Distribution theories based on representations of locally compact Abelian topological groups

Distribution theories based on representations of locally

compact Abelian topological groups

Citation for published version (APA):

Elst, ter, A. F. M. (1988). Distribution theories based on representations of locally compact Abelian topological groups. (EUT-Report; Vol. 88-WSK-06). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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Department of Mathematics and Computing Science

Distribution theories based on representations oflocally compact

Abelian topological groups by

A.F.M. ter Elst

EDT-Report 88-06 3-12-1987

AMS Subject Classifications (primary) 16A80, 22010, 46A12

(secundary) 15A69, 43A65, 46A05, 46F05, 46H25, 46M05

I. 1.1. 1.2.

1.3.

II. ll.l. 11.2. 11.3. III. III.1. III.2. III.3. IV.

V.

VI. VI. 1. V1.2. VII. VIII. VIII.I. VIlI.2. Appendix A.

B.

C. Introduction Some notations

The spacesSuo

c

andTu.

c

The spaceSuo

c

The spaceT

u. c

The pairing betweenSuo

c

andTu,

c

Seminonns onSu,

c

and bounded sets inTu.

c

The Stone-representant for a unitary representation Seminonns onSuo

c

and bounded sets inTu.

c

A growth condition

Nice properties of the spacesSuo

c

andTu.

c

Nice properties of the spaceTu. c

Nice properties of the spaceSuo

c

Su, c andTu, c spaces which are Montel or nuclear

Examples

Continuous linear maps

Conjugate linear homeomorphims The strong dual ofSu,

c

andTu,

c

The spaces

Su.c

and

1U,c

Cartesian products

Tensor products and kernel theorems Tensor products

Kernel theorems Topological groups . Topological vector spaces

A suitable topology for the tensor product of two locally convex topological vector spaces which have additional structure

Acknowledgement Index of symbols List of properties References Page 1 3 4 4 6 8 12 12 16 20

26

28

31 37

49

62

69

69

71 75 81 81 85 102 104 107 110 111 113 114

Inthe monograph [EG2] the topological spacesSH,A enTH,A' which are both inductiveand pro-jective limit of Hilbert spaces, have been studied extensively. Here the labelHdenotes a separ-able Hilbert space, the label A denotes a positive self-adjoint operator in H. IfH =L2(JR) and

A

=-

1-

d 2

2 it turns out thatSH A

=

C * L 2(JR) as sets, with C

=

{x i--7

l.

2 t 2 :t

>

O} a

-'I

d x ' 1t t +x

subset of LI(JR) and * the convolution product between LI(JR) and L2(JR).Ifwe put a suitable topology on C * L 2(JR) we can prove thatSH,A

=

C * L 2(JR) as locally convex topological vec-tor spaces. The map! i--7 !*. from L1(lR) into L(L2(lR» is a --representation ofthealgebra

L1(JR) by operators on the Hilbert space L 2(JR) and this map corresponds to the regular representation of JR in L2(JR). (See [HRI], Theorem 22.10.) So in fact the spaceSL

2

(1R) ,

"-1-

:::2

is detennined by the group JR, a subset C of LI(JR)and a representation of this group in a Hilbert space.

The aim of this report is to study the results in [EG2] from the viewpoint of commutative har-monic analysis and also to explore the possibilities for generalizations which naturally arise in that new context. We start with a locally compact Abelian group G, a subset C ofLI(G), a representationU of G in a (not necessarily separable) Hilbert spaceH and we suppose that the pair (C ,U)satisfies some technical conditions (pI, P2).

Inchapter I we define the locally convex topological vector space of smoothed emelementsSu,c' the trajectory space Tu,C and a duality between them. The spaceSu,c is an inductive limit of Hilbert space and T

u.

c is a projective limit of Hilbert spaces. Assuming a weak condition(P2 ')

we prove thatSuoc

=

Has sets if and onlyifSuoc

=

Has topological vector spaces.

Inchapter II we define a projective limit topology on the vector space Suoc and we consider a special type of bounded subsets ofTu.c. It turns out that the projective limit topology forSuoc is equal to the inductive limit topology forSu,c if and only if all bounded subsets of T

u.

c are of this special type. Further, we consider an additional condition on the set C (P3) and then we prove thatSuoc is a projective limit of Hilbert spaces.

Assuming the conditions PI, P2, P3, in chapter III we show that the following conditions are equivalent: the pair (C ,U) has property P4; Tu.c is bomological; Tu.c is reflexive; Su,c is complete; Suoc is sequentially complete; every bounded sequence inSuoc has a weakly conver-gent subsequence; all bounded subsets ofSuoc look as if the inductive limit were strict; and several other conditions. Finally we present necessary and sufficient conditions for Suoc and

T

u,

c to be nuclear or Montel.

Inchapter IV we include several (old and new) examples. We show that allSH,A' TH,A' t(H, A)

anda(H, A)spaces, whereHis a Hilbert space andAa positive self-adjoint operator inH,fit in the theory.

Inchapter V we consider continuous linear maps between spaces of type Suoc and Tu.c. The theorems are nearly similar to the theorems in [EG2], Section lA, but the proofs are (of course) different.

homeomorphic with the strong dual ofT

u,

c. Assuming property P3, it is shown thatT

u,

Cis

con-jugate linear homeomorphic with the strong dual ofSu,c if and onlyifa symmetry property (P4) holds.

Inchapter

vn

it is investigated whether Cartesian products of twoSu.c spaces (resp. twoTu.c spaces) are again of typeSu.c (Tu,C).Here property P5 plays an essential role.

Inchapter VIII we introduce explicit completions of a-topological tensor product ofSUhCI and

Tu₂,C₂' (Four possibilities.) It is shown that the completed a-topological tensor product of two

T

u,

c-spaces is again of type

T

u.

c. We formulate and prove four kernel theorems. The latter is a generalisation of the corresponding chapter (III) in[EG2].

Appendix A contains a summary of the concepts of commutative harmonic analysis as used in this report. Appendix B summarizes the prerequisites from topological vector space theory. In Appendix C we introduce a so-called a-topology on the tensor product of two locally convex topological vector spaces which have enough seminorms which correspond to semi inner pro-ducts. This a-topology corresponds to the Hilbert space tensor product of two Hilbert spaces. Inthe final part of this introduction we sketch the relation between the results of the underlying report and the papers[EGK]and[EK].These papers also expand the results of[EG2],be it in dis-tinct directions.

The set<I>of nonnegative functions on

R

Il_{in [EGK] can}

becompared more or less with the sets of complex valued bounded functions

j

on G in this report. An important difference is that <I> may contain unbounded functions. On the other hand, G need not necessarilybe equal to 1R1l

•

The condition A.I in[EGK]is superfluous and the (strong) condition A.III in[EGK](cf. property P3 in this report) is always assumed. By condition A.I1 in [EGK],for every

x

E JR." there exists ~E <I>such that~x),p 0, whereas in this report all functions

j,

with

f

E C, maybe0 on a large

subsetofG.

In[EK]the operators

a

E Rare all bounded as is the case with the operatorsU(j)in this report.

Further, the operatorsaE Rare required tobepositive Hermitian and bounded by 1. Also a

sub-semigroup property is required which is not assumed in[EGK]nor in this report. A condition like property P3 is considered at the end of the paper[EK].

In both papers [EGK] and [EK] no theorems are proved which are similar to the theorems in chapters 5-8 in this report.

Some notations

Let A be a set and let V be a subset of A. 1hen VC

is the complement of V in A, so VC

=

{aE A :a~ V}. I

v

is thecharacteristic function ofV,thus

Iv:A -+ JR

{

lifaEV'

lv(a)

=

(a E A)

Oifa~V.

Let

I

bea complex valued bounded function onA.Then II

111

00:=sup ( I

I

(a)I :a E A}. INis the

set of positive integers, JV=={1,2,3,···}.

Let Xbea topological space and letV bea subset of X. Then cIoV==Vdenotes theclosure ofV. By Cc(X)we denote the set ofallcomplex valued continuous functions Ion X with compact sup-port, Le. there exists a compact subsetK of X such that

I

(x)=0 forallx E KC•Further, Co(X)

denotes the Banach space of all complex valued continuous function

I

on X which vanish at infinity, Le. forall£

>

0 there exists a compact subsetKof X such that II(x)I

<

£forallxE KC•

The nonn onCo(X) is 111100 •ByC(X)we denote the set ofallcomplex valued continuous

func-tions onX.

The abbreviation a.e. meansalmost everywhere.

Let Gbea locally compact Abelian topological group and let

I,

g E L1(G).Then 11/111denotes

the norm of

I

and

1*

gdenotes the convolution product of

I

and g. The adjoint

j

of

f

is the function (equivalence class) with

j(x)

=

l(x-

1

), a.e.

x

E G. For every subset C ofL l(G)let

C

denote the set

{! :

lEe}.

j

is theFourier transform off.Thedualgroup of G is

G.

LetHbea Hilbert space. Theinner product inHis denoted by ( , ) andthe normby II II. LetTbe

a densely defined operator fromH into a Hilbert space. ThenD

<n

denotes thedomain ofTand T* denotes theadjoint operator. IncaseTis a bounded operator, we denote by IITII thenormof

Chapter I. The spaces SU, c and T

u.

c 1.1. The spaceSu,c

Let G be a locally compact Abelian topological group with a Haar measure~and letUbe a (con-tinuous) representation of G in a Hilbert spaceH.For definitions concerning topological groups the reader is referred to Appendix A. We suppose that every representation is continuous. For every

f

ELI(G)define a continuous operator onHby

(U(f)u,v)=

J

f(x)(UJ(;u,v)d~x) (u,vEH).

G

(See [HRI], Theorem 22.3.) The operators U (f),

f

ELI(G)have the following properties: II U (f)u II S II fill\IuII ,

U(f)*

=

uif>,

(f,

g

E LI(G), uE H)

U(f*g)

=

U(f)U(g)

=

U(g)U(f).

Let C be a subset ofLI(G).Suppose the pair(C, U)possesses the following properties. PROPERTIES1.1.

PI. For all

f,

gEethere exists h E C such that 1 and 2 hold: 1) f=hor there existsfl E LI(G)such thatf= h

*

ft,

2) g

=

hor there existsgl E LI(G)such thatg

=

h

*

gl.

n.

There exist a net (f).kJ in C such that for all x E Hholds lim U (f'A)x

=

x. 'A

Throughout this report we suppose that the conditions PI and

n

are satisfied by the pair(C, U).

REMARK. These conditions are weak, but sufficient for the purpose of this chapter. In practice one meets sets C which satisfy much stronger properties such as:

PI'. For all

f,

gEethere existhE C and

It ,

gI ELI(G)such that

f

=

h

*

It

andg

=

h

*

gI.

n'.

There existsaLI(G)-bounded net (f'AkJ in C such that for all x E Hholds lim U(f'A)x=X.

'A

n".

There exists a LI(G)-bounded net (f'A)'A.eJ in C such that for all g E LI(G) holds limA

*

g

=

ginLI(G).

).

Of course: PI' implies PI and

n'

implies

n.

Also

nit

implies

n'.

Indeed, let

E:= {xE H :lim U(f'A)x =x} andF :={U(f)x :fE LI(G), x E H} Then E is closed in Hand

'A

FeE.SinceUis continuous,F is dense inH.SoE

=

H.

REMARK. For all xE Hwe have limU(f'A)*X

=

x. Indeed, U(f'A) -Iis a nonna! operator, so 'A

II (U(f'A)-/)* xII =11 (U(f).)-/)xll for all I.E J. Hence, lim II U(f'A)* x-xll=

lim II(U(f>..) -/)*xII=limII(U(f~)-/)xII

=

O.

~ ~

REMARK. Let

C

:=

if :

I

e C}. Then the pair(C, U) satisfies propertiesPI and

n.

The pair

(C, U)satisfies propertiesPI',

n'

and

n"

if and only if the pair(C, U)satisfies propertiesPI',

n'

resp.

n".

DEFINITION 1.2. Let/e L1(G).Define

Nf := {x e H : U(f)x =O} , the kernel ofU(f), Rf:= U(f)(H). the range ofU(f),

Of:=U(f) I

Ny :

Ny-+ Rf .

Note thatNf= NjforallI e L1(G).

LEMMA 1.3.

Let Ie L1(G). Then Of is a bijection. So there exists a unique norm II II_{f on} Rf such thatRf

becomes a Hilbert space and Of is a unitary map. The identity map fromR_f intoHis continuous. Proof.

Of is injective. Let x e H. Since U(f) is continuous. the set Nfis closed, hence there exist v e Nf and we Ny such thatx

=

v+ w. Then U (f)x

=

U (f)w

=

Ofw. So Of is swjective. Let cp e _{Rf .}ThenIIcpll~ II U(f)1I11Oi(cp)II~11/11₁11cpllf'so the identity map from_Rf intoH is

continu-ous.

0

Inthe next lemma the spacesRfandRgare compared ifI

=

g

*

hfor someh e L I(G).

LEMMA 1.4.

Let I, g. he L1(G). Suppose I=g

*

h. Then _{Ng cNf} and RfcRg. For all cpe _Rf holds O;I(cp)

=U(h)(Oi(cp».

The identity map from_Rf into_Rgis continuous.

Proof.

Letx eNg. Then U(f)x=U(h)U(g)x=0, sox e _{Nf .} Similarly: RfcRg. Let cp e Rf and let

x:=Of\Cp). Assertion: U(h)xeN{ Let _veNg. Then U(g)U(h)v=U(h)U(g)v=O, so

U (h)ve NgC Nf and 0

=

(x, U (h)v)

=

(U (h)x, v). This proves the assertion. Since Og(U (h)x)

=

U(g) U(h)x

=

U(f)x

=

cp we have proved that O;I(cp)

=

U(h)Oi(cp).Hence, forall cp e _Rf we obtainIIcpllg=11O;I(cp)lI~ IIU(h)II IIOi(cp)lI~ IIh11₁11cpllf.

0

DEFINITION 1.5.

LetSu c:= u R_{f ·}

• feC

By Lemma 1.4 and propertyPI,Su,Cis a linear vector space. The topologyO'ind forSu,Cis the inductive limit topology generated by the Hilbert spacesRf' Le. the finest locally convex topol-ogy forSuoCfor which all natural maps fromRfintoSu,Care continuous(fE C).

For the tenninology of locally convex topological vector spaces we refer to Appendix B.

LEMMA 1.6.

The identity map fromSuoc intoH is continuous. Hence the topologyO'ind forSuoc is Hausdorff. The setSu,Cis dense inH.Let IE C. Then the mapU (f) : H ~Su,c is continuous.

Proof.

The first assertion follows from Lemma 1.3 and the definition ofO'ind'Since H is Hausdorff, also

(Su C,O'ind)is Hausdorff. By property

n,

limU(f,.)x

=

xfor allx EH,soSu c is dense inH.Let

, A '

IE C and letPbe the projection ofHontoNJ.ThenU (f)

=Or

P : H ~ Su,c is continuous.

0

THEOREM 1.7.

The space(Su.C,O'ind)is bomological, barrelled and a Mackey space.

Proof. O'indis the inductive limit topology ofbomological and barrelled spaces, hence(Su,C,O'ind) is bomological and barrelled ([Wil], Theorem 13-1-13). Every homological space is a Mackey space. (See [Will, Theorem 8-4-9.)

1.2. The spaceT

u,

C

Besides the spaceSuoc we introduce the spaceTu,Cin this section. DEFINITION 1.8.

A C-trajectoryis a map eI> from C intoH with the property that

eI>(f*g)=[U(g)]* eI>(f) for alllEe, gEL1(G)such that

I

*gEe.

LetT

u,

C be the vector space of all C-trajectories. The Hausdorff topology'tproj forT

u,

c is the locally convex topology for Tu,C generated by the seminonns tl' defined by tleI» := II eI>(f) II, eI>E Tu.c,/E C.

Itfollows that for all

I

E C, the map eI>~ eI>(f) fromTu, cintoHis continuous.

LEMMA 1.9.

o

Proof.

First, suppose there exists 1ELI(G)such that

f

=

g

*

1. ThenU (g)* «'P(j)=

U (g)* U(1)* <l>(g)

=

U (j)*<I>(g). Ingeneral, by property PI, there existshE C such that 1) and

2)holds:

1) 1=hor there exists

h

ELI (G)such that

f

= h

*

it,

2) g=hor there existsg1ELI (G)such thatg=h

*

g1.

If1= horg=h,then the lemma is trivial or already proved. Suppose

I

*'

handg

*'

h.Let11and

g1as in 1)and2).ThenU (j)* «'P(g)= U (h)* U (jl)* U (g1)*<I>(h) =

U(h)* U(gl)* U(jt>* <I>(h)=U(g)* <I>(j).

0

The Hilben spaceHcan be embedded inT

u.

Cin a natural manner.

DEFINITION 1.10. Define

emb:H~Tu.c

[emb(x)](f)

=

U(f)* x (x E H, I E C).

THEOREM 1.11.

The map emb is injective and continuous. For all <I>E T

_u

Cholds <I>=lim emb(<I>(f),» inT

_{u c.}

. A. .

So the rangeemb(H)of the map emb is dense inT

u. c.

Proof.

LetxE Handembx=O.By propenyP2, then alsox=limU (f).,)* x=lim [emb(x)](f),)=O. So

A. A.

emb is injective.Itis trivial that emb is continuous. Let<I>

E

T

u.

Cand let

lEe.

Then by Lemma 1.9, limtlemb<l>(fA.)-«'P)=limIIU(f)* «'P(fA.)-<I>(f) II=limIIU(f),)* <I>(f)-<I>(j) II=O. So

A. A. A.

<I>=lim emb(<<'P(jA.» andemb(H)is dense inTu c.

A. .

COROLLARY 1.12.

I. Forall«'PE Tu. c,fE C andgEL I(G)such thatf

*

g=0 holdsU(g)*<I>(f)=O.

II. Forall<I>E Tu. c andIE C holds«'P(j)E

Nt.

Proof.

I. Let IE C, g E L1(G) and suppose1* g=0. Let E:= {«'PETu,c: U(g)* «'P(f)=0}. By definition of'tproj' the set E is closed in Tu.

c.

It is trivial that E contains embeR). So E=Tu. c .

II. Letx E H andlEe. Forall vE Nf , ([emb(x)] (f), v) = (U(f)* x,v) = (x, U(f)v)= O. So [emb(x)] (f)E

Ny.

Since

Ny

is closed andemb(H)is dense inTv.c. the corollary follows. I]

THEOREM 1.13.

The space(T

_u,

c,'tproj) is complete.

Proof.

Let«(f)a)aeA be a Cauchy net inTv,c. ForalllEe, «(f)a(f»aeA is a Cauchy net inH, so there exists (f)f E H such that lim(f)a(f)=(f)f. Define (f) :C ~H by CfJ(f):=(f)f' lEe. LetlEe,

a

gE L 1(G)and suppose

I

*

gEe. Then(f)(f*g)= lim(f)a(f*g)=limU(g)*(f)a(f)=

a a

U(g)* (lim(f)a(f» =U(g)*~). So (f) E Tv,c. Since limtt<(f)-(f)a) =0 for all lEe,

a a

o

The next lemma turns out to be useful. (Cf. Theorem 1.18.)

LEMMA 1.14.

Let

I,

gEe,hE L 1(G)and suppose that/=g

*

h.Thentf~II h 111 tg.Hence, letBe Tv,c be a 'tprorneighbourhood ofO. Then there exist

I

E C andc

>

0 such that {(f)E Tv.c:tt<CfJ)~c} c B.

Proof.

For all (f)E Tv,c we get tt<(f)=IIU(h)*(fJ(g)II~IIU(h)III1(fJ(g)II~lIhIl1tg«fJ).From this and

property PI, the second assertion follows.

0

1.3. The pairing betweenSv,CandTv.C

In this section we define a duality

< ,

>

betweenSu,C andTu,C with the properties thatGind and 'tproj are compatible with

< ,

>

and

<

cp,embx

>

=(ell,x) for allellESu,C andx EH. It turns out that<U (f)x,(f)

>

=(x,(f)(f» foralllEe,XEHand(f) E Tv c. (See Theorem 1.16 II.)

LetellESu,c and(f) E Tu,c. There existslEesuch thatellE _{Rf .}We show that it makes sense to define<cp,(f)

>

:=(Oi(cp),(f)(f».LetgEeand suppose alsoellERg.Then we havetoprove that

(Oi(eIl),(f)(f»=(Q

g

1(eIl),(f)(g». First suppose there exists 1E L1(G) such that

I=g

* I. By Lemma 1.4:

In the general case there existsh E C such that 1) and 2) hold: 1)

1=

hor there exists

II

ELI(G)such that

I

=

h

*

II,

2) g

=

hor there existsgI ELI(G)such thatg

=

h

*

gI.

In any case:(Qi(ell), <1>(f))

=

(Q

_h

l(ell),<1>(h))

=

(Q;I(eIl),<1>(g )). DEFINITION 1.15.

Define

< ,

> :

Su, exTu,C -7C

The pairing

< , >

is sesquilinear.

THEOREM1.16.

I. LetellE Suoc andxE H.Then

<

eIl,embx

>

=

(ell,x).

II. Let

I

E C,x E Hand <1>E Tu, c.Then

<

U (f)x,<1>

>

=

(x,<1>(f)). Proof.

I. Let

lEe

be such that ellE Rf . Then

<

<I>,embx>= (Qi (<I>),(embx)(f)) =

(Qi(<I», U(f)* x)

=

(U(f) Qi(<I», x)

= (<1>,

x).

II. LetPbe the projection ofH onto

Nj.

ThenQ"i (U (f)x) =Px. Hence by Corollary 1.12 II:

<U (f)x,<I>

>

=

(Qjl(U (f)x),<1>(f))

=

(Px, <1>(f))

=

(x, P<I>(f))=(x,<fJ(f)).

0

LEMMA 1.17.

The pairing

< , >

is nondegenerate. Proof.

Let ellE Su,e and suppose

<

eIl,<1>

>

= 0

for all <1>E Tu,

c.

Then

111\111

2

=

(<1>,1\1)

=

<

eIl,emb

1\1

>

=o.

So

1\1=0.

Let <1>E TU,e and suppose

<1\1,<1»

=

0

for allellE Su, c. Let

lEe.

For all xE H we obtain by Theorem 1.16 11,0

=

<

U(f)x,<1>

>

=

(x,<1>(f)). So <1>(f)

=O.

0

THEOREM1.18.

The topology(jindforSu,c is compatible with the dual pair

<

Su,c'Tu,c

>

and the topology'tproj

forTu,c is compatible with the dual pair

<

Tu,c'Su,c

>.

Proof.

For

lEe

letif :

R

_f -7Su, c bethe natural map.

Let <1>E T

u, c.

Define

[(cjl):=<cjl,~> (cjl e Su.d.

Let / e C. For all cjl e Rf: 110 ilcjl)I= I< cjl,<1> > I= I(ni(cjl),<1>(f))IS;"<1>(f)1111 n/\cjl) 11=

II~(f)1I11cjlllf. so [0 if:R_f ~C is continuous. Hence [ : Suo e ~C is continuous. (See [Will,

Theorem 13-1-8.)

Conversely, let [ :Suo e ~C be continuous and linear. Let / e C. The map [0 U (f) : H ~ C is

continuous and linear by Lemma 1.6. So (Riesz) there exists a unique <1>(f)e H such that

[ 0 U (f)x=(X,~(f)) for allx e H. We prove that~is a C-trajectory. Letg e C,h eLl(G) and

suppose /:=g

*

h e C. Then for all xe H we have (x, U(h)* <1>(g))= (U(h)x,<1>(g))=

[0 U(g)U(h)x= [0 U(f)x= (x,<l>(f)). We conclude that <1>(f)=U(h)*~(g)and ~e Tu e.

Finally, let cjl e Suo e. Let/ e C be such that cjl e R_{f .}Then [(cjl) = (njl(<I»,<1>(f)) = <cjl,<1». So the topology(JindforSu,eis compatible with the dual pair <Suoe,Tu.e>.

Let cjl e Suo e. Define

[: Tu.e ~C

[(~):= <<1>,<1» (<1> e T

_u.

d.

The map is [ is linear. There exists /e C such that cjlE R_{f .} Then 11(<1»1 = I <cjl,<1>

>

I= I(ni(<I»,<1>(f))IS;II ni(cjl) IIt/~) for all ~ E T_u.e. So1is continuous.

Conversely, let1 : Tu.e ~ Cbe continuous and linear. Since 1is continuous, by Lemma 1.14, there exist/ E C and c

>

0 such that Il(~)IS;ctl<1» for all<1>E Tu.e. Define ex:U(j)(H)~ C by ex := [0 emb0 nt. The map ex is linear. Let the topology forU (j)(H) be the induced

topol-ogy ofH. Letxe U (j)(H). Then I a(x) IS; cII[emb(ni(x))](f) II = c IIxII. So ex is continuous. By Riesz' theorem there exists a unique ve U (j)(H)=Nt such that a(x)=(x, v) for all

xe U(j)(H).

.l

-We show that [(<1» = <U (f)v,~> for all ~e Tu.e. Let xEN

i'

Then [(embx) =ex(U (f)x) =

(U(f)*x,v)= (v,[embx](f))= <U(f)v,embx>. Let xENj. Then Il(embx)/$

ctlembx)=O, so 1(embx) =0= (U(j)x,v)= <U(f)v,embx>. So for all xEH we obtain

1(embx) =

<

U (f)v,embx >. Because [ and ~ ~ <U (f)v,<1> > are continuous functions on

Tu.e andemb(H)is dense inTu•e,forall~e Tu.eholds [(<1» = <U(f)v,<1».

0

InLemma1.6 it is proved that the setSuo e is dense in the Hilbert spaceH. Itmay happen that

SUoeequalsH as a set In a particular case, it can be proved that (Su.e,(Jind),as a topological vec-tor space, is equal to the Hilbert spaceH.

THEOREM 1.19.

SupposeSuoe=H as sets and suppose property P2' holds. Then (Su,e,(Jind)=H as topological vector spaces.

Proof.

The identity map from Su, e into H is continuous by Lemma 1.6, so we have to prove that the

let> : H ---+C

let>(x):=<x, <I>> (xE H) .

Let (/;..heJbe the net as in propertyP2'and letM

>

0be such thatII

1..

111<M for allAE J. LetxE H. Then lemb(x)(Y)=<y,embx>=(Y, x) for ally E H. So lemb(x) is continuous(xE H).

Let <I»E T

u.

c. Assertion: the set {lemb(et>(f,» :AE J} is pointwise bounded. Indeed, let

x

E H.

Since H=Su,

c

as sets. there exist

1

E C and y E NJ such that

x

=

U (j)y. Then for all AE J: llemb(et>(f,»(x)1 = I<x,emb<l>(j..)> I = l(y, [emb<l>(j..)](j»\ = l(y, U(j)* <I»(j')..) I= l(y, U(j..)*<I»(f»1 S lIyII IIU(f..)II II<I»(f)IIS lIyII IIhlll II<I»(j)IIS Mllyllll<l>(j)11. This proves

the assertion. By the uniform boundedness theorem ([Wil] , Theorem 3-3-6), the set {lembet>(f,) :AE J} is uniformly bounded. LetM0 be an upper bound for this set. Then for all

x E H : Ilet>(x)I = I <x,<I»

>

I=lim I<x,emb<I>(j..)

>

I=lim IlembdV')(x) ISM0IIxII. Solet>is

..

..""'V'

continuous for all <I»E Tu,

c.

Leta.E Su,c'. There exists <I>E Tu,c such thata(l\I)= <1\1,<I>

>

for all1\1E Suoc. Thena0 i =let>

is continuous. Since a Hilbert space is bomological, the mapi :H---+Suo c is continuous.

0

We fInish this chapter with a lemma which will be used in Chapter 8, but belongs here. LEMMA 1.20.

Let

1

E C,gE L1(G) and suppose

1

*

g =0. Then U(g)*

x

=0 for all

x

E Nj.

Proof.

Let E:= (xE H: U(g)* x=O}. Then E is closed in H. Let YE H. Then

U(g)* U(j)* Y = U(f* g)* Y =0, so U(j)* YE E. Hence U(j)* (H) cEo By [Wei], Theorem

Chapter II. Seminorms onSu,cand bounded sets inTU,c II.I. TheStone-representant for a unitary representation

LetUbe a representation of a locally compact Abelian group G in a Hilbert spaceH. The group G and the representationUare taken fixed throughout this chapter.

DEFINITION 2.1.

LetAbe a locally compact topological Hausdorff space,

m

a measure onA, / an index set, for all iE/letAj be an open subset ofA with induced topology and let tj :

G

~Aj be a topological

homeomorphism. LetWbe a unitary operator fromHontoL2(m).

The tuple(A,m, /,Aj,tj,W)willbe called aStone-representantforUif and only if:

Aj ( )A_j =0 ifi

*

j (i,j E I),

A=UAj, jel

The mapY ~ m(tj(Y», Ya Borel measurable subset of

G,

is a finite regular measure on

G

(i E I),

o

<

m(Ai)

<

00 (i E I),

m

(Z)=

L

m

(Z () Aj) (Z

c

ABorel measurable),

iel

For all

!

ELI(G)let

i

be the continuous function onA such that f(ti(Y» =](y)for all iE/ andallYE

G.

Then

W U(f)W-1~ =i.~forall!E L1(G) and~EL2(m).

This definition is useless without the next theorem. THEOREM 2.2.

There exists a Stone-representant forU.

Proof.

See[HRII], Remark 33.6and[HRI], Theorem C.37. REMARK. The Stone-representant is not unique.

o

REMARK.If,in addition, the Hilbert spaceH is separable, then the measuremis regular.

Proof. Fori E /letei:= (m(Ai)rt IA;'Then(ei)iel is an orthonormal set inL2(m),soI is count-able. We may suppose that the set / is infinite, and hence suppose that / = IN.

Every compact subsetKofAis contained in only finite many open subsetsAj ,hence

m

(K)

<

00.

Let Z be a Borel measurable set,m(Z)

<

00and let e

>

O. For alln E INthere exist openU,.

c

G

such that m(t,.(U,.»< m(Z()A,.)+

er"

and U,.:::>t;;l(Z()A,.), by the regularity of y ~ m(t,.(Y». Let U:= v t,.(U,.). Then V is open inA,Z c U and m(U)

=

L

m(t,.(U,.»'5.

00

L

m(ZnAII)+£rll=m(Z)+£.

11=1

Som(Z)

=

inf{m(U) : Uopen, Zc U}.

Let U be an open subset ofA. Suppose

m

(U)

<

00. Let £

>

O. There exists N E IN such that

N

m(U)~

L

m(U nAil)+

~.

Forall

n

E IN,

n

~

Nthere exists compactK n

c

t;;I(U

n

An)such

n=1

2

N

thatm(U

nAII)~

_m(tn(KII

»

+~.

LetK := u _tn(Kn). Since eachAll is open inA,the setK is

2n

11=1 compact inA, K

c

Uand N m(U)~

L

m(tll(Kn»+£=m(K)+£. 11=1 Hence,

m(U)=sup{m(K): K compact,K

c

U} .

Similar, if

m

(U)

=

00,

m(U)

=

sup(m(K) : K compact,K

c

U} .

Somis a regular measure onA.

REMARK. The measure

m

is locally finite, Le. for every measurable set Z

c

A with

m

(Z)

>

0, there exists a measurable Z1 C Z such that 0

<

m

(Z1)

<

00.

DEFINITION 2.3.

Let BorCG,C) be the set of all complex valued Borel measurable functions on G and let Borb(G,C)be the subset of all bounded elements of Bor(O,C). Define similarly Bor(O, JR.) and

Borb

(0

,JR.).

With the aid of the Stone-representant for U we can extend the set operators U(j) with

fE LtcG) to a set operatorsU[F] withF E Borb(G,C).

DEFINITION 2.4.

Let(A,

m,

I, Aj,'tj, W)bea Stone-representant forUand letFbea Borel measurable function on

O.

Define the Borel measurable function F onAby F(tj(Y» :=F(y) (i EI, yEO) .

For everyFE Borb(O,l')define a continuous operatorU [F, A]onHbyU [F, A]:=W-1MFW,

withMF the multiplication operator by F onL2(m).

The definition of F depends onthechoice ofA,but from the context it will follow on which setA

REMARK. It is clear that forallF, K E Borb(G,C),allAE Candall!E L1(G) U [F

+

K, A]

=

U [F, A]

+

U [K, A] , U [AF, A]

=

AU[F, A] , U[FK, A]

=

U[F, A]0 U [K, A], U[/1,A ]

=

U [F, A]* , U

ri,

A ]

=

U (f) ,

for every Stone-representant (A, M, I, Aj ,'tj,W) for U.

LEMMA 2.5.

Let(A,

m,

I,Aj,'tj,W)be a Stone-representant forU. I. LetVbean open subset of

G.

Let

X:=clo{U[F,A](H):FE Co(G),O~F~ Iv}. ThenU[1

v,

A ]is the projection ofHontoX.

II. Let F1,F2"" be a uniformly bounded sequence in Borb(G,C) and suppose

F('Y) := lim FII('Y)exists for every'YE

G.

ThenU [F, A] =s - lim U [FII,A].

11-+"0 11-+"0

Proof.

I. LetXo :=U[Iv,A](H). ThenXo is closed inH and X cXo. Let!E X oand let~:=WI

Then Iv'~=~ in L2(m). We have to prove that ~E clo{F1'\:FE Co(G), O~F~Iv. 1'\E L2(m)}=W-1(X).We may as well assume that~is real valued and~ ~ O.Moreover, by Lebesgue's dominated convergence theorem, we may also assume that there exista,~

>

0

such that~=~·Iz with Z={a E A:a~ ~(a)~~} and ZrlAj c'tj(V)for all iE I. Then

m

(Z)

<

00. Let E

>

O. There exist

n

E IV and i_{1 , ' •• ,}illE I such that

m (Z)

<

±

m(ZrlAj})

+

~.

Sincem0 'tj is a regular measure on

G,

there exists compact

j=1 2~

A l E

K cG such that K c'ti} (ZflAj.) and m(Z rlAj.)-m('tj,(K»

<

- - 2 for all

J J J 2n~

j E {I, ., .,n}. Then K C V. By Urysohn's lemma there exists FE Cc(G) such that

IK~F~ Iv.

II

Let K1:=U 'tj(K). Then K1 C Z and m(Z) - m(K1)

<

E~-2. Let 1'\:= IXt .~. Then

j=1 J

II~-F1'\1I2 = II ~-F IKt ~II = IIIz~-IKt ~1I2 ~ ~2I1Iz-IK, 112~ ~2. E~-2 = E. SOXo cX. II. This follows from Lebesgue's dominated convergence theorem.

o

THEOREM 2.6.

Then_{U[F, AI] =U [F, A 2]}forallF E BorbG,C).

Proof. Let fE LI(G). Then _{uli,Ad=u(f)=uli,A 2].} Since {j:fE LI(G)} is dense in

Co«h

we obtainU [F, AI]

=

_{U [F, A 2]}foraUF E Co(G)by Lemma 2.5 II. LetV

c

Gbe open.

Then we have seen that {U [F, Ad(H): FE Co(G),O~ F~ Iv}

={U [F,A 2](H): FE Co«h

O~ F~ Iv}. Hence by Lemma 2.5 I,U[lv,

Ad

=

U[lv,A2 ].

LetB :={V : V

c

G

Borel measurable and U[lv,A

d

=U[lv,A2]}' ThenB contains all open subsets of G andBis a a-algebra by Lemma 2.5 II, soBcontains all Borel measurable subsets of G. Using Lemma 2.5 II again, we obtainU [F,

Ad

=

U [F, A2]for allFE Borb(G,C). [) We naturally arrive at the following definition.

DEFINITION 2.7.

Let FE Borb(G,C). Define U[F]:=U[F,A] where (A,m,I,Aj,tj,W) is any Stone-representant forU.

We summarize some properties ofU [. ].

PROPOSITION 2.8.

LetF, K E Borb(G,C), AE C andfE LI(G).Then IIU [F]II~\I F1100 , U [F+K]=U [F]+U[K], U [AF]

=AU

[F] , U[FK]=U[F]o U[K], U [F]=U [F]* , U

I1J

=

U(f), U[1a]=I. LEMMA 2.9.

LetF E Bor(G,C) and let(A,

m,

I,Aj,tj,W) be a Stone-representant for U. For all

n

E IN let

Vn :={YE

G:

n-l~ IF(y)1

<n}.

Then the orthogonal sum

L

U[F.lvJ= W-IMFW,

nelN

whereMF denotes the (unbounded) multiplication operator by F onL2(m).

Proof.

In

the following definition we extend

U [.]

to Bor(G,

C).

This definition makes sense by Lemma

2.9.

DEFINITION 2.10.

Let

F E

Bor(G,

C).

Define a closed, densely defined operator

U [F]

on

H

by

U [F]

:=

W-1MFW

where

(A,

m, /,

Aj,'tj,W)

is any Stone-representant for

U

and

MF

denotes the multiplication

operator by F on

L2(m).

REMARK.

U [F]

is a self-adjoint operator on

H

for every

F e

Bor(G,

JR).

11.2.

Seminonns on

Su.c

and bounded sets in

Tu.c

Let C

be

a fixed subset of

L1(G)

which satisfies properties

PI

and

n.

Corresponding to the set C

we define

three

subsets of Bor(G,C) and seminonns on

Su.c.

DEFINmONS 2.11.

Let

C*

:={FE

Bor(G,C): for

allfE

Cis

F·

jbounded} ,

C::= {F EC*:

for

all'YE

G

isF('Y)il:0} ,

C:

:={FE

C* : there exists

£

>

0

such that for

all'Y E

G holds

IF('Y)I

>

E} .

For

allF E

C* define

SF :Su.c-? JR

REMARK. Since}=! for

allfE

C, it is obvious that (6)*

=

C* , (6): = C: and (6); =

C;.

We

-*

-*

-*

- *

-

1# -

*

write C

,C"

and C

p

for

(C) ,(C)u'

resp.

(C)p.

REMARK. C* is a vector space.

Borb(G,C)

c

C'.

For

all FE CI#

and

K E Borb(G,C),

also

FK

E

C*

andF

E C*.

LEMMA 2.12.

I.

Let

FE el#.

Then

SF

is O'ind-continuous on

Su,c.

I. Let f E C. For all C/lE Rf holds SF(C/l)=11U [F]U(f)Oi(C/l)1I= IIU [Fh0i(C/l)IIS

IIU[Fhllil0i(C/l)IIS IIFjII00IIC/lllf.By [Will, Theorem 13-1-8,SFis <rind-continuous.

II. Trivial because16 E

e;.

0

DEFINITION 2.13.

Let <rpmj be the locally convex Hausdorff topology forSu,c generated by the seminormsSFwith

FE

e;.

COROLLARY 2.14.

The following inclusion holds for the topologies forSu,c:

Proof.

Lemma2.12I.

We introduce a specialtypeof elements ofTu.c. DEFINITION 2.15.

LetFE eNandxE H.DefineU[F].x:

e

~H

(U[F].x)(f):=U[F]U(f)*x (fE e).

LEMMA2.16.

LetF E eNandxE H.Then:

I. U [F].

x

isa

e

-trajectory.

II. IfxED (U[F]),thenU[F].x

=

emb(U[F]x).

III. LetC/lE Su.c.Then

<

C/l, U [F].x

>

= (U [F]C/l ,x).

IV. LetKE Borb(G,C).ThenU[F]. U[K]x=U[FK].x.

V. SupposeF E e~.ThenU [F].x=0 if and only ifx=O.

VI. The mapU [F] •fromH intoTu,cis continuous.

o

Proof.

I. LetfEe, gEL1(G)and supposef*g E C.Then

(U[F].x) (f*g)= U[F] U(f*g)* x=U[F f {]x=U[{] U[Ff]x=U(g)* (U[F].x)(f).

II. Supposexe D(U[FJ).Forallfe Cholds

(U[F].x)(f)

=

U [F] U(f)* x

=

U(f)* U[F]x

=

(emb(U[F]x» (f).

III. Letfe C and

ell

e Rf.Then

<

ell,

U[F].x

>

=(ni (ell),

U[F] U(f)* x)

=(ni(ell)'

U[F !]x)

=

(U(F

h

0:iCell) ,

x)

=

CU[F]

ell,

x). IV. Forall

f

e C we have

(U[F]. U[K]x) (f)= U[F] U(f)* U[K]x=U[FK] U(f)*x=(U[FK].x) (f).

V. Let(f',,»),eJ be the net in propertyPZ. SupposeU[F].x =0.

Then U[F](U(f~)*x)=(U[F].x)(f~)=O,so U(f~)*x=O for all AeJ. Hence

x

=

limU(f~)*x =0.

VI. Let

f

e C. Forall xe Hwe obtain

-

-tf(U[F].x)

=

II U[F!]xliS;II F!1100II xII.SoU[F].is continuous.

DEFINITION2.17.

LetF e C~. By Lemma 2.16 V there can be defined a unique nonn _{II IIF}onU[F].H such that

U [F]. Hbecomes a Hilbert space andU [F].is a unitary map fromHontoU [F] • H.

LEMMA 2.18.

I. The identity map fromU[F] .Hinto (Tu,c,'tproj)is continuous forall F e C~.

II. LetF, K e C~. Then IF I

+

IK I e C:, U[F].H

c

U[ IF I

+

IK f ].H and the

iden-tity map fromU [F] • HintoU [I F I

+

I K I ] • H is continuous.

Proof.

1. This follows from Lemma 2.16 VIandDefinition 2.17.

II. It is clear that I F I

+

I K t e

C~.

Define L e Borb(6, C) by L:

=

I F I

~

I K I . Then ( I F I

+

I K I ). L

=

F.Letxe H.Then by Lemma 2.16. IV,

U [F]. x

=

U [ 1F I

+

I K I ] • U [L ]xE U [ I F I

+

I K I ]. Hand

IIU[F].xll'FI+IKI =11 U[L]xllS; IIxli=U U[F].xIlF.

IJ

DEFINITION 2.19.

LetT:

=

u U[F].H.

Fee:

By Lemma 2.18 II,T is a linear vector space. The topology'tindforT is the inductive limit topol-ogy generated by the Hilbert spacesU [F] • H , F e C~.

every 'tprorbounded subset ofTu,c is contained in a set of the formU [F] • BwithFEe; andB a bounded subset ofH.Then, in particular,T

=

Tu,c as sets and 'tind is a locally convex topology for the vector space TU,C'

LEMMA 2.20.

Tis a subset ofTu,c.The identity map from (T,'tind) into(Tu,c ,'tproj) is continuous. Hence the topology 'tind forTis Hausdorff.

Proof.

From Lemma 2.16I it follows thatT

c

Tu,c and by Lemma 2.18I and[WillTheorem 13-1-8, the identity map from (T, 'tind) into (Tu,c ,'tproj) is continuous. Since'tproj is Hausdorff, also 'tind

is a Hausdorff topology.

0

The main result of this section is the following theorem.

THEOREM2.21.

The following conditions are equivalent.

1. (Su,c,O'ind)=(Su,c ,O'proj) as topological vector spaces.

II. For every 'tproj -bounded subsetBofTu,cthere existFEe;and a bounded subsetB0ofH such thatB

=

_{U[F].B o.}

Proof.

I =:> II. LetBbea bounded subset ofTu,c, LetBO bethe polar ofB.By[WillTheorem 8-4-12,

Ba is an absorbing subset ofSu,c,which is also closed and absolutely convex. SoBO is a barrel inSu,c and since(Su,c, O'ind) is barrelled,BO is a O'ind-neighbourhood of 0 in Su,c, Because O'ind=O'proj, there exist FEe; and £

>

0 such that

{ell

E Su,c: SF(eIl)$ e} cBO.

Let <I>EB. For all

ell

ESu,c with IIU[FlcIlll$

e

holds I

<

ell,

<I>

>

I$ I, so

1

I

< ell,

<I>

>

I=:; -II_£ U[F]eIlllforall

ell

ESu_'

c.

Define1:U[F](Sud_, ~Cby

I(U[F]eIl):=

<

ell ,

<1» (eIlE SU,d. Let the topology for U[F](Su,d be the induced topology ofH.Then the map1is continuous and linear. By Riesz' theorem there exists XII>E U[F](Su,dcH such that l(x)=(x,xII» for all XE U[F](Su,d. Then

\I XII>II

=

11111=:;

e-

1•Forall

ell

ESu,cwe obtain

-

-<

ell,

U[F]. XII>

>

=(U[F]

ell,

XII» =I(U[F]

ell)

=

<

ell,

<I>

>,

so U[F]. XII>

=

CI» by Lemma 1.17.

II

=>

I. Let V be a O'ind-neighboumood of 0 inSu.c. Since O'ind is a regular locally convex topology forSu.c there exists a closed absolutely convex O'ind-neighboumoodW of0 such that W

c

V. Then W

=

WO

0 by [Will Theorem 8-3-8, hence (W

O

r

is absorbing. So «(Wil] Theorem 8-4-12) WO

is 'tprorbounded in T

_{u.c .}

By assumption there exist FE

C:

and a bounded subsetB

_o

ofHsuch thatWO

=

U(F]. B

_o.

Then(U[F]. Bor

=

Woo

=

We V. Let M

>

0 be such thatIIxliSM for all

x

E B o.

Let~E Su,cand supposeIIU(F]~IIS M-I•Then for allx E B_o:

-

-I<~,UfF].x> I

=

I(U[F]c1l,x)I S IIU[Fl,UUxllS 1,soc1lE: (U[F].Bor c V.

Then{~E:Su.c : si(')SM-IleVandVisaO'proj neighbourhood. 11.3. A

growth

condition

Let C be a subset ofLI(G)and suppose the pair (C ,U)satisfies properties PI and

n.

Inthis sec-tion we consider an addisec-tional condisec-tion on the set C. We then prove that every 'tprorbounded subsetBofTU,cis equal toU [F] • B0for someF E C; and some bounded subsetB0ofH.

Suppose the set C has the following property.

PROPERTY 2.22.

P3. There exist a sequence of Borel measurable disjoint setsQ1 ,Q2, . . . in

G

and a sequence of positive real numbers b_{l ,} b2 , •.• such that

G

=

~

Qn and

i

b;/ <00 and for all

n=1 11=1

f

E C there existgE Cand

a

>

0 suchthatfor all n E INholds: b_{n sup}{I }(y) I :yE Qnl:S;ainf{Ig(y)I : yE Qnl.

This property is inspired by (EGK] Definition 1.1, A IIIandimproved by F.J.L. Martens. REMARK. Property P3 is equivalent with the following property.

P3'. There exist a sequence of Borel measurable disjoint setsQI ,Q2, ... in

G,

a sequence of

00 00

positive real numbersb1 ,b_{2 , •..} and v

>

0 such that

G

=

u Qn and

1:

b;v

<

00and for

,.=1 n=1

all

f

E C there existgE Cand

a

>

0 such that forallnE INholds:

Of course, P3 implies P3'. Suppose P3' holds. Then, inductively, for allkE INwe obtain: for all

fEe

there existg E C and 3

>

0 such that for all nE N holds:

b~sup{ I}(y) I :YE QnlS3inf{ 19(y) I :yE Qnl. I

Takek

>-.

v

REMARK.Incase the set C is supposed to satisfy property P3 then we write

Q

1 ,

Q

2, . , . and b1 ,b_{2 , . , .} forthesequences as in property P3 without explicity defining them again, if no con-fusion can arise.

REMARK. The set C satisfies property P3 if and onlyifthe set

C

satisfies property P3.

We start with some useful lemmas. Let (A ,m ,I , Ai ,'ti, W) be a Stone-representant for U,

which is taken fixed throughout the remaining part of this chapter. DEFINITION 2.23.

Let nE IN. Define the projection operator Pn:= U[1QJ, define the layer

L"

of Qn in A by

Ln:={'ti(Y):i E I,yE Qn}.

Note thatPn

=

0 if and only ifm(L_{n )}

=

0 (nE IN).

LEMMA 2.24.

Let!E C,n E IN,£

>

0and suppose If(a) I~£for a.e.a E Ln.Then:

I. Pn(H)is a subset ofRf.

II. The map

ni

PnfromHintoHis bounded andII

ni

PnII$

1..

£

III.

ni

(<I» E Pn(H)forall<I> E Pn(H).

IV. U(g)

ni

P_n=

ni

P_n U(g)forall

g

E L1(G).

Proof.

LetV :={yE

G:

}(y)=o} and letP := U[1v]. ThenPis the projection ofHontoN_{f .}Define

A:A~€ { (}(a)rl A(a):=

o

if If(a) I ~£andaE L_{n ,} else. (a E A)

Then A is Borel measurable and II A1100$ £-1. LetMA be the multiplication operator by A on

L2(m).

I. Let xEH. Then U(f)W-1MAWX=w-l(i.A.Wx)=w-l(1L,..WX)=Pnx, So Pn(H) cRf .

II. LetXE H. ThenW-1 M A Wx E Ntsince

tv.

(A.Wx)=Oa.e. So

ni

Pnx =W-1 M A Wx and

ni

_{Pn =W-}1_{M A W.}Then II

ni

_Pn11$ II A 1100$ £-1.

III. Let cp e P,.(H).Then

P,. ni(cp)=P,. n iP,.(cp) =P,. W-l MA Wcp= W-I(lL,..A.Wcp)=W-I(A. Wcp) =

n iP,.cp = n,i(cp). So ni(cp) e P,.(H).

IV. Letx e H.ThenU(g)ni P,.x=W-I(g.A.Wx)=ni P,. U(g)x.

o

LEMMA 2.25.

For allne INwithP,. ~0 there exists! e C and£

>

0 such that I](y) I

>

£for all Ye Q".

Proof.

Let ne IN and suppose m(L,.)~O. There exists ie I such that m('ti(Q,.» ~O. Let ~:=1<t;(Q.)'

Then~e L2(m) and~ ~O. Let(f,j"M:Jbethe net in property

n.

SupposeA.(Y)= 0 for all A. e J and ye Q,.. Then 0,*W-l ~ = limU(f,jW-1~ = limW-l(i..~) = 0, contradiction. So there exists

..

..

A.e J and ye Q,. such that J..(y)~O. By property P3 there exist ge C and

a>

0 such that b,. sup (IJ..(oo)I :00e Q,.}::;a inf{ Ig(oo) I :00e Q,.}.Then Ig(oo) I~ a-I IJ..(y) Ib,.

>

0 for all

00e Q,..

0

COROLLARY2.26.

Letne IN.ThenP,. (H)

c

Su,c,

Proof.

IfP,.= 0, thenP,.(H)= {OJ

c

Su,c, IfP,. ~0, the corollary is proved by Lemmas 2.25 and 2.24

L

0

LEMMA 2.27.

Let<J> e Tu,candn e IN.Define'¥ :C ~H

'¥(f):=P,. <J>(f) (fe C).

Then'¥e emb(H). More explicitely, suppose there exist ge C and£

>

0 such that Ig(y)I~t

for all ye Q,..Then'¥= emb(niP,. <J>(g».

Proof.

We may assume thatP,.

'*

O. By Lemma 2.25 there existg e C and£

>

0 such that Ig(y) I

>

e for all ye Q,.. Then Ig(y) I

>

e for all ye Q". Since the pair

(C ,

U) satisfies properties PI-P3, we obtain by Lemma 2.24 I thatP,.<J>(g) e Rg.So nil P,.<J>(g)is well defined.

Let! e C. Then by the Lemmas 2.24N, 1.9 and 2.24 I,

[emb n iP,. <J>(g)] (f)=U(f)* n i P,. <J>(g)= n i P,. U(f)* <J>(g)= nil P,. U (g)* <J>(f) =

n i U(g)p,. <J>(f)= n i n_{g P,.}<J>(f)=P,. <J>(f)='¥(f).So'¥=emb(niP,. <J>(g».

0

DEFINITION 2.28.

Let c;l)E TuC and nE IN. DefineP71 • c;l)E H by the relationP71c;l)(f)=[emb(P71 •c;l»)](f) forall

fE

C.

LEMMA 2.29.

N

I. Let <l>E Tu,c.Then c;l)

=

lim

L

emb(Pn.c;l») in(Tu,c ,'rproj).

N __71=1

II. P71 • emb

x

=

P71Xforall

x

E Hand

n

E IN.

III. P71 • c;l)E Pn(H)forallc;l)E Tu,cand

n

E IN.

IV. The map c;l) ~ P71 • c;l) from(Tu.c ,'tproj) intoHis continuous forallnE IN.

in N

c;l)= lim

L

emb(P11 • c;l»)

N-+oo11=1

So

I. Let c;l)E Tu,c,For every

fEe

we obtain in theHilbertspaceH:

00 N

<l>(f)=

L

_{Pn <l>(f)=} lim

(L

emb(Pn·c;l»))(f).

71=1 N __ 71=1

(Tu,C''tproj).

II. Let n E IN and x E H. Then Pn(embx)(f)=Pn U(f)'" x

=

U(f)"'Pnx=(embPnx)(f) for all

fEe.

By Definition 2.28, P71 • embx

=

P71

X.

LetnE IN. For the proof of III andIV we may suppose that P71

*"

O. By Lemma 2.25 there exist

gEe

and£

>

0such that Ig(y) I~£forall'fE Qn.

III. Let c;l)E Tu,c.By Lemmas 2.27 and 2.24 III,P71 • c;l)

=

ni

P11c;l)(g)E Pn(H). IV. By Lemmas 2.27 and 2.24 II we obtain forall<l>E Tu,c :

IIPn.c;l)1I

=lIni

P_n<l>(g)II~II

nil

Pllllllc;l)(g)II~ £-1 tg(<l».So the map c;l) ~ PII • c;l) from

(Tu,c ,'tproj) intoHis continuous.

o

The main result of this section is the following theorem. THEOREM 2.30.

LetBeTu,cbe a 'tprorbounded set,B

*"

0. Then there exist

FEe;

and a bounded setB0c H such thatU[F]. B

_o

=BandU[F]. IBo:(B

_o,

II II)~(B ,'tproj IB)is a homeomorphism.

Proof.

Let

n

E IN. By Lemma 2.29IV, the map c;l) ~ PII • c;l) from(Tu,c ,'tproj)intoH is continuous,

sorn :=sup{1IPI I .c;l)1I : c;l)E B}

<

00.

DefineF :

G

~ JR

F :=

L

(1+bn

r

n ) lQ.'

71=1

bn sup{I}(y)1:yE Qn}S~inf{1g(y) I :yE Qn} foralln E IN.

Then 1F}IS 1I}1I00

+

~ sup{tg(W) :4> EB} <00. Indeed, for all nE IN, YE Qn such that

}(y)

*'

0and forall4> E Bwe obtain by Lemma 2.24II:

(l_+bn"Pn.4>II)1}(y) I S 1I}1100

+

bnll0 iPnllll4>(g)1I I }(y) I S IIjlloo

+

bn(~-1 bnIj(y) 1)-1 •

• 11W(g)1I I](y) I S IIjlloo

+

~ sup{tg('¥):'¥EB}. So the assertion is proved, since 1F(y)j(y) I

=

sup{(l+bnll Pn.WII) Ij(y)1:WEB}.

00 1

For all 4>EB define x~:=1:: 1 b Pn .4>EH. Let Bo:={X~:4>EB}. Then

n=1 +n'n

00 1 00 1

IIx~1I2=

1::

2 HPn.4>1I2S

1::

2

<ooforall<l>E B,soBoisboundedinH.

n=1 (l+bn'n) n=1 bn

Let 4> EB.ForallfE Cwe obtain by Definition 2.28:

Pn <I>(f) =Pn U(f)* (Pn.4» _{=Pn U[F] U(f)*}x~ =Pn«U[F].x~)(f) for all nE IN. So

<1>=U[F].x~.By Lemma 2.16 V,U[F]. 1₈₀ :Bo~Bis a bijection.

Obviously U[F]. Iso:B_o~B is continuous. (See Lemma 2.16 VI.) The inverse map is also continuous. Indeed, let(<I>a)a.eMbea net inB, let WEB and suppose lim 4>a

=

4> in'tproj lB.

a

00 4 E2

Assertion: limx~. =x~ inH. Let

E>

O.

Let N E IN be so large that

1::

2

ST'

For all

a n=N+1 bn

nE IN,nS N,andPn

*'

0,there existfn E CandEn

>

0such that Ij nIIQ.~Enby Lemma 2.25. There exist 00 E M such that for all aEM, a~00, and all nE IN, nSN, Pn*'O holds:

tt.

(<I>a-4»S(l+bn'n)En E(2Nrt .Then for allaEM,a~ 00 we obtain by Lemma 2.24 II:

N 00

IIXII>._x~1I2

=

1::

IIPn(x~. -XII» 112

+

1::

IIPn(xll>._x~)112

n=1 n=N+1

So the assertion and the theorem is proved.

COROLLARY 2.31.

(Su.c, (Jind)

=

(Su.c, (Jproj)as topological vector spaces.

Proof.

Theorems 2.30 and 2.21.

o

Let Tbe the subspace ofTu.e asinDefinition 2.19. COROLLARY 2.32.

T=Tu,e as sets. We have even the stronger equality:Tu,e= U U [F] • Has sets.

Fee~

The identity map from(Tu.e, 'tind) into(Tu,e,'tproj) is continuous.

In the next chapter we give conditions such that(Tu,e ,'tind) is equal to(Tu.e ,'tproj)

as

a topolog-ical vector space.

COROLLARY 2.33.

LetBbe a subset ofTu.e.Then:

I. B is bounded in(Tu,e ,'tproj) if and only if there existF E C~ and boundedB0c H such thatB

=

U[F]. B

_o.

II. Bis compact in(Tu,e ,'tproj) if and only if there exist

FEe:

and a compactB0

c

H such

thatB=U[F].Bo.

III. B is sequentially compactin(Tu,e ,'tproj) if and only if there existF E C~ and a

sequen-tially compactB0

c

Hsuch thatB=U [F] • B

o.

IV. Bis sequentially compact in(Tu,e ,'tproj) if and only ifBis compact in(Tu,e ,'tproj)' V. Let (<I>n)neIN be a sequence in(Tu,e ,'tproj)' Then (cI>n)neIN is a Cauchy sequence if and

only if there existFEe: and a Cauchy sequence (xn)neIN inH such thatcI>n

=

U[F] • Xn

for all

n

E IN.

VI. Every'tprorbounded sequence in Tu,ehas a weakly convergent subsequence.

Proof.

Every compact and every sequentially compact set is bounded. Every Cauchy sequence is bounded. ForallF E C~,the mapU[F].fromH into(Tu.e, 'tproj) is continuous. So I, II, III, V

Chapter

m.

Nice properties of the spacesSu,c and Tu,c

LetGbe a locally compact Abelian group and letU bea representation ofGin a Hilbert spaceH.

Let

e

be a subset ofL1(G) which satisfies properties PI-P3. We fix the groupG, the representa-tionUand the set

e

throughout this chapter.

As we have seen Tu,c carries the topology 'tproj in a natural way. In the previous chapter, a second topology,'tmd, has been introduced onT

u,

c. (Note that we used property P3 to define the topology 'trod.) It turns out that the assertion (T

u,

c ,'tind)=(T

u,

c ,'tproj) as topological vector spaces is equivalent withthe assertion thatSu,c is complete. As we shall see it is also equivalent with a lot more assertions such

as:

(T

u,

C ,'tproj) is bomological; (T

u,

c ,'tproj) is reflexive;

Su,c = ("\ D(U[F]). In particular it turns out to be equivalent with the following property of

Fee;

the pair(e ,U).(Cf.

[tEl

property AIV'.) PROPERTY 3.1. P4.

'VFeBor(G,C) [ ('VKeC' (F.K is bounded]) =>3feC

3

c>o(U[lheG:IF(y)1 >clj(y)l}l = 0] .

REMARK. It is trivial that the following property implies property P4.

P4'. 'VFeBor(G,C) [('VKec' [F .Kisbounded]) =>3feC 3c>o [1 F I:s;c 1

j

Il] .

REMARK. The pair(e , U)satisfies property P4 resp. P4' if and only if the pair

(C ,

U)satisfies property P4 resp. P4'.

LEMMA 3.2.The following conditions are equivalent. I. Property P4 holds.

II. 'VFeBor(O,C) [('VKec; [F.Kisbounded])=>

3

fec

3

c>o

[U[lheG:IF(Y)I>Clj(Y)I}]=O~.

m.

'VFeBor(G,C) [('VKec' [F.Kis bounded])=> 3feC3c>o

"t~eH

[IIU{F]xll:S;

c

B

U(f)X"~

. IV. "tFeBor(O,C) [('VKec' [F.Kis bounded]) =>

3feC 3c>o

'V~eH

'VyeHWI U(f)xll::;; 1 A lIyII::;; 1)=> J(U[F]y,x) I::;;

c~

.

Proof. Note that a functionF E Bor(G ,C)is bounded ifF • K is bounded for allKEeN,since

10 E eN.So

m.

and IV. make sense.

II=>I. Trivial.

I=> III. LetF E Bor(G , €) and suppose F • K is bounded for all KEeN. By property P4 there exist fEe and

c

>

0 such that U[ly]

=

0 with

Y:= (rE

G:

IF(r) I>c Ij(r)I}. DefineL: G ~ JR

o ifj(r)

=

0 OryE Y,

L(r)

=

(rE

G)

jg;

ifj(r)

*

0 and rE Y.

ThenLE Borb(G,€)andIIL1100$;c. Let

x

E H.By Proposition2.8:

II U[F]x II=II U[ly.] U[F]xll=11U[ly. F]xll = II U[Lilxll$; II U[L]II IIU rhx II$;

c IIU(f)xll.

III => I. LetF E Bor(G, €) and supposeF. K is bounded for all KEeN. By assumption,

there exist fEe and c

>

0 such that II U [F ]x II $; c II U (f)x II for all x E H. Let

(A ,

m ,

I , Ai ,ti ,W)be a Stone-representant forU.

Let e

>

0 and define Z£:=(aE A : IF(a)I

>

2c If(a) I

>

e}. Suppose m(Z£)

*

O.

There existiE I and

n

E IVsuch that0

<

m(Z£I'lti(QII»

<

00.LetAI :=Z£n ti(QII)'

Define~:A ~€ I

~

~(a):=

o

if

a

E Al , (aE A)

Then ~ is bounded, so ~EL2(m). But O*2cIlIAIII$;IIF;II=IIU[F]W-I~II$; ellU(f) W-I ;II

=

_{c IIIAI}II. Contradiction. Hencem(Z£)

=

O.

Similar, define Zo :=(a EA : IF(a) I

>

0 and If(a) I=OJ. Thenm(Zo)=O. Define

00

Y:=(rE G : IF(r)I

>

2c Ij(r)I}. ThenZo u U ZII-I=U ti(Y)' hence U[ly]=O. 11=1 iel

So III impliesI.

III=>IV.LetFE Borb(G. C), fEeandc >0 and suppose thatII U[F]xll$; ellU(f)xll forall

x E H. Let x, y E H and suppose that II U(f)xll$; I and lIyll$; 1. Then

I(U[F]y,x)1= l(y, U[F]* x)I$;lIyIlIlU[F]* xll=