Vol 2, No 9 (2011)

(1)

#

_{$$$% &}

_%

On Hilbert Modules over Locally m-Convex

_H

*

₋

_Algebras

M. Khanehgir*

Department of Mathematics, Faculty of Science, Islamic Azad University-Mashhad Branch, Mashhad, Iran, P. O. Box 413-91735

E-mail: [email protected]

(Received on: 22-08-11; Accepted on: 04-09-11)

--- ABSTRACT

I

n this paper a Hilbert

E

−

module

W

is defined where

( , (| . | )

E

_{λ λ}_∈Λ

)

is a locally m-convex

H

*

−

algebra. With

each

λ ∈ Λ

,

we associate a Hilbert module

W

λ over an

H

*

−

algebra

E

λ. We obtain relationship between these

spaces and the initial space. Moreover the existence of orthonormal bases in a Hilbert

E

−

module is proved. We topologized the space of bounded

E

−

linear operators via suitable

family of seminorms.

2000 AMS subject classification: 46L08, 46k05, 46c05, 46c50.

Key Words: Hilbert module, Locally m-convex

H

*

−

algebra.

---

1. Introduction

A locally multiplicatively convex algebra (l.m.c.a in short) is a topological algebra

( , )

E

τ

whose topology

τ

is

determined by a directed family

(| . | )

_{λ λ}_∈Λ of submultiplicative seminorms. Such an algebra will usually denoted by

( , (| . | )

E

_{λ λ}_∈Λ

)

. If, in addition,

E

is endowed with an involution

x

* such that

| | =|

x

_λ

x

*

|

_λ, for any

,

x

∈

E

λ

∈ Λ

then

( , (| . | )

E

_{λ λ}_∈Λ

)

is called an l.m.c.

*

−

algebra. Let

( , (| . | )

E

_{λ λ}_∈Λ

)

be a complete l.m.c.a. It is known that

( , (| . | )

E

_{λ λ}_∈Λ

)

is the inverse limit of the normed algebras

(

E

_λ

, (| . | )

_{λ λ}

′

_∈Λ

)

, where

E

_λ

=

E N

/

_λ with

= {

:| | = 0}

N

_λ

x

∈

E

x

_λ , and

| | =| | .

x

_λ

′

x

_λ An element

x

of

E

is written

x

= (

x

_{λ λ}

) = (

π

_λ

( ))

x

_λ, where

:

E

λ λ

π

→

is the canonical surjection. The algebra

( , (| . | )

E

_{λ λ}_∈Λ

)

is also the inverse limit of the Banach algebras

,

E

λ the completion of

E

_λ ,s. The norm in

E

λ will also be denoted by

| . |

_λ

′

.

In the following we define the locally m-convex

H

*

−

algebra spaces. This notion was introduced in [5] as a natural extension of the classical

H

*

−

algebras of W. Ambrose ([1]). Here we consider the case where the algebra is complete and it is endowed with a continuous involution.

' %

H

*

−

*

H

−

*

( , (| . | )

E

_{λ λ}_∈Λ

)

( .,. )

_{λ λ}_∈Λ

, ,

x y z

∈

E

λ

∈ Λ

:

2

( ) | | =

i

x

_λ

x x

,

_λ

,

*

( )

ii

xy z

,

_λ

=

y x z

,

_λ

,

*

( )

iii

yx z

,

_λ

=

y zx

,

_λ

,

λ

∈ Λ

E

_λ

=

E N

/

_λ

x

_λ

,

y

_{λ λ}

=

x y

,

_λ

(2)

H

−

" # $ % # % & !'

E

λ ! " #

(

E

λ

,| . | )

′

_λ

*

H

−

( , (| . | )

E

_{λ λ}_∈Λ

)

_H

*

₋

₍

_E

_{,| . | )}

λ _λ

′

# $%&#

2.3

'

φ

:

E

→

_←lim_λ

E

λ

χ ϕ

_µ

o

=

π

_{λ µ}_,

o

π

_λ#

| . |

_λ

≥

| . |

_µ lim

:

E

λ

E

µ

µ λ

χ

_←

→

( )

ϕ

lim_←_λ

E

λ

≅

E

' * ' +

1.1 [7]

)

{

E

λ

:

λ

∈ Λ

}

E

,

H

*

( , (| . | )

E

_{λ λ}_∈Λ

)

'

*

#

E

#

lan E

( ) = {0},

( ) = {

:

= {0}}

lan E

x

∈

E xE

E

λ#

λ

∈ Λ

( )

E

τ

E

τ

( ) = {

E

ab a b

: ,

∈

E

}

- #

τ

( )

E

*

−

τ

_λ

(.),

λ

∈ Λ

E

₍

_{a a}

*

_{) =| | ,}

_a

2

λ λ

τ

a

E

λ

∈ Λ

,

( )

E

τ

−

λ

E

tr

_λ

.,.

_λ

E

_{tr ab}

₍

_{) =}

_{a b}

_,

*

λ λ

a b

,

∈

E

*

H

−

E

λ

,

λ

∈ Λ

(

E

λ

) = {[

a

N

_λ

][

b

N

_λ

] : ,

a b

E

}

τ

+

∈

. *

τ

(

E

λ

)

E

λ#

*

τ

λ

(.)

τ

λ

| . |

′

_λ

E

λ

* 2

([

a a

N

]) =| [

a

N

] | ,

λ _λ _λ _λ

τ

+

′

a

∈

E

τ

(

E

_λ

)

E

_λ

,

λ

∈ Λ

)

τ

(

E

_λ

)

E

_λ

*

−

τ

_λ

(.),

* 2

(

a a

N

) =|

a

N

|

λ λ λ λ

τ

+

′

a

∈

E

λ

∈ Λ

#

tr

λ

τ

(

E

λ

)

*

([

][

]) =

([

][

]) =

,

tr

λ

a

+

N

_λ

b

+

N

_λ

tr

λ

b

+

N

_λ

a

+

N

_λ

a b

_λ '

tr

_λ

τ

(

E

_λ

),

λ

∈ Λ

,

*

(

) =

(

) =

,

tr ab

_λ

+

N

_λ

tr ba

_λ

+

N

_λ

a b

_λ

/

χ τ

_λ

: ( )

E

→

τ

(

E

_λ

)

χ

_λ

(

ab

) =

ab

+

N

_λ

π

_{λ µ}_,

: (

τ

E

_λ

)

→

τ

(

E

_µ

)

,

(

a

N

)(

b

N

) = (

a

N

)(

b

N

),

λ µ λ λ µ µ

π

+

| . |

_λ

≥

| . |

_µ

{ ( ), ;

τ

E

τ χ

_λ

}

,

{ (

τ

E

_λ

),

τ π

_λ

;

_{λ µ}

, ,

λ µ

∈ Λ

, | . |

_λ

≥

| . | }

_µ

,

{ (

τ

E

λ

),

τ π

λ

;

_{λ µ}

, ,

λ µ

∈ Λ

, | . |

_λ

≥

| . | }.

_µ

In this paper we will intoduce a Hilbert module

W

over an l.m.c.

H

*

−

algebra

( , (| . | )

E

_{λ λ}_∈Λ

)

and for each

λ

∈ Λ

we will associate a Hilbert module

W

λ over an

H

*

−

algebra

E

λ. We shall see that

W

≅

lim_←_λ

W

λ. Then we will

discuss about orthonormal bases in these spaces. Also we will topologize

L V W

_E

( ,

)

and

B V W

_E

( ,

)

, the set of

adjointable

E

−

linear operators and the set of bounded

E

−

linear operators from Hilbert

E

−

module

V

into Hilbert

E

−

module

W

, respectively, via suitable families of seminorms. Throughout this paper

E

is an l.m.c.

H

*

−

algebra

and

W

is a Hilbert

E

−

module except some results about unitary operators in Hilbert

H

*

−

modules at the end of the paper. The paper is organized as follows.

In section 2 we will introduce Hilbert modules over l.m.c.

H

*

−

algebras and their properties are studied. The existence of orthonormal bases in these spaces is proved.

(3)

H

−

2. Hilbert modules over l.m.c.

H

*

−

algebras and orthonormal bases

Hilbert modules over l.m.c.

H

*

−

algebras generalize the notion of Hilbert

H

*

−

modules by allowing the

( )

E

τ

−

valued product in an l.m.c.

H

*

−

algebra.

Definition: 2.1 Let

( , (| . | )

E

_{λ λ}_∈Λ

)

be a Hausdorff l.m.c.

H

*

−

algebra. A pre-Hilbert

E

-module is a left module

W

over

E

provided with a mapping

[. | .] :

W W

×

→

τ

( )

E

(called

τ

( )

E

−

valued product) where

τ

( ) = {

E

ab a b

: ,

∈

E

}

which satisfies the following conditions:

( ) [

i

α

x y

| ] = [ | ]

α

x y

∀ ∈

α

C

,

∀

x y

,

∈

W

,

( ) [

ii

x

+

y z

| ] = [ | ] [ | ]

x z

+

y z

∀

x y z

, ,

∈

W

,

( ) [

iii ax y

| ] = [ | ],

a x y

∀ ∈

a

E

,

∀

x y

,

∈

W

,

*

( ) [ | ] = [ | ]

iv x y

y x

∀

x y

,

∈

W

,

( )

v

∀ ∈

x W x

,

≠

0,

∃ ∈

a

E a

,

≠

0,

such that

[ | ] =

x x

a a

*

,

( )

vi

for each

λ

∈ Λ

,

W

is a semi-definite pseudo-inner product space with

( , ) =

x y

_λ

a b

,

* _λ(or

tr ab

_λ

(

)

) where

[ | ] =

x y

ab

∈

τ

( )

E

.

We say that

W

is a Hilbert

E

−

module if it is complete with respect to the topology determined by the family of

seminorms

x

_λ

=

( , ) ,

x x

_λ

x W

∈

,

λ

∈ Λ

.

Given a Hilbert

E

−

module

W

, then for

λ

∈ Λ

,

ξ

_λ

= {

x W

∈

:

x

_λ

= 0}

is a closed submodule of

W

. Indeed, for non zero

x

in

ξ

_λ and

a

∈

E

,

if

[ | ] =

x x

b b

* for some non zero

b

∈

E

, then

ax

_λ

=|

ab

*

|

_λ

≤

| | | | = 0

a

_λ

b

_λ . For

λ

∈ Λ

,

W

_λ

=

W

/

ξ

_λ is an inner product space with

( , ) =

x y

_λ

tr x

_λ

[

+

ξ

_λ

|

y

+

ξ

_λ

]

(or

a b

,

* _λ where

[ | ] =

x y

ab

)

and its completion

W

λ is a Hilbert space.

Example: 2.2 Let

( , (| . | )

E

_{λ λ}_∈Λ

)

be an l.m.c.

H

*

−

algebra. Then

E

is a Hilbert module over itself with

( )

E

τ

−

valued product defined by

[ | ] =

a b

ab

*. Also it is easy to verify that a closed submodule of a Hilbert

E

−

module is again a Hilbert

E

−

module. Note that analogue of Lemma

2.2

of

[1]

holds for l.m.c.

H

*

−

algebras.

More precisely if

x

is a non zero element in an l.m.c.

H

*

−

algebra

E

, then

x x xx x

*

,

*

,

*are also non zero. The proof of the following proposition can be based on the direct application of previous comments about Hilbert

E

−

module

W

.

Proposition: 2.3 Let

W

be a Hilbert module over an l.m.c.

H

*

−

algebra

( , (| . | )

E

_{λ λ}_∈Λ

)

. For each

λ

∈ Λ

,

W

λ is a

Hilbert module over the proper

H

*

−

algebra

E

λ with

π

_λ

( )(

a x

+

ξ

_λ

) =

ax

+

ξ

_λ and

[

x

+

ξ

_λ

|

y

+

ξ

_λ

] =

π

_λ

([ | ])

x y

for every

a

∈

E

and for every

x y

,

∈

W

. Let

σ

W_λ be the canonical map from

W

onto

W

λ

,

λ

∈ Λ

. For ₁ ₂

1 2

,

, | . |

_λ

| . |

_λ

λ λ

∈ Λ

≥

, there is a canonical surjective linear map ₁ ₂

1 2

:

W

λ

W

λ

λ λ

σ

→

such

that

1 2

(

1

( )) =

2

( ).

W W

_x

W

_x

λ λ λ λ

σ

Also ₁ ₂

1 2 1 2

{

_W

,

_E

;

W

, | . |

| . | , ,

}

λ

σ

_{λ λ} _λ

≥

_λ

λ λ

∈ Λ

is an inverse system of Hilbert

*

H

−

modules in the following sense:

1 2

(

1

( )

1

( )) =

1 2

(

1

( ))

1 2

((

1

( ))

W W W W

a

x

a

x

λ λ λ λ λ λ λ λ λ λ

σ

π

σ

π

σ

and

*

1 2 1 1 2 1 2 1 2 1 1 2 1 2

(

W

((

W

( )),

_x

W

((

W

( )))

_y

=

(

( )),

_a

(

( ))

_b

λ λ λ λ λ λ λ λ λ λ λ λ λ λ

σ

π

(4)

H

−

" # $ % # % & !

lim 2 3 1 2

=

1 3

, | . |

1

| . |

2

| . | ;

3

=

,

W W W W

W

id

W

λ

λ λ λ λ λ λ λ λ λ λλ _λ λ

σ

≥

σ

← is a Hilbert

E

−

module with

(

π

_λ

( )) (

a

_λ

σ

_λW

( )) = (

x

_λ

σ

W_λ

(

ax

)) ,

_λ

and

*

(

σ

W_λ

( ))

x

_λ_∈Λ

, (

σ

_λW

( ))

y

_λ_∈Λ

= (

π

_λ

( ),

a

π

_λ

( ) )

b

_{λ λ}_∈Λ

Where

[ | ] =

x y

ab

. Indeed, lim_←_λ

W

λ maybe identified with

W

. So every coherent sequence in

{

W

λ

:

λ

∈ Λ

}

determines an element of

W

.

For the basic facts about Hilbert

H

*

−

modules we refer to [2] and [4]. In particular with the assumption of the

previous proposition, we have the following three relations in Hilbert

E

λ

−

module

W

λ(

λ

∈ Λ

),

2

₌

_([

_|

_{]) =}

_([

_|

_]).

x

+

ξ

_λ _λ

tr

λ

x

+

ξ

_λ

x

+

ξ

_λ

τ

λ

x

+

ξ

_λ

x

+

ξ

_λ

| [

x

+

ξ

_λ

|

y

+

ξ

_λ

] |

′

_λ

≤

τ

λ

([

x

+

ξ

_λ

|

y

+

ξ

_λ

])

≤

x

+

ξ

_λ _λ

y

+

ξ

_λ _λ

.

(

a

+

N

_λ

)(

x

+

ξ

_λ

)

_λ

≤

|

a

+

N

_{λ λ}

|

′

x

+

ξ

_λ _λ

.

As an immediate consequence of the above relations and the previous comments we obtain:

2

₌

_{([ | ]) =}

_{([ | ]).}

x

_λ

tr

_λ

x x

τ

_λ

x x

| [ | ] |

x y

′ ≤

_λ

τ

_λ

([ | ])

x y

≤

x

_λ

y

_λ

.

| |

.

ax

_λ

≤

a

′

_λ

x

_λ

Proposition: 2.4. Let

W

H

*

−

algebra

E

and

( ) = {

:

=

| | <

}

b E

a

∈

E

a

_∞

sup

_λ

a

_λ

∞

and

b W

(

) = {

x W

∈

:

x

_∞

=

sup

_λ

x

_λ

<

∞

}

. Then

b E

( )

is an

*

H

−

algebra and

b W

(

)

is a

b E

( )

−

Hilbert module.

Proof: Clearly, the sets

b E

( )

and

b W

(

)

are complex vector spaces and

b W

(

)

is a left

b E

( )

−

module. Because,

when

W

is identified with lim_←_λ

W

λ

,

we see that

b W

(

)

corresponds to the set of bounded coherent sequences. The

Cauchy-Schwarz inequality, applied to Hilbert

E

λ

−

module

W

λ, yields for

x y

,

∈

b W

(

)

, the inequality

2

( , )

x y

_∞

≤

( , )

x x

_∞

( , )

y y

_∞

,

so that the restriction of

b W

(

)

of the

τ

( )

E

−

valued product on

W

is a

( ( ))

b E

τ

−

valued product on

b W

(

)

. Obviously,

( ( ), ( .,.

b E

_λ

|

_{b E}_{( )}_×_{b E}_{( )}

)

_λ_∈Λ

)

and

( (

b W

),[.,.] |

_{b W}₍ ₎_×_{b W}₍ ₎

)

take more properties being as a subset of

E

and

W

respectively. To proof of completeness in

[7],

Satz

3.1,

also applied

here and show that

b E

( )

and

b W

(

)

are complete for norm

1 2

=

,

a

_∞

a a

_∞ and

1 2

= ( , )

x

_∞

x x

_∞,

respectively, for every

a

∈

E x W

,

∈

. Q.E.D.

Definition: 2.5 A non zero projection

e

in an l.m.c.

H

*

−

algebra

E

is called minimal, if

eEe

=

Ce

.

Also element

u

in a Hilbert

E

−

module

W

is said to be a basic element if there exists a minimal projection

e

∈

E

such that

[ | ] =

u u

e

. An orthonormal system in

W

is a family of basic elements

{ } ,

u

_{α α}

α

∈

I

satisfying

[

u

_α

|

u

_β

] = 0

for all

α β

,

∈

I

,

α

≠

β

. An orthonormal basis in

W

is an orthonormal system generating a dense submodule of

W

.

If

V

is a subset of a Hilbert

E

−

module

W

, we define

V

⊥

= {

w W

∈

| [ | ] = 0

w v

∀ ∈

v V

}

. Clearly

V

⊥ is a

closed submodule of

W

. If

V

is a submodule of

W

then

V

⊥

= {

x W

_∈

| ( , ) = 0,

x v

_λ

_{∀ ∈}

v V

,

_{∀ ∈ Λ}

λ

}

(5)

H

−

Our next result is a generalization of Lemma 1.3 in [4].

Lemma 2.6. Let

W

be a Hilbert

E

−

module and let

u

in

W

be such that

e

= [ | ]

u u

is an idempotent in

E

. If the

closed submodule generated by

u

is complemented or

e

does not belong to

N

_λ, for each

λ

∈ Λ

then

[ | ] = [ | ]

w u

w u e

for all

w W

∈

.

We can also generalized Corollary

1.4

of

[4]

to Hilbert modules over l.m.c.

H

*

−

algebras.

Corollary: 2.7 If

{ }

u

_{α α}_∈_I is an orthonormal system in a Hilbert

E

−

module

W

in which for every

α

∈

I

the closed submodule generated by

u

_α is complemented or

u

_α does not belong to

ξ

_λ

,

for each

λ

∈ Λ

, then

*

[

[ |

]

|

[ |

]

] = [ | ]

[ |

][ |

]

J J J

w

w u u

_α _α

w

w u u

_α _α

w w

w u

_α

w u

_α

α∈ α∈ α∈

−

for every finite subset

J

of

I

and for all

w

in

W

.

Our next result is a generalization of Proposition

1.5

of

[4]

.

Proposition: 2.8 Let

W

be a Hilbert

E

−

module, let

u

be a basic element in

W

, and let

M

denote the closed submodule of

W

generated by

u

. If

M

is orthogonally complemented in

W

then the mapping

w

→

[ | ]

w u u

is the

orthogonal projection from

W

onto

M

. As a consequence we have

M

=

Eu

.

Our next result is a generalization of Theorem

1.6

of

[4]

which provides a very useful characterization of

orthonormal bases in a Hilbert

E

−

module

W

.

Theorem: 2.9 Let

{ }

u

_{α α}_∈_I be an orthonormal system in a Hilbert

E

−

module

W

in which for every

α

∈

I

, the closed submodule generated by

u

α is complemented, then the following statement are equivalent.

( )

i

For all

w w

₁

,

₂ in

W

the family

{[

w u

₁

|

_α

][

w u

₂

|

_α

] }

* _α_∈_I is summable in the space

( ( ), (

τ

E

τ

_{λ λ}

)

_∈Λ

),

with sum

equale to

[

w w

₁

|

₂

]

.

( )

ii

For every

w

in

W

,

we have

[ | ] =

[

₁

|

][

₂

|

]

*

I

w w

_α

w u

_α

w u

_α

∈ (Parseval's identity) in the space

( ( ), (

τ

E

τ

_{λ λ}

)

∈Λ).

( )

iii

For every

w

in

W

, we have

=

[ |

]

I

w

_α

w u u

_α _α

∈ (Fourier expansion).

( )

iv

{ }

u

_{α α}_∈_I is an orthonormal basis in

W

.

Remark: 2.10 Parseval's identity leads to the equality

| [ |

] | =

2 2

I

w u

α λ

w

λ

α∈ for every

λ

∈ Λ

. Indeed,

2 * * 2

| [ |

] | =

([ |

][ |

] ) =

(

([ |

][ |

] )) =

([ | ]) =

.

I I I

w u

_α _λ

tr

_λ

w u

_α

w u

_α

tr

_λ

w u

_α

w u

_α

tr

_λ

w w

w

_λ

α∈ α∈ α∈

The existence of basic elements in a Hilbert module over an l.m.c.

H

*

−

algebra can be guaranteed by an argument similar to Proposition 1.7 of [4]. A slightly modification of Theorem

1.9

of

[4]

gives the following theorem in a

Hilbert

E

−

module.

Theorem: 2.11 Let

S

be a subset of a Hilbert

E

−

module

W

. If

S

is a maximal orthonormal system then it is an orthonormal basis in

W

and converse is true when each closed submodule generated by every element of

S

is complemented in

W

.

Proof: Let

S

be a maximal orthonormal system in

W

and let

M

be the closed submodule generated by

S

. If

M

≠

W

, then there exists

x W

∈

−

M

. It implies that

0

0 x

_λ

≠

, for some

λ

₀

∈ Λ

. Now if

M

⊥

= {0}

then

0

(

M

₊

ξ

_λ

)

⊥

_⊆

M

⊥

= {0}

. Hence in the Hilbert

E

λ₀

−

module

W

λ₀, we have

0

=

0

(6)

H

−

" # $ % # % & )

contradiction. So

M

⊥

≠

{0}

and therefore there exists a basic element

u

in

M

⊥. Obviously

S

∪

{ }

u

is an

orthonormal system strictly containing

S

, which is a contradiction. The converse is obvious by Fourier expansion. Q.E.D.

Corollary: 2.12 Every non zero Hilbert module over an l.m.c.

H

*

−

algebra has an orthonormal basis.

Theorem: 2.13. In a Hilbert

E

−

module

W

, { }

v

_{α α}_∈_I is an orthonormal system if and only if

{

v

_α

+

ξ

_{λ α}

}

∈_I is an

orthonormal system in the Hilbert

E

λ

−

module

W

λ when

v

ξ

_λ for each

α

∈

I

and for each

λ

∈ Λ

. Also if

{ }

v

W

and

v

ξ

_λ for each

α

∈

I

then

{

v

_α

+

ξ

_{λ α}

}

_∈_I is an orthonormal basis in the Hilbert

E

λ

−

module

W

λ. So we have an analogue of Fourier expansion

and Parseval's identity associated to this orthonormal basis in Hilbert

E

λ

−

module

W

λ.

Proof: Suppose that

v

is a basic element in

W

. If for some ₀

0

,

v

_λ

λ

∈ Λ

∈

ξ

, then

0

v

+

ξ

_λ is not a basic element in

0

W

λ . It is clear that for

µ

∈ Λ

in which,

0

| . |

_λ

≥

| . | ,

_µ

v

∈

ξ

_µ and

v

+

ξ

_µ is not a basic element in

W

µ. Now if

v

does not belong to

ξ λ

_λ

,

∈ Λ

then we have

[

v

+

ξ

_λ

|

v

+

ξ

_λ

]

E v

λ

[

+

ξ

_λ

|

v

+

ξ

_λ

] =

π

_λ

([ | ])

v v E

λ

π

_λ

([ | ])

v v

=

π

_λ

([ | ] [ | ])

v v E v v

=

π

_λ

( [ | ])

C v v

= [

C v

+

ξ

_λ

|

v

+

ξ

_λ

].

Hence

v

+

ξ

_λ is a basic element in

W

λ. Conversely, if for each

λ

∈ Λ

,

v

+

ξ

_λ is a basic element in

W

λ then we

have

[

v

+

ξ

_λ

|

v

+

ξ

_λ

]

E v

λ

[

+

ξ

_λ

|

v

+

ξ

_λ

] = [

C v

+

ξ

_λ

|

v

+

ξ

_λ

]

and this implies that

([ | ] [ | ]

v v E v v

C v v

[ | ]) = 0,

λ

π

−

for every

λ

∈ Λ

. So

[ | ] [ | ]

v v E v v

−

C v v

[ | ]

⊆

N

λ for every

λ

∈ Λ

and therefore

[ | ] [ | ]

v v E v v

−

C v v

[ | ] = 0

.

It is easy to verify that,

{ }

v

_{α α}_∈_I is an orthonormal system in

W

if and only if

{

v

_α

+

ξ

_{λ α}

}

_∈_I is an orthonormal system

in

W

λ

,

when

v

ξ

_λ for each

α

∈

I

and for each

λ

∈ Λ

. Now suppose that

{ }

v

_{α α}_∈_I is an

orthonormal basis in

W

and for each

α

∈

I v

,

_α does not belonge to

ξ

_λ then as we mentioned before,

{

v

_α

+

ξ

_{λ α}

}

_∈_I

is an orthonormal system in

W

λ. We are going to show that it generates a dense submodule of

W

λ. For this, let

x

+

ξ

_λ

∈

W

λ. Since

{ }

v

W

,

So we have

=1

,

{ }

;

,

n

i_k _i I i_k _i

k k k

C

v

_α

v

_{α α}

k

I N

v

_α

x

γ

_∈

γ

∃

∈

∃

∈

→

as

n

tends to

∞

. It implies that

=1

(

)

,

n

i_k _i k k

v

_α _λ

x

_λ

γ

+

ξ

→ +

ξ

when

n

tends to

∞

. Thus if for all

α

∈

I v

,

ξ

_λ then

{

v

_α

+

ξ

_{λ α}

}

_∈_I is an orthonormal basis in

(7)

H

−

3. Space of bounded operators

Definition: 3.1 Let

V

and

W

be two Hilbert modules over an l.m.c.

H

*

−

algebra

( , (| . | )

E

_{λ λ}_∈Λ

)

. An operator

:

T V

→

W

is called

E

−

linear if it is linear and satisfies

T ax

(

) =

aT x

( )

for all

a

∈

E

and for all

x V

∈

. We

say that

E

−

linear operator

T

is bounded if for each

λ

∈ Λ

,

and for each

x V

∈

there exists

K

_λ

> 0

in which

( )

T x

_λ

≤

K

_λ

x

_λ.

Put

1 2

( ) = ( , )

V

P

λ

x

x x

_λ and

1 2

(

) = (

,

)

W

P

λ

Tx

Tx Tx

_λ, where

( , )

x x

_λ and

(

Tx Tx

,

)

_λ are denoted positive semi-definite

pseudo-inner products in

V

and

W

respectively. So the

E

−

linear operator

T

is bounded if for each

λ

∈ Λ

,

and

for each

x V

∈

there exists

K

_λ

> 0

in which

P

Wλ

(

Tx

)

≤

K P

_λ Vλ

( )

x

.

The set of all bounded

E

−

linear operators from Hilbert module

V

into

W

is denoted by

B V W

_E

( ,

)

and when

=

V

W

is denoted by

B V

_E

( )

. It is easy to see that the map

P

λ

,

λ

∈ Λ

,

defined by

( ) =

{

W

(

) :

,

V

( ) 1}

P T

λ

sup P

λ

Tx

x V P

∈

λ

x

≤

is a seminorm on

B V W

_E

( ,

)

.

Theorem: 3.2 Let

V

and

W

be Hilbert modules over an l.m.c.

H

*

−

algebra

( , (| . |

E

_λ_∈Λ

))

. Then

( )

i

B V W

_E

( ,

)

is a complete locally convex space with topology determined by the family of seminorms

{

P

_{λ λ}

}

_∈Λ.

( )

ii B V

_E

( )

is a locally

C

*

−

algebra with the topology determined by the family of seminorms

{

P

_{λ λ}

}

_∈Λ.

Proof: Suppose that ₁ ₂ ₁ ₁

1 2 ₁

,

, | . |

| . | ,

(

,

),

E

S

B

V

λ

W

λ

λ λ _λ

λ λ

∈ Λ

≥

∈

set of bounded

E

λ₁

−

linear operators. We have

2

1 2 1 1 2 1 2 1 2 2 2

(

σ

_{λ λ}W

( (

S

σ

V_λ

( ))),

x

σ

_{λ λ}W

( (

S

σ

_λV

( )))

x

_λ

≤

S

_λ

(

σ

_λV

( ),

x

σ

V_λ

( )) .

x

_λ

Therefore the following map is a bounded

E

λ₂-bounded.

* 2 2

1 2

(

π

_{λ λ}

) ( ) :

S

V

λ

→

W

λ

2

( )

1 2

( (

1

( ))).

V W V

x

S

x

λ λ λ λ

σ

So we yield a bounded operator _*

1 2

(

π

_{λ λ}

)

from ₁ ₁

1

(

,

)

E

B

V

λ

W

λ

λ into 2 2

2

(

,

)

E

B

V

λ

W

λ

λ . Also

* 1 2

1 2 1 2

{

(

,

), (

) ; | . |

| . | , ,

}

E

B

V

λ

W

λ _{λ λ} _λ _λ

λ

π

≥

λ λ

∈ Λ

is an inverse system of Banach spaces. We are going to show that

( ,

)

E

B V W

and lim

(

,

)

E

B

V

λ

W

λ

λ _λ

← are isomorphic. Suppose that

λ

∈ Λ

,

T

∈

B V W

E

( ,

)

. One can see that,

(

V

)

W

T

ξ

_λ

⊆

ξ

_λ and so there exists a unique operator

T

_λ

:

V

_λ

→

W

_λ in which

σ

_λW

oT

=

T o

_λ

σ

_λV. Moreover

T

_λ is a

bounded

E

λ

−

linear operator. It has a continuous extension

T

λ

:

V

λ

→

W

λ. Thus we can define the following

continuous linear operator

*

(

) :

_E

( ,

)

(

,

)

E

B V W

B

V

λ

W

λ

λ _λ

π

→

T

λ

where

σ

W_λ

oT

=

T o

λ

σ

V_λ. Also, for ₁ ₂

1 2

,

, | . |

_λ

| . |

_λ

λ λ

∈ Λ

≥

we have _* _* _*

1 2 1 2

(

π

_{λ λ}

) (

o

π

_λ

) = (

π

_λ

)

. Now we can define

the following isomorphism operator

:

B V W

E

( ,

)

lim

B

_E_λ

(

V

λ

,

W

λ

)

λ

φ

←

→

*

( ) = ((

T

_λ

) ( )) .

T

_λ

(8)

H

−

" # $ % # % & )!

For each

T

∈

B V W

_E

( ,

),

φ

( )

T

_λ

=

P T

λ

( )

. Linear operator

φ

is surjective. Indeed, let lim

([

])

(

,

)

E

T

_λ _λ _λ

B

V

λ

W

λ

∈Λ

∈

← . We define linear operator

T

as follows

:

T V

→

W

(

V

( ))) .

x

T

λ

σ

_λ

x

_λ

For

λ λ

₁

,

₂

∈ Λ

that

1 2

| . |

_λ

≥

| . |

_λ we have

*

1 1 2

1 2

(

1

( ))) = (

1 2

) (

)(

2

( )) =

(

2

( )).

V V V V

T

λ

x

T

λ

x

T

λ

x

λ λ λ λ λ λ λ

σ

π

σ

So

T

is well-defined. Also it is a bounded

E

−

module map and

φ

( ) = ((

T

π

_λ

) ( ))

_*

T

_λ∈Λ. Completeness of

lim

(

,

)

E

B

V

λ

W

λ

λ _λ

← implies that

B V W

E

( ,

)

is complete.

For

λ

∈ Λ

,

P

λ is a submultiplicative seminorm on

B V

_E

( )

and _* ₁ ₂

1 2 1 2

{

B

_E

(

V

_λ

), (

_{λ λ}

) ; | . |

_λ

| . | , ,

_λ

}

λ

π

≥

λ λ

∈ Λ

is an

inverse system of

C

*

−

algebras and linear operator

:

B V

E

( )

lim

B

_E_λ

(

V

λ

)

λ

φ

←

→

*

((

) ( ))

T

π

_λ

T

_λ

is an isomorphism of topological algebras. Also,

φ

( )

T

_λ

=

P T

λ

( )

and since

(

)

E

B

V

_λ

λ 's are

*

C

-algebras, so

( )

E

B V

is a locally

C

*-algebra. Q.E.D.

Definition: 3.3 We say that

E

−

linear operator

T

has an adjoint if there exists

E

−

linear operator

T

*

:

W

→

V

in

which

[

Tx y

| ] = [ |

x T y

*

]

for each

x

∈

V

and

y

∈

W

. The set of adjointable

E

−

linear operators from Hilbert

E

−

module

V

into Hilbert

E

−

module

W

is denoted by

L V W

_E

( ,

)

and for each

λ

∈ Λ

,

the set of adjointable

operators from

V

_λ into

W

_λ is denoted by

L

_E

(

V W

_λ

,

_λ

)

λ . Let

T

∈

L V W

E

( ,

)

. For each

λ

∈ Λ

,

since

(

v

)

W

,

T

ξ

_λ

⊆

ξ

_λ we can define

*

(

_λ

) :

L V W

_E

( ,

)

L

_E

(

V W

_λ

,

_λ

)

λ

π

→

*

(

) ( )(

_{T x}

V

) = ( )

_{T x}

W

,

λ λ λ

π

+

ξ

+

ξ

Obviously

(

_λ

) ( )

_*

T

L

_E

(

V W

_λ

,

_λ

)

λ

π

∈

and

|

| = (

) ( )

_* _L ₍_V _,_W ₎

E

T

_λ _λ

T

λ λ λ

π

defines a seminorm on

L V W

_E

( ,

)

, where

( , )

.

_L _V _W

E_λ λ λ is the operator norm in

L

Eλ

(

V W

λ

,

λ

)

.

We topologize

L V W

_E

( ,

)

via these seminorms. By similar argument just like previous theorem

L V W

_E

( ,

)

may be

identified with lim

(

,

)

E

L

V

λ

W

λ

λ _λ

← . In particular

lim

( )

(

)

E E

L V

_λ

L

V

λ

←

≅

and we conclude that

L V

_E

( )

is a locally

*

C

−

algebra. The connecting maps of the inverse system

{

L

_E

(

V W

_λ

,

_λ

)}

_λ

λ ∈Λ will be denoted by

* 1 2

1 2 1 2

(

π

_{λ λ}

) ,

λ λ

,

∈ Λ

, | . |

_λ

≥

| . | ,

_λ where

*

1 2 ₁ 1 1 ₂ 2 2

(

_{λ λ}

) :

L

_E

(

V W

_λ

,

_λ

)

L

_E

(

V

_λ

,

W

_λ

)

λ λ

π

→

*

1 2 2 1 2 1

(

) ( )(

_{T x}

V

) =

W

( (

_{T x}

V

)).

λ λ λ λ λ λ

(9)

H

−

So _{* |.|} _|.|

1 2 ₁ ₂

{

L

_E

(

V W

_λ

,

_λ

); (

_{λ λ}

) }

λ

π

λ ≥λ is an inverse system of normed spaces and

{

L

E_λ

(

V W

λ

,

λ

); (

π

λ λ1 2

) }

* |.|λ₁≥|.|λ₂ is an

inverse system of Banach spaces. Also,

lim

( ,

)

(

,

).

E _E

L V W

_λ

L

V W

_λ _λ

λ

←

≅

So

L V W

_E

( ,

)

is a complete locally convex space. On the other hand by [2] each

(

)

E

T

B

V

_λ

λ

∈

belongs to

(

)

E

L

V

_λ

λ .

From this we obtain that each

T

∈

B V

_E

( )

belongs to

L V

_E

( ).

Definition: 3.4 Let

W

H

*

−

algebra,

E

. Let

v w W

,

∈

be basic vectors and let

the operator

F

_{v w}_,

:

W

→

W

be defined with

F

_{v w}_,

( ) = [ | ]

x

x w v

. The linear span of the set

{

F

_{v w}_,

: ,

v w W

∈

}

is

denoted by

F W

_E

(

)

and an operator

T

belonging to

F W

_E

(

)

is called a generalized finite rank operator. Observe that

(

)

(

)

E E

F W

⊆

B W

and *_, _, _, _, _, _*

,

=

,

=

,

=

v w w v v w Tv w v w _{v T w}

F

TF

F

F T

F

, for each

v w W

,

∈

, for each

T

∈

B W

_E

(

)

.

Therefore

F W

_E

(

)

is a selfadjoint two-sided ideal in

B W

_E

(

)

.

Definition: 3.5 An operator

T

∈

B W

_E

(

)

is said to be a generalized compact operator if there exists a sequence of

generalized finite rank operators

{ }

F

_n such that

lim

n

F

n

=

T

. The set of all generalized compact operators is denoted

by

K W

_E

(

)

. By definition

K W

_E

(

) =

F W

_E

(

)

is a closed two-sided ideal in

B W

_E

(

)

. Moreover,

K W

_E

(

)

may be

identified with lim

(

)

E

K

W

λ

λ _λ

← .

We terminate with a result about unitary operators in Hilbert

H

*

−

modules.

Definition: 3.6 Let

E

be a proper

H

*

−

algebra. We say that Hilbert

E

−

modules

V

and

W

are unitary equivalent

if there is a unitary element

U

in

L V W

_E

( ,

)

, namely,

UU

*

=

id

_W and

U U

*

=

id

_V.

If

U

∈

L V W

_E

( ,

)

is unitary then it is clear that

U

is a surjective

E

−

linear map and also that

U

is isometric,

since

U x

( )

2

= [ ( ) |

tr U x U x

( )] = [

tr U U x

*

( ) | ] = [ | ] =

x

tr x x

x

2. Our next result will be the converse

assertion, that if

U E

:

→

F

is an isometric, surjective

E

−

linear map then

U

is unitary. For this we need the following lemma.

Lemma: 3.7 Let

E

be a proper

H

*

−

algebra and

a

∈

E

. If

ac

=

bc

for each

c

∈

E

then

a a

*

=

b b

* .

Proof: We have

ac

2

=

bc

2, so that

ac ac

,

=

bc bc

,

and

a ac c

*

,

=

b bc c

*

,

.

Hence

(

a a b b c c

*

−

*

) ,

= 0

for each

c

∈

E

. From this and by Lemma

3.1

of [1] we have

* * * *

=1

)

=

=1

| (

) ,

|= 0

sup

_c

a a b b c

−

sup

_c

a a b b c c

−

.

Thus

(

a a b b c

*

−

*

)

= 0

, where

c

= 1

, so that

(

a a b b c

*

−

*

) = 0

for arbitrary

c

∈

E

. Therefore

* *

(

a a b b E

−

)

= 0,

so that

a a b b

*

−

*

= 0

. Q.E.D.

Proposition: 3.8 With

E V W

, ,

as before, let

U

be an

E

−

linear map from

V

to

W

. The following conditions are equivalent:

( )

i

U

is an isometric surjective

E

−

linear map;

(10)

H

−

" # $ % # % & )*

Proof. Suppose that

( )

i

holds. For

x

in

V

,

[ ( ) |

U x U x

( )] =

b b

* and

[ | ] =

x x

c c

* for some

b c

,

in

E

. For each

a

in

E

, we have

* 2 *

ab

=

tr a U x U x a

( [ ( ) |

( )] )

=

tr U ax U ax

([ (

) |

(

)])

=

U ax

(

)

2

=

ax

2

=

tr ax ax

([

|

])

=

tr a x x a

( [ | ] )

*

=

tr a c c a

( (

*

) ))

=

ac

* 2.

Thus

ba

*

=

ca

* for each

a

in

E

. By previous lemma

b b

*

=

c c

* . This implies that

[ ( ) |

U x U x

( )] = [ | ]

x x

for each

x

in

E

and by polarization identity

[ ( ) |

U x U y

( )] = [ | ]

x y

for each

x y

,

in

E

.

Now let

x

∈

V

and

z

∈

W

. Since

U

is surjective, there is a

y

∈

V

such that

( ) =

U y

z

. We have

[ ( ) | ] = [ ( ) |

U x

z

U x U y

( )] = [ | ] = [ |

x y

x U

−1

( )]

z

.

Hence

U

*

=

U

−1. This implies that

U

satisfies

( )

ii

; and the implication

( )

ii

( )

i

is obvious as already discussed. Q.E.D.

References:

[1]

W. AMBROSE, Structure theorems for a special class of Banach algebras, Trans. Amer. Math. Soc. 57(1945), 364-386.

[2]

D. Bakic and B. Guljas, Operators on Hilbert

H

*

−

modules , J. Operator Theory, Vol. 46(2001), 123-137.

[3]

F. F. Bonsall, and J. Duncan, Complete normed algebras, Ergebrise der Mathematik Band 80, Springer Verlag (1973).

[4]

M. CABRERA, J. MARTINEZ, A. RODRIGUEZ, Hilbert modules revisited: Orthonormal bases and Hilbert-Schmidt operators, Glasgow Math. J. 37(1995), 45-54.

[5]

M. Haralampidou, On locally convex

H

*

−

algebras, Math. Japonica 38(1993), 451-460.

[6]

A. El Kinani, On locally pre-

C

*

−

algebra structures in locally m-convex

H

*

−

algebras, Turk J. Math. 26(2002), 263-271.

[7]

K. SCHM

U

DGEN,

U

ber LMC*

−

Algebren, Math, Nachr, 68(1975), 167-182.