}}\in \mathcal{L}_{A}(E, A)$, whereby $Q_{x_{0}}(x)=<x_{0}$ ,$\xi\eta$$x_{0}$

satisfies Properties El$\tilde{y_{0}}\in E$by.

is the product on$x>\mathrm{a}\mathrm{n}\mathrm{d}Q_{\tilde{y0}}(x)=<\tilde{y_{0}}$,$A_{1}$. Define$x>\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y}$Then we have the following theorem.
Theorem 4.1. (1)$(E, \phi, \delta_{x0}, Q_{x0})$

is a compatible right counital A-coring.

(2) Suppose that there esist elements $(v_{i}, w_{i})\in E\cross E(i=1, \cdots K)$ such that$\hat{\delta}_{x_{0}}(\overline{y_{0}})=\sum_{i=1}^{K}v_{i}\otimes_{\psi}w_{i}$

.

.

91
Then (E,is a compatible counital A-coring.

$\psi, \hat{\delta}_{x_{0}}, Q_{\tilde{y0}})$5. CORING STRUCTURES ON THE SET OF COMPACT OPERATORSLet $H$ be an infinite-dimensional separable Hilbert space. We consider $H$ to beaHilbert$\mathbb{C}$-module, in particular the inner product is linear in the second variable.

We denote by$A$-module$A$

the C’-algebra$\mathcal{K}(H, H\otimes H)$$\mathcal{K}(H)$of compact operators on $H$ . Let $E$ be aHilbert. The right action of

for $x\in E$ , $a\in A$ and $\xi\in H$ and the$<x$ , $y>=x’ y$ for $x$ , $y\in E$ .
$\phi(a)x=(a\otimes I_{H})x$

$E$

$A$

on

is defined by (xa)(\mbox{\boldmath $\xi$}) $=x(a(\xi))$$E$-valued inner product ofDefine -homomorphisms$*$$\phi$

and$\psi$$E$is defined by

$A$

to

of

$\mathcal{L}_{A}(E)$byand $\psi(a)x=(I_{H}\otimes a)x$ for $a\in A$ and $x\in E$ respectively. We

denote by $F$ the Hilbert
and the

$A$

$A$

$A$

-module$\mathcal{K}(H,$$H\otimes H$(&H). The right action of

on

$A$

$F$

-valued inner product of $F$ are defined by the same formulas as those in. There exist unitary operators$M\in \mathrm{C}\mathrm{a}\{\mathrm{E}\otimes_{\phi}E,$

$F$

) and

$\overline{M}\in(E\otimes_{\psi}E, F)$

such

that
$M(x\otimes_{\phi}y)=(x\otimes I_{H})y$,$\overline{M}(x\otimes_{\psi}y)=(I_{H}\otimes x)y$for , $y\in E$ respectively. Define $W=M^{-1}\overline{M}$. Then we have the following:$x$

Theorem 5.1. The operator $W$ is the unique multiplicative unitary operator in$\mathcal{L}_{A}(E\otimes_{\psi}E, E\otimes_{\phi}E)$.

Now we introduce acoring structure on
unit$\{u_{n}\}_{n=1}^{\infty}$of A. is said to be increasing if

Definition 5.2. Let$\delta$$(E, \phi, \psi)$$u_{n}\geq 0$. Recall that an approximate

and $u_{n+1}\geq u_{n}$ for every .

be acoproduct of $(E, \phi)$ . For $n=1,2$ ,$n$$\cdots$, let$Q_{n}$

be anelement of Ca(E, ) such that $Q_{n}(\phi(a)x)=aQn(x)$ for $a\in A$ and $x\in E$ and let$A$$\{u_{n}\}_{n=1}^{\infty}$

be an increasing approximate unit offor every . Then$n$$(6, \{Q_{n}\}_{n=1}^{\infty}, \{u_{n}\}_{n=1}^{\infty})$$A$such that $u_{1}\neq 0$ and $u_{n}\neq u_{n+1}$

is called acoring structure on$(E, \phi, \psi)$

if it

92
satisfies the following equations for every ;$n$$\mathrm{t}$(&p$I_{E}$$Q_{n}$)$\delta=\mathrm{t}’(Q_{n}\otimes_{\phi}I_{E})\delta=\psi(u_{n})$.

$Q_{n}\psi(u_{n})=Q_{n}$Then$\{Q_{n}\}$

Let

$T$

,

is called an approximate counit.

be an element of$\mathcal{L}(H, H\otimes H)$. We will say that$T$

has Property$\mathrm{D}$if itsatisfies the following conditions:(i) $(T\otimes I_{H})T=(I_{H}\otimes T)T$ .
(ii) There exists afamily$\{K_{n}\}_{n=1}^{\infty}$

subspaces of $H$ such thatmal basis

we setof$K_{n}$

for $n=1,2$ ,, $Te_{i}>$ , then(a) for $i=k_{n-1}+1$ ,and there exists acomplete orthonor-$H=\oplus_{n=1}^{\infty}K_{n}$$\{e_{k_{n-1}+1}, \cdots, e_{k_{n}}\}$$\lambda_{j,\ell}^{i}=<e_{j}\otimes e_{\ell}$of mutually orthogonal non-trivial finite-dimensional$\lambda_{i,i}^{i}\neq 0$$\{\lambda_{j.\ell}^{i}\}$and$\cdots$, where $k_{0}=0$ , such that, if

satisfies the following conditions;$\lambda_{j,\ell}^{i}=\lambda_{\ell,j}^{i}=0$

for every$j\in \mathrm{N}$and$\ell=$$k_{m}+1(m=0,1,2, \cdots)$ except for $j=\ell=i$ ,(b) if $\dim K_{n}\geq 2$ , for $i=k_{n-1}+2$ ,$\cdots$,$k_{n}$,$\lambda_{i,k_{n-1}+1}^{i}=\lambda_{k_{n-1}+1,i}^{i}=\lambda_{k_{n-1}+1,k_{n-1}+1}^{k_{n-1}+1}$

and$\lambda_{j,\ell}^{i}=\lambda_{\ell,j}^{i}=0$

for every$j\in \mathrm{N}$,and $\ell=k_{m}+1(m=0,1,2, \cdots)$ except

for $(j, \ell)=(i, k_{n-1}+1)$ .
Then we have the following theorem:
one-tO-One correspondence bettneen the set of coring

Theorem 5.3. There exists$a$structures $(5, \{Q_{n}\}, \{u_{n}\})$ on$(E, \phi, \psi)$and the set of elements $(T, \{K_{n}\}, \{e_{k_{n-1}+1}\})$

which satisfy Property D. The correspondence is given as follows: If $(T, \{K_{n}\}, \{e_{k_{n-1}+1}\})$

93
has Property

$D$

, set

$H_{n}=\oplus_{i=1}^{n}K_{i}$,$\xi_{n}=\sum_{i=1}^{n}\eta_{i}\in H_{n}$define$\delta$and$f_{n}\in H$$Q_{n}$,

where$\eta_{i}=(\lambda_{k.+1,k\dot{.}+1}^{k\dot{.}+1}.-1)^{-1}e_{k+1}-1-1:-1\in K_{i}$’by $f_{n}(\xi)=<\xi_{n}$ , $\xi>$ , then,is the projection onto$u_{n}\in \mathcal{K}(H)$ased$H_{n}$are given by the following equations;$\delta(x)=M^{-1}(I_{H}\otimes T)x$

$Q_{n}(x)=(I_{H}\otimes f_{n})x$

Question. Suppose that

$T$

,

.

has Property D. Does

determine$T$$\{K_{n}\}$and$\{e_{k_{n-1}+1}\}$

uniquely?
The following theorem shows the relation between the coring structures and themultiplicative unitary operator $W$ defined above:

Theorem 5.4. Let

be a coring structure on$(\delta, \{Q_{n}\}, \{u_{n}\})$$(E, \phi, \psi)$and let$T$bethe operator which corresponds to $(6, \{Q_{n}\}, \{u_{n}\})$ by Theorem 5.3. Put $x_{n}=Tun$ .

Then$(E, \phi)$$x_{n}$

is an element of$E$

and satisfies Property $El$ . Let$\delta_{n}$be the coproduct ofdefined by$\delta_{n}(x)=W(x_{n}\otimes_{\psi}x)$.

Then the following equation holds;$\delta=\lim_{narrow\infty}\delta_{n}$with respect to the strict topology on$\mathcal{L}_{A}(E, E\otimes_{\phi}E)$.

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DEpARTMENT OF AppLIED MATHEMATICS,

FACULTY OF SCIENCE, OSAKA WOMEN’S UN1-

VERSITY, SAKAI CITY, OSAKA 590-0035, JApAN$E$-rnail address: ouchi(Oappmath. osaka-wu . ac.jp