GENERALIZED POSITIVE DEFINITE FUNCTIONS AND COMPLETELY MONOTONE FUNCTIONS ON FOUNDATION SEMIGROUPS

GENERALIZED POSITIVE DEFINITE FUNCTIONS AND COMPLETELY

MONOTONE FUNCTIONS ON FOUNDATION SEMIGROUPS

*

M. Lashkarizadeh Bami

†

Department of Mathematics, Institute for Studies in Theoretical physics and Mathematics, Tehran, P. O. Box 19395-1795, Islamic Republic of Iran

Abstract

A general notion of completely monotone functionals on an ordered Banach

algebra

B

into a proper

H

*

-

algebra

A

with an integral representation for such

functionals is given. As an application of this result we have obtained a

characterization for the generalized completely continuous monotone functions on

weighted foundation semigroups. A generalized version of Bochner’s theorem on

foundation semigroups is also obtained.

*

1991 Mathematics Subject Classification. Primary 43 A35, 43 A10; This research was in part supported by grant from IPM.

†

E-mail: [email protected]

Introduction

In the present, paper we shall introduce the concept of a completely monotone functional on an ordered Banach algebra B into a proper H*-algebra A and we shall give an integral representation for such functionals with respect to A-valued measures on , the space of all positive multiplicative linear functionals on B. As an application of the theory we shall obtain an integral representation for the generalized w-bounded conti-nuous completely monotone A-valued functions with respect to positive A-valued measures on , the space of all w-bounded continuous nonnegative semicharac-ters on a foundation semigroup S with a Borel measurable weight function w. We will also give a generalization of our earlier version of Bochner’s

theorem [4; Theorem 4.2].

) (B

+

Δ

+

Γw

Keywords: Locally compact semigroups; Positive definite functions; H*-algebras; Spectral measures

1. Preliminaries

Recall that (see, [11], [12], [13], [17]) a proper

H*-algebra is a Banach algebra A whose norm is a Hilbert space norm and which has an involution:

on A such that for

all

*

x

x→ (_y,_x*,_z)=(_xy,_z)=(_x,_zy*)

A z y

x, , ∈ . Let be the trace class of A. It is a Banach algebra with respect to a norm

} , : { )

(A = xy x y∈A

τ

(.)

τ which is related to the norm . of A by

2 * ₎ (a a = a

τ for all a∈A. There is a trace tr defined on τ(A) such that tr(ab)=tr(ba)=(a,b∗) for all

A b

a, ∈ , where (., .) denotes the scalar product on A. if

b b

a= * _{for some b}∈_{A then a is called positive and we}

write . It is obvious that if and only if for all

0

≥

a a≥0

0 ) ,

(ax x ≥ x∈A. A right module H over A is called a Hilbert module if there is a -valued function ( , ) on H×H with the following properties

) (A

τ

1. (ξ+η,ϕ)=(ξ,ϕ)+(η,ϕ) for all ξ,η,ϕ∈H. 2. (ξ,η)*=(η,ξ) for all ξ_,η∈_H.

if ξ=0.

5. tr(ξ,η)2≤τ(ξ,ξ)τ(η,η) for all ξ,η∈H.

6. H is complete in the norm ξ =(τ(ξ,ξ))12.

The function ( , ) is called a generalized scalar product. There is a linear structure on H such that H is an ordinary Hilbert space with respect to the scalar product ξ,η =tr(η,ξ). An A-linear operator on H is an additive linear mapping such that

for all

H H T: →

a T a

T(ξ )=( ξ) ξ∈H,a∈A; T is bounded in the in the sense that Tξ ≤M ξ for some and every

0

≥

M .

H

∈

ξ For each bounded A-linear operator T its adjoint T* is A-linear and has the property that

for all .

) , ( ) ,

(_Tξ η = ξ _T*η ξ_,η∈_H

By a real ordered Banach algebra we shall mean a real Banach algebra Mr with a closed partial order satisfying the following:

≥

(i) x≥ y⇒x+z≥y+z, for all z∈Mr. (ii) x≥0,y≥0⇒xy≥0.

(iii) x≥0⇒ax≥0,for all nonnegative real numbers α. Note that an order “ ” on an ordered Banach algebra is called closed if for every two sequences (x

≥

n) and (yn)

in B from and and

(n∈

x

xn⎯⎯→n y y

n n⎯⎯→ n

n y

x ≥ ) it follows that . A complex

Banach algebra B of the form M y x≥

r⊕iMr, where Mr is a real ordered Banach algebra, is called an ordered Banach algebra. On an ordered Banach algebra B, we put P(B)={b∈B:b≥0} and P1(B)={b∈P(B):

} 1

=

b . A linear functional f on B is called positive if

for all . In the case where B is commutative, we shall denote by the space of all bounded multiplicative linear functionals on B and by

the space of all positive functionals in .

0 ) (b ≥

f b∈P(B)

) (B

Δ

) (B

+

Δ Δ(B)

Definition 1.1. Let B be a commutative ordered Banach algebra. For every n∈ (the set of nonnegative integers) we define the operator Δn on B* (the dual of B) by

) ( ) ( 0f b = f b

Δ

) ( ) ( ) ( ) ( ) ;

( 1 0 0 1 1

1f b b =Δ f b −Δ f bb = f b −f bb

Δ

and for every n≥2

) , , ; (

) , , ; ( )

, , ; (

1 1 1

1 1 1 1

− −

− −

Δ −

Δ = Δ

n n

n

n n

n n

b b bb f

b b b f b

b b f

K K K

). , 2 , 1 ; , , , ,

(_f∈_B* _{b b1K b} ∈_{B n}= _K

n A linear functional

is called completely monotone if

*

B f ∈

0 ) , , ;

( ₁ ≥

Δ_nf b b _Kb_n

for all n∈ and b,b₁,_K,b_n∈P₁(B).

An operator-valued transformation L (the space of all bounded linear operators on a Hilbert space H) is called completely monotone if for every

→

B

U: (H)

H

∈

ξ the mapping ϕ_ξ:b_a U_bξ,ξ (b∈B) defines a completely monotone functional on B.

We now recall some definitions concerning topological semigroups.

Throughout this paper S will denote a locally compact, Hausdorff topological semigroup.

Definition 1.2. On a commutative topological

semigroup S with (the space of bounded continuous complex-valued functions on S) inductive identity, for each n∈

) (S Cb

+ we define the operator Δn on by

), ( ) (

0f x = f x

Δ

) ( ) ( ) ( ) ( ) ;

( _{1 0 0 1 1}

1f x h =Δ f x −Δ f xh = f x − f xh

Δ

and for every n≥2

), , ; (

) , , ; ( )

, , ; (

1 1

1 1 1 1

− −

− −

Δ −

Δ = Δ

n n n

n n

n n

h xh f

h h x f h

h x f

K K K

). , 2 , 1 , , , , ), (

(f∈C_b S xh_1K h_n∈S n= _K A function is called completely monotone if

(n∈

) (S C

f ∈ _b Δ_nf ≥0

) (cf. [5; p. 43]).

Definition 1.3. An operator-valued transformation L is called completely monotone if for every

→

S

T: (H)

H

∈

ξ the mapping

) ( , x S T

xa xξ ξ ∈

is completely monotone on S.

Definition 1.4. Let B be an ordered commutative Banach algebra and H be a Hilbert module over a proper H*-algebra A. A linear mapping is called a completely monotone A-functional if for every n∈

A B f : →

+ for every (n + 1)-positive elements b,b

K

, ; (b b₁ f n

Δ ,b_n)≥0

1,…,bn of B where,

) ( ) ( 0f b = f b

Δ

) ( ) ( ) ( ) ( ) ;

( _{1 0 0 1 1}

1f b b =Δ f b −Δ f bb = f b − f bb

Δ

). , , ; (

) , , ; ( )

, , ; (

1 1 1

1 1 1 1

− −

− −

Δ −

Δ = Δ

n n

n

n n

n n

b b bb f

b b b f b

b b f

K K K

Definition 1.5. Let S be a commutative topological semigroup with an identity. A mapping is called completely monotone if for all nonnegative integers n and all where

A S f: →

0 ) , , ;

( ₁ ≥

Δ_nf xh _K h_n S h h

x, ₁,_K, _n∈ )

( ) (

0f x = f x

Δ

) ( ) ( ) ( ) ( ) ;

( _{1 0 0 1 1}

1f xh =Δ f x −Δ f xh = f x −f xh

Δ

and for every n≥2

). , , ; (

) , , ; ( )

, , ; (

1 1 1

1 1 1 1

− −

− −

Δ −

Δ = Δ

n n

n

n n

n n

h h xh f

h h x f h

h x f

K K K

Definition 1.6. Let B _{a Banach * algebra and} A be a proper H*-algebra. A linear mapping is called a positive A-functional if

A B f : →

0 ) (

1 1 *

* ≥

∑∑

= =

n

i n

j

j j i i f bb a

a

for all b1,…,bn in B and a1,…,an in A.

Definition 1.7. Let S _{be a *-semigroup. Then a mapping} A

S→ :

ϕ is called positive definite if

0 ) (

1 1 * * ≥

∑ ∑

= =

n i

n

j aiϕ xixj aj

for all x1,…,xn in S and a1,…,an in A.

Recall that a Borel measurable mapping w:S→ +

(the set of nonnegative real numbers) with

and such that w and )

( ) ( )

(xy w xw y

w ≤ (x,y∈S)

w 1

are locally bounded (i.e., bounded on compact subsets of S) is called a weight function on S. A function

→

S

f: is called w-bounded if there is a such that

0

>

k

) ( ) (x kw x

f ≤ , for all x∈S.

Recall also that M(S,w) denotes the set of all complex, regular, signed measures μ (not necessarily bounded) of the form μ=μ₁−μ₂+i(μ₃−μ₄) where μ i is a positive regular measure on S with

i = 1,2,3,4 (see, for example [2], [7], [9]). Note that for an element and a Borel set B, μ (B) is well-defined whenever B is relatively compact. For every , the equation

) , (

1

i

S L

w∈ μ

) , (S w M

∈

μ

) , (S w M

∈

μ

)), ( ( )

.

(w fwd f C S

fd _b

S

S =

∫

∈

∫

μ μ

defines a measure , the space of all bounded regular complex measures on S. With the norm

) ( . M S wμ∈

)), , ( (

. M S w

w

w= μ μ∈

μ

where w.μ denotes the total variation of w.μ, the space M(S,w) defines a Banach lattice, and with the convolution product

)), ( ),

, ( , , (

) ( ) ( ) ( )

)( (

00 S

C f w S M

y d x d xy f f

S S

∈ ∈

= ∗

∫ ∫

ν μ

ν μ ν

μ

(1)

where C00(S) denotes the set of all functions in Cb(S) with compact support, defines a Banach algebra. From part (iii) of Theorem 4.6 of [7], we conclude that (1) also holds for every w-bounded Borel measurable function f on S.

We also recall (see, for example, [1], [6], [18]) that (or ) denotes the set of all measures

for which the mappings

) (S

M_a L~(S) )

(S M

∈

μ x_aδ_x∗μ and

x

x_a μ∗δ (where denotes the Dirac measure at x) from S into M(S) are weakly continuous. As in [7], we can define M

x

δ

a(S,w) (or ) as the set of measures for which . Then, M

) , ( ~

w S L

) , (S w M

∈

μ w.μ∈M_a(S) a(S,w) is a closed, two-sided L-ideal of M(S,w). Finally, we call S a foundation semigroup if is dense in S. A mapping

)} ( :

) (

{supp ∈M_a S

∪ μ μ →

S :

χ is called a semicharacter if χ(xy)=χ(x)χ(y) for all x,y∈S. We denote by the set of all w-bounded continuous semicharacters on S, and by the set of nonnegative semicharacters in . If S is commutative and foundation, then is homomorphic to

whenever has the compact open topology and has the Gelfand topology. In particular; is a locally compact Hausdorff space (see, Theorem 2.10 of [8]).

w

Γ

+

Γ_w

w

Γ

w

Γ Δ(M_a(S,w))

w

Γ

)) , ( (Ma S w

Δ

w

Γ

An operator-valued transformation L is called w-bounded (continuous, respectively) if for every

→

S

U: (H)

H

∈

η

ξ, the map: x_a U_xξ,η is w-bounded (continuous, respectively). Finally if L is such that

→

S

U: (H) )

, (x y S U

U

U_xy = _{x y} ∈ , then U is called a

2. Generalized Representations and Positive-Definite Functions on Weighted

Foundation Semigroups

We start this section with the following result which is indeed a generalization of our earlier result (Theorem 4.4 of [7]).

Theorem 2.1._{Let S be a foundation *-semigroup with}

identity and with a Borel measurable weight function w

such that _{. Let T be a}

*-representation of M

) ( ) ( )

(x w x x S

w ∗ = ∈

a(S,w) by bounded A-linear operators on a Hilbert module H over a proper H* -algebra A such that for every 0≠ξ∈H there exists a measure μ∈M_a(S,w) such that T_μξ≠0. Then there exists a unique w-bounded continuous *-representation V of S by A-linear operators on H such that

(

η,Tμξ

)

=

∫

_S

(

η,Vxξ

)

dμ(x)

(

ξ,η∈H,μ∈Ma(S,w)

)

.

(2)

Proof. Recall that by Theorem 1 of [11] H with the inner product .,. where ξ,η =tr(η,ξ) defines a Hilbert space and by Theorem 4 of [11], the adjoint operator T* of T defines a bounded A-linear operator on H. So by Theorem 5.4 of [7] there exists a w-bounded continuous *-representation V of S by bounded operators on the Hilbert space (H,.,.) such that

(

( , ), ,

)

. )

( ,

, V d x M S w H

T _a

S x ∈ ∈

=

∫

ξ η μ μ ξ η η

ξ

μ

(3)

Now let R(A) denote the space of the right centralizers of A. From Lemma 2 of [14] and Theorem 1 of [11] for every U∈R(A) we have

(

) (

)

(

,

)

( ). )

( ,

, ,

,

∫

∫

′ =

=

′ =

′ =

S x

S Vx U d x trU V d x

U T T U tr T trU

μ ξ η μ

η ξ

η ξ ξ η ξ

η _{μ μ μ}

So by Theorem 2 of [16]

(

η,Tμξ

)

=

∫

_S

(

η,Vxξ

)

dμ(x)

(

μ∈Ma(S,w),ξ,η∈H

)

.

This proves formula (2).

We shall now use formula (3) and prove that if is

A-linear for every , then is A-linear for every To see this from (3) for every

,

μ

T

) , (S w Ma

∈

μ Vx

. S x∈

) , (S w M_a

∈

μ ξ,η∈H, and a∈A we have

( )ξ μ η ( )ξ

(

ξ η

)

η,V a d (x) ,T_μ a trT_μ( a),

S x = =

∫

(

) (

)

∫

∫

=

=

= =

∗

∗

S x

S a Vx d x V ad x

a T tr a T tr

). ( ) ( , ) ( ,

, ,

) (

μ ξ η μ

ξ η

η ξ η

ξ _μ

μ

Since both the mappings: x→ η,V_x(ξa) and

a V

x→ η,( _xξ) are w-bounded and continuous and S is a foundation semigroup, from Lemma 4.8 of [7] we conclude that V_x(ξa)=(V_xξ)a(x∈S,a∈A).

The following result is indeed a generalization of our earlier version of Bochner’s theorem [4; Theorem 4.2].

Theorem 2.2. (Generalized Bochner’s theorem on foundation semigroups). Let S be a commutative foundation topological *-semigroup with identity and with a Borel measurable weight function w. Let A be a proper H*-algebra over a Hilbert module H. Then a mapping is w-bounded continuous and positive definite if and only if there exists a unique positive A-valued measure

) ( :S τ A

ϕ →

ϕ

λ on Γw such that

) ( ) ( ) ( ) (

* xd x S

x

w

∈ =

∫

_Γ χ λ χ

ϕ ϕ .

Proof. Since ϕ is w-bounded and continuous, by Theorem 1 of [16] there exists a w-bounded weakly continuous *-representation V of S by bounded A-linear operators on a Hilbert A-module K with some

such that and

K

∈

0

ξ

) , ( )

( ξ₀ ξ₀

ϕ x = Tx Vx ≤w(x) for every

. S x∈

Using the integration theory on page 120 of [13] and Lemma 2 of the same reference, we conclude that the mapping Φ:M_a(S,w)→τ(A) given by

(

,

)

( )( , )) )

( ) ( )

( = x d x = _S 0Vx 0 d x ∈MaSw

Φ μ

∫

ϕ μ

∫

ξ ξ μ μ

is well-defined. It is also easy to see the Φ defines a positive A-_{functional on the Banach *-algebra} Ma(S,w). Therefore, by Theorem 3 of [15] there exists a positive

-valued measure λ on such that

) (A

τ& Δ(M_a(S,w))

∫

Δ

= Φ

)) , (

( ˆ( ) ( ). )

(

w S Ma

dλσ

σ μ μ

Using Theorem 2.10 of [7], we conclude that

(

( ) ( )

)

( ) ( ( , )).

)

( _* xd x d x M_a S w

S

w

∈ =

Φ μ

∫ ∫

_Γ χ μ λ μ

By Fubini’s theorem

(

( ) ( )

)

( )( ( , )). )

( )

(xd x _* xd d x M_a S w

S

S w

∈ =

∫ ∫

∫

ϕ μ Γ χ λ χ μ μ

Since both functions ϕ and →

∫

_Γ* ( ) ( ) are

w

d x

w-bounded and weakly continuous and S is a foundation semigroup, we infer that

). ( ) ( ) ( )

(x _* xd x S

w

∈ =

∫

_Γ χ λ χ ϕ

The uniqueness of λ follows in the same lines as those of Theorem 4.2 of [4].

3. Completely Monotone Functionals on Ordered Banach Algebras

Our starting point of this section is the following:

Theorem 3.1. Let B be a commutative ordered Banach algebra with a bounded approximate identity in P

) (e_α 1(B). Let k be the set of all completely monotone

functionals f in B* such that f ≤1. Then K is a convex and weak*-compact subset of A*. If f is an extreme point of K, then for all and

for all 0 ) (a ≥

f a∈P(B)

) ( ) ( )

(ab f a f b

f = a,b∈B.

Proof. It is clear that K is a convex and weak*-closed subset of the unit ball of B* and so by the Banach Alaoglu theorem is weak*-compact. Let f be an extreme point of K. Then it is clear that for all

. Since P

0 ) (a ≥ f

) (B P

a∈ 1(B) spans B, to prove that

for all

) ( ) ( )

(ab f a f b

f = a,b∈B, it suffices to show

that for all For every

we define by . It is

easy to see that

) ( ) ( )

(ab f a f b

f = a,b∈P1(B).

B

a∈ _{f B}*

a∈ fa(b)=f(ab) (b∈B)

), , , , ; ( )

, , ; )(

(f f_a b b₁ b_{n n 1}f bb₁ b_n a

n − K =Δ + K

Δ

for all n∈ , and Thus, is also completely monotone. So

). ( , , ,

,bb₁ b P₁ B

a _{K n}∈ f−f_a

, 0 ) )( ( ) )(

(f −f_a e_αb =Δ₀ f − f_a e_αb ≥

and

, 0 ) ; )( ( ) )( ( ) )(

(f −f_a e_α − f −f_a e_αb =Δ₁ f − f_a e_α b ≥

for all From these two inequalities it follows that

). ( ,b P₁ B

a ∈

) ( 1 ) ( ) (

) )( ( ) )( ( 0

α α

α

α α

ae f ae

f e f

e f f b e f

f _{a a}

− ≤ −

= −

≤ −

≤

for all α and all a,b∈P₁(B). Since (eα) is a bounded approximate identity for B, it follows that

)). ( , ( ) ( 1 ) )( (

0≤ f −f_a b ≤ − f a a b∈P₁ B (4) Using the fact that f is completely monotone, we

conclude that

)), ( , ( ) ( ) (

0≤Δ₀f ab = f ab a b∈P₁ B

and

)). ( , ( ) ( ) ( ) ; (

0≤Δ₁f a b = f a − f ab a b∈P₁ B

Thus

)). ( , ( ) ( ) (

0≤ f ab ≤ f a a b∈P₁ B (5)

We shall now consider three cases.

Case 1. f(a)=0. So by (5), f(ab)=0. Hence

)). ( , ( ) ( ) ( 0 )

(ab f a f b a b P₁ B

f = = ∈

Case 2. f(a)=1. Then by (4),

)) ( , )( ( )

(b f ab a b P1 B

f = ∈ and so

)). ( , )( ( ) ( ) ( )

(ab f b f a f b a b P₁ B

f = = ∈

Case 3. 0< f(a)<1. In this case we write

) ( ) ( ) ( 1 )) ( 1 (

a f

f a f a f

f f a f

f a + a

− − −

= .

From (4) it follows that (f − f_a) (1−f(a))∈K, and (5) implies that f_a f(a) also belongs to K. Since f is an extreme point of K, it follows that f_a f(a)= f. So

) ( ) ( )

(ab f a f b

f = for all This completes the proof.

). ( ,b P₁ B

a ∈

Theorem 3.2. Let B be a commutative ordered Banach algebra with a bounded approximate identity (eα) in

P1(B). Then a linear transformation L(H) (H is

a Hilbert space) is completely monotone if and only if there is a positive operator-valued measure E on

such that

→

B U:

) (B

+

Δ

). , , ( , (.) ) ( ,

)

( b d E H b B

U

B

bξ η =

∫

_Δ+ σ σ ξ η ξ η∈ ∈ (6)

Moreover, U is a representation if and only if E is a spectral measure.

Proof. Let L(H) be completely monotone.

Without loss of generality, we may assume that

→

B U:

). (b B b

U_b ≤ ∈ For every ξ∈H with ξ =1 we define the linear functional Lξ on B by

). ( , )

(b U b B

L_ξ = _bξ ξ ∈

It is clear that defines a completely monotone

functional on B with

ξ

L

. 1

≤

ξ

L By the integral form of

) (B

+

Δ such that

). ( ) ( ) ( ) ( )

( bd , b B

b L

B ∈

=

∫

+

Δ σ μξξ σ

ξ

So if 0≠ξ∈H is arbitrary, then there exists a unique positive regular measure μξ,ξ with μξ,ξ ≤ ξ 2

and ). ( ) ( ) ( , )

( bd , b B

U

B

bξ ξ =

∫

_Δ+ σ μξξ σ ∈

By the polarization identity for every ξ,η∈H and we have B b∈

(

)

(

)

(

(

)

,

(

)

,

)

. , , 4 1 , η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ η ξ i i U i i i U i U U U b b b b b − − − + + + − − − + + = Thus ), , , ( ) ( ) ( , )

( bd , b B H

U

B

bξ η =

∫

_Δ+ σ μξη σ ∈ ξ η∈

where

(

ξ ηξ η ξ ηξ η ξ ηξ η ξ ηξ η

)

η

ξ μ μ μ μ

μ _, = _{+ , +} − _{− , −} +i _{+i , +i} −i _{+i , +i}

4 1

.

Now let B denote the σ-algebra of all Borel subsets of Define the operator-valued measure E on B by

)) ( (Δ₊ B

). (B + Δ )) ( (Δ₊ B

)). ( ( , , ( ) ( , )

(M , M H M B

E ξ η =μξη ξ η∈ ∈B Δ₊

It is easy to see that E is positive, in the sense that

0 , ) (M ξ ξ ≥

E for all ξ∈H and Moreover,

)). (

( B

M∈B Δ₊

). , , ( , (.) ) ( , )

( bd E b B H

U

B

b =

∫

∈ ∈

+

Δ σ ξ η ξ η

η

ξ σ

For simplicity, we abbreviate this equality as

). ( ) (

)

( bdE b B

U

B

b =

∫

∈

+

Δ σ σ

Now for every we denote by the restriction of the Gelfand transform of b to that is

for all Since by the Gelfand

representation theorem separates the points of from the Stone-Weierstrass theorem it follows that it is dense in the space of all continuous complex-valued functions on

vanishing at infinity. Now if U is multiplicative, then

for every

B

b∈ bˆ

), (B + Δ ) ( ) ( ˆ _b

bσ =σ σ∈Δ₊(B). } : ˆ {b b∈B

= P ), (B + Δ )), ( ( 0 B

C Δ₊

) (B + Δ B b

a, ∈ we have

. ) ( ˆ ) ( ˆ ) ( ) ( ˆ ) ( ˆ ) ( ) ( ) ( ) (

∫

∫

∫

∫

+ + + + Δ Δ Δ Δ = = = = B B b a ab B B dE b dE a U U U dE ab dE b a σ σ σ σ σ σ σ σ σ

Since for every fixed b∈B, each of the functions and

are bounded and

linear on P, and P is dense in then for every Borel subset M of we have

∫

Δ+( ) ( )

ˆ

B ab dE

aa σ σ

∫

+

Δ ( )ˆ( )

ˆ

B a dE

aa σ σ

)), (

(Δ₊ B ˆ( ) (ˆ ) )

( ∈P

∫

Δ₊ B bσ dEσ a

)), ( (

0 B

C Δ₊

) (B + Δ

∫

∫

∫

+ + + Δ Δ Δ = ) ( ) ( ) ( ) ( ˆ ) ( 1 ) ( ˆ ) ( 1 B B M B M dE a dE dE a σ σ σ σ σ σ σ

where 1M denotes the characteristic function of the set M. A similar argument shows that for every two Borel subsets M and N of Δ₊(B)

. ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( ) ( ) (

∫

∫

∫

+ + + Δ Δ Δ = B N B M

B M N

dE dE dE σ σ σ σ σ σ σ

That is E(M∩N)=E(M)E(N). So E is a spectral measure on Δ₊(B). The proof is now complete.

The following theorem gives a characterization of the completely monotone functionals on commutative ordered Banach algebras.

Theorem 3.3. Let B be a commutative ordered Banach algebra with a bounded approximate identity in P1(B).

Then a bounded linear mapping U of B into a proper H*-algebra A is completely monotone if and only if there is a positive -valued measure E on such that

) (A

τ Δ₊(B)

(

, ( )

)

( )

(

, (.)

)

( , , ).

)

( bd E b B H

b U

B ∈ ∈

=

∫

+

Δ σ ξ η ξ η

η

ξ σ

Moreover, U is a positive homomorphism if and only if E is a generalized spectral measure.

Proof. For every ξ∈H with tr(ξ,ξ)=1 we define

). ( ), ( )) ( , ( )

(b tr U b U b b B

L_ξ = ξ ξ = ξ ξ ∈

From ) , , ; ( ) , ( )) , ; ( ,

( _{1 n n 1 n}

n U bb Kb = Δ U b b Kb

Δ ξ ξ ξ ξ

and the fact that U is bounded and completely monotone we conclude that defines a completely linear

functional on B. So by Theorem 3.2 there exists an operator-valued measure E by bounded operators on the Hilbert space

ξ

L

(H, .,.) such that

). ( , (.) ) ( ),

( ) (

)

( bd E b B

b U b L

B ∈

=

=

∫

+

Δ σ ξ η

η

ξ σ

ξ

For every T∈R(A) by Lemma 2 of [14] we have

( )

(

)

(

( )

)

( )

( )

( )

( )

(

,

( )

.

)

. , .

, ,

,

) (

) (

∫

∫

+ +

Δ Δ

=

′ =

′ =

′ =

B B

E trT d b

T E d b

T b U b U T tr b U trT

ξ η σ

η ξ σ

η ξ

ξ η ξ

η

σ σ

Now it is easily seen that the mapping: given by:

defines a -valued measure on Therefore, by Lemma 2 of [14] and Theorem 2 of [16] we have

→ Δ₊(B) )

(A

τ M →(η,E(M)ξ) (M∈B(Δ₊(B)) )

(A

τ Δ₊(B).

( )

(

,

)

( )

(

,

( )

.

)

( ).

)

( bd E b B

b U

B ∈

=

∫

+

Δ σ η ξ

ξ

η _σ

Thus, the proof is complete.

We are now in a position to state and prove the main result of this paper. Note that if H is a right Hilbert module over a proper H*-algebra A, then a mapping is called w-bounded and continuous if for every

A S T: →

H

∈ η

ξ, the mapping is a

w-bounded continuous complex-valued function on S.

) , ( T_x tr

x_a ξ η

Theorem 3.4. Let S be a commutative foundation semigroup with identity and with a Borel measurable weight function w continuous at the identity. Let H be a Hilbert module over a proper H*-algebra A. Then a mapping is w-bounded continuous and completely monotone if and only if there exists a unique positive A-valued measure E on such that

A S T: →

+

Γw

( , T ) ( )xd

(

,E ( ).

)

(x S, , H).

w

x =

∫

_Γ+χ ξ η ∈ ξ η∈

η

ξ _χ

T is a homomorphism if and only if E is a generalized A-valued spectral measure.

Proof. From the continuity of w at the identity of S it follows that has a bounded approximate identity in (see [9]). It is also easy to see that the equation

) , (S w M_a

)) , ( (

1 M S w

P a

( )

(

U

)

(

T

) ( )

d x

(

M_a

(

S w

)

H

S x ∈ ∈

=

∫

ξ η μ μ ξ η μ

η

ξ, , , , ,

defines a completely monotone A-valued bounded functional on the ordered Banach algebra

Therefore, by Theorem 3.3 there exists a unique positive A-valued measure E on such that

). , (S w Ma

)) , ( (M_a S w

+

Δ

)

) ( )

( )

∫

( )

(

(

)

+

Δ

=

)) , (

( ˆ , .

,

w S

M_a d E

U μ μ χ ξ η

η

ξ χ

(ξ,η∈H,μ∈M_a(S,w)).

Now an application of this equality and Theorem 2.10 of [8] with the aid of Fubini’s theorem gives

( )

(

)

( ) ( )

(

( )

)

( )

(

,

( )

.

)

( )

. . , ,

∫ ∫

∫ ∫

+ +

Γ Γ

= =

S S

x d E d x

E d x d x U

w w

μ η ξ χ

η ξ μ χ μ

η ξ

χ χ

Now since both mappings

∫

_Γ+ ( ) ( )

w

E d x

x_a χ ξ, _χ .η

and xa ξ,ηTx are w-bounded and continuous and S

is also a foundation semigroup, we conclude that

( , T ) ( )xd

(

,E ( ).

)

( , H,x S).

w

x =

∫

_Γ+χ ξ η ξ η∈ ∈

η

ξ χ

Remark. The following example shows that the

conclusion of the preceding theorem is not valid in general for non-foundation semigroups.

Example 3.5. Let S = [0,1]. Then with the usual topology of the real line and the multiplication

) , )( ,

min(x y x y S

xy= ∈ S defines a non-foundation semigroup. If we choose on S, then

where 1 denotes the function which is identically one on S. It is clear that the mapping (m denotes the Lebesgue measure on [0,1]) given by

1

=

w Γ_w+={1}, )) , ( ( :_{S L}2 _{S m}

T →L

( )

(

, ,

)

, ˆ_{f x S f L}2 _{S m}

x f

T_x = ∈ ∈

where denotes the characteristics function on [0,x], defines a completely monotone operator-valued transfo-rmation of S by operators on the Hilbert module

(see [3]). If the formula (6) is valid for T, then we arrive at the contradiction that for every x in S, where I denotes the identity operator on

xˆ

) , ( 2 _{S m}

L

I T_x=

). , ( 2 _{S m}

L

References

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217-227, (1998).

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13. Saworotnow, P. P. Bochner-Raikov theorem for a generalized positive definite function, Ibid., 38, 117-121, (1971).

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