Directory UMM :Journals:Journal_of_mathematics:OTHER:

Schur Class Operator Functions

and Automorphisms of Hardy Algebras

Paul S. Muhly

1

and Baruch Solel

2

Received: June 12, 2007

Communicated by Joachim Cuntz

Abstract.

Let

E

bea

W

∗

-orrespondeneoveravonNeumann alge-bra

M

andlet

H

∞

(

E

)

betheassoiatedHardyalgebra.If

σ

isa faith-ful normalrepresentationof

M

onaHilbert spae

H

, thenonemay form thedual orrespondene

E

σ

andrepresentelementsin

H

∞

(

E

)

as

B

(

H

)

-valuedfuntionsontheunitball

D

(

E

σ

)

∗

. Thefuntionsthat oneobtainsarealledShurlassfuntionsandmaybeharaterized in terms of ertain Pik-like kernels. We study these funtions and relate them to systemmatries and transfer funtions from systems theory. Weusetheinformationgainedtodesribetheautomorphism groupof

H

∞

(

E

)

intermsofspeialMöbiustransformationson

D

(

E

σ

)

. Partiularattentionisdevotedtothe

H

∞

-algebrasthatareassoiated tographs.

2000 Mathematis Subjet Classiation: 46E22, 46E50, 46G20, 46H15,46H25,46K50,46L08,46L89,

Keywords and Phrases: Hardy Algebras, Tensor Algebras, Shur lassfuntions,

W

∗

-orrespondene,nonommutativerealization the-ory,Möbiustransformations,freesemigroupalgebras,graphalgebras, Nevanlinna-Pikinterpolation

1

SupportedinpartbygrantsfromtheNationalSieneFoundationand fromthe U.S.-IsraelBinationalSieneFoundation.

2

1

Introduction

Let

M

bea

W

∗

-algebraandlet

E

bea

W

∗

-orrespondeneover

M

. In[31℄we builtanoperatoralgebrafromthisdatathatwealledtheHardyalgebraof

E

and whih wedenoted

H

∞

(

E

)

. If

M

=

E

=

C

-the omplexnumbers,then

H

∞

(

E

)

isthelassialHardyalgebraonsistingof allbounded analyti fun-tionsontheopen unit dis,

D

(seeExample 2.4below.) If

M

=

C

again,but

E

=

C

n

, then

H

∞

(

E

)

is thefree semigroupalgebra

L

n

studied by Davidson andPitts[17℄,Popesu[32℄andothers(seeExample2.5.) Oneoftheprinipal disoveriesmadein [31℄,andthesoureofinspirationforthepresentpaper,is thatattahedtoeahfaithfulnormalrepresentation

σ

of

M

thereisadual or-respondene

E

σ

, whihisa

W

∗

-orrespondeneovertheommutantof

σ

(

M

)

,

σ

(

M

)

′

, and theelementsof

H

∞

(

E

)

denefuntions onthe openunit ballof

E

σ

,

D

(

E

σ

)

. Further,thevaluedistributiontheoryofthesefuntionsturnsout to be linked through our generalization of the Nevanlinna-Pik interpolation theorem [31, Theorem 5.3℄ with the positivity properties of ertain Pik-like kernelsofmappings betweenoperator spaes.

Inthesettingwhere

M

=

E

=

C

and

σ

isthe

1

-dimensional representationof

C

onitself,then

E

σ

is

C

again. Therepresentationof

H

∞

(

E

)

intermsof fun-tionson

D

(

E

σ

) =

D

isjusttheusualwaywethink of

H

∞

(

E

)

. Inthissetting, our Nevanlinna-Pik theorem is exatlythe lassialtheorem. If, however,

σ

is arepresentationof

C

on aHilbert spae

H

,

dim(

H

)

>

1

, then

E

σ

maybe identiedwith

B

(

H

)

andthen

D

(

E

σ

)

beomesthespaeofstritontrations on

H

, i.e., all those operators of norm stritly less than

1

. In this ase, the valueof an

f

∈

H

∞

(

E

)

at a

T

∈

D

(

E

σ

)

is simply

f

(

T

)

,dened throughthe

usual holomorphi funtional alulus. OurNevanlinna-Piktheorem givesa solutiontoproblems suh asthis: given

k

operators

T

1

, T

2

, . . . , T

k

allofnorm lessthan

1

and

k

operators,

A

1

, A

2

, . . . , A

k

,determinetheirumstanesunder whihoneanndaboundedanalytifuntion

f

ontheopenunitdisofsup norm at most

1

suh that

f

(

T

i) =

A

i

,

i

= 1

,

2

, . . . , k

(See [31, Theorem6.1℄.) On the other hand, when

M

=

C,

E

=

C

n

, and

σ

is one dimensional, the spae

E

σ

is

C

n

and

D

(

E

σ

)

is the unit ball

B

n

. Elements in

H

∞

(

E

) =

L

n

arerealizedasholomorphifuntionson

B

n

thatlieinamultiplierspae stud-iedin detailbyArveson[5℄. Moreaurately,thefuntionalrepresentationof

H

∞

(

E

) =

L

n

in termsofthesefuntions expressesthisspaeasaquotient of

H

∞

(

E

) =

L

n

. The Nevanlinna-Pik theoremof [31℄ontainsthoseof

David-son andPitts [18℄, Popesu[34℄, and Ariasand Popesu [4℄, whih dealwith interpolationproblemsforthesespaesoffuntions(possiblytensoredwiththe boundedoperatorsonanauxiliaryHilbertspae). Italsoontainssomeofthe results of Constaninesu and Johnson in [16℄ whih treatselementsof

L

n

as funtionsontheballofstritrowontrationswithvaluesintheoperatorson a Hilbert spae. (See their Theorem 3.4 in partiular.) This situation arises when onetakes

M

=

C

and

E

=

C

n

, but takes

σ

to besalar multipliation onanauxiliaryHilbert spae.

those funtionson

D

(

E

σ

)

that arisefromevaluatingelementsof

H

∞

(

E

)

. For this purpose, weintrodueafamilyof funtions on

D

(

E

σ

)

that weallShur

lassoperatorfuntions(seeDenition3.1). Roughlyspeaking,thesefuntions aredened sothat aPik-likekernelthatonemayattahtoeahone is om-pletely positive denite in the sense of Barreto, Bhat, Liebsher and Skeide [14℄. In Theorem 3.3 weuse their Theorem 3.2.3 to give aKolmogorov-type representation of the kernel, from whih we derive an analogueof a unitary

systemmatrix

A

B

C

D

whosetransferfuntion

A

+

B

(

I

−

L

η

∗

D

)

−

1

L

∗

η

C

turns out to be the given Shur lass operator funtion. We then prove in Theorem 3.6that eahsuhtransferfuntion arises byevaluatinganelement in

H

∞

(

E

)

at pointsof

D

(

E

σ

)

andonversely, eah funtion in

H

∞

(

E

)

hasa representationintermsofatransferfuntion. Themeaningofthenotationwill bemadepreisebelow,but weuseitheretohighlighttheonnetionbetween our analysis and realization theory as it omes from mathematial systems theory. Thepointto keepinmindis thatfuntionson

D

(

E

σ

)

thatomefrom elementsof

H

∞

(

E

)

arenot, apriori, analytiin anyordinarysenseand itis not at alllear what analyti features theyhave. Our Theorems3.1 and3.6 together with[31, Theorem 5.3℄ showthat theShur lassoperator funtions are preisely the funtions one obtainswhen evaluating funtions in

H

∞

(

E

)

(ofnormatmost

1

)atpointsof

D

(

E

σ

)

. Thefatthateahsuhfuntionmay berealizedasatransferfuntion exhibitsasurprisinglevelofanalytiitythat isnotevidentin thedenition of

H

∞

(

E

)

.

Ourseondobjetiveisto onnettheusualholomorphipropertiesof

D

(

E

σ

)

with the automorphisms of

H

∞

(

E

)

. As aspae,

D

(

E

σ

)

is theunit ball of a

J

∗

-triplesystem. Consequently,everyholomorphiautomorphismof

D

(

E

σ

)

is theomposition ofaMöbiustransformation andalinearisometry [20℄. Eah ofthese implementsanautomorphismofthealgebraofallbounded, omplex-valued analyti funtions on

D

(

E

σ

)

, but in our setting only ertain of them implement automorphisms of

H

∞

(

E

)

- those for whih the Möbius part is determinedbyaentralelementof

E

σ

(seeTheorem4.21).Ourproofrequires thefatthattheevaluationoffuntionsin

H

∞

(

E

)

(ofnormatmost

1

)atpoints of

D

(

E

σ

)

arepreiselytheShurlassoperatorfuntionson

D

(

E

σ

)

. Indeed,the whole analysisisanintriate point-ounterpoint interplayamongelements of

H

∞

(

E

)

, Shur lass funtions, transfer funtions and lassial funtion theory on

D

(

E

σ

)

. In the last setion, we apply our general analysis of the automorphisms of

H

∞

(

E

)

to the speial ase of

H

∞

-algebras oming from direted graphs.

also use (See Setion 3 and, in partiular, the proof of Theorem 3.3.) The analysis of Ball et al. makes additional ties between the theory of abstrat Hardyalgebrasthat wedevelophereandlassialfuntiontheoryontheunit dis.

2

Preliminaries

Westartbyintroduingthebasidenitionsandonstrutions. Weshallfollow Lane[24℄forthegeneraltheoryofHilbert

C

∗

-modulesthatweshalluse. Let

A

bea

C

∗

-algebraand

E

bearightmoduleover

A

endowedwithabi-additive map

h·

,

·i

:

E

×

E

→

A

(referred to asan

A

-valuedinner produt) suh that, for

ξ, η

∈

E

and

a

∈

A

,

h

ξ, ηa

i

=

h

ξ, η

i

a

,

h

ξ, η

i

∗

=

h

η, ξ

i

, and

h

ξ, ξ

i ≥

0

,with

h

ξ, ξ

i

= 0

only when

ξ

= 0

. Also,

E

is assumed to be ompletein thenorm

k

ξ

k

:=

kh

ξ, ξ

ik

1

/

2

. We write

L

(

E

)

for the spae of ontinuous, adjointable,

A

-modulemapson

E

. Itisknowntobea

C

∗

-algebra. If

M

isavonNeumann algebraandif

E

isaHilbert

C

∗

-moduleover

M

,then

E

issaidtobeself-dualin aseeveryontinuous

M

-modulemapfrom

E

to

M

isgivenbyaninnerprodut withanelementof

E

. Let

A

and

B

be

C

∗

-algebras. A

C

∗

-orrespondenefrom

A

to

B

is aHilbert

C

∗

-module

E

over

B

endowed witha struture of aleft moduleover

A

viaanondegenerate

∗

-homomorphism

ϕ

:

A

→ L

(

E

)

.

Whendealingwithaspei

C

∗

-orrespondene,

E

,froma

C

∗

-algebra

A

toa

C

∗

-algebra

B

, itwill be onvenientsometimes to suppress the

ϕ

in formulas involvingtheleft ation and simplywrite

aξ

or

a

·

ξ

for

ϕ

(

a

)

ξ

. This should ausenoonfusioninontext.

If

E

is a

C

∗

-orrespondene from

A

to

B

and if

F

is a orrespondene from

B

to

C

, then thebalanedtensor produt,

E

⊗

B

F

is an

A, C

-bimodule that

arriestheinner produtdenedbytheformula

h

ξ

1

⊗

η

1

, ξ

2

⊗

η

2

i

E⊗

B

F

:=

h

η

1

, ϕ

(

h

ξ

1

, ξ

2

i

E

)

η

2

i

F

TheHausdorompletion ofthis bimodule isagaindenotedby

E

⊗

B

F

. InthispaperwedealmostlywithorrespondenesovervonNeumannalgebras that satisfy some natural additional properties as indiated in the following denition. (Forexamplesand moredetailssee[31℄).

Definition 2.1

Let

M

and

N

bevonNeumannalgebrasandlet

E

beaHilbert

C

∗

-moduleover

N

. Then

E

isalleda Hilbert

W

∗

-moduleover

N

inase

E

is self-dual. Themodule

E

isalleda

W

∗

-orrespondenefrom

M

to

N

inase

E

isaself-dual

C

∗

-orrespondenefrom

M

to

N

suhthatthe

∗

-homomorphism

ϕ

:

M

→ L

(

E

)

, giving the left module struture on

E

, is normal. If

M

=

N

weshallsay that

E

isa

W

∗

-orrespondene over

M

.

Wenote thatif

E

isaHilbert

W

∗

-moduleoveravonNeumannalgebra,then

L

(

E

)

isnotonlya

C

∗

-algebra,butis alsoa

W

∗

Definition 2.2

An isomorphism of a

W

∗

-orrespondene

E

1

over

M

1

and a

W

∗

-orrespondene

E

2

over

M

2

is a pair

(

σ,

Ψ)

where

σ

:

M

1

→

M

2

is an isomorphism of von Neumann algebras,

Ψ :

E

1

→

E

2

is a vetor spae isomorphism preserving the

σ

-topology and for

e, f

∈

E

1

and

a, b

∈

M

1

, we have

Ψ(

aeb

) =

σ

(

a

)Ψ(

e

)

σ

(

b

)

and

h

Ψ(

e

)

,

Ψ(

f

)

i

=

σ

(

h

e, f

i

)

.

When onsidering the tensor produt

E

⊗

M

F

of two

W

∗

-orrespondenes, one needs to takethe losure of the

C

∗

-tensor produt in the

σ

-topologyof [6℄ in order to get a

W

∗

-orrespondene. However, we will not distinguish notationallybetweenthe

C

∗

-tensorprodutandthe

W

∗

-tensorprodut. Note alsothatgivena

W

∗

-orrespondene

E

over

M

andaHilbertspae

H

equipped with anormalrepresentation

σ

of

M

,weanform theHilbert spae

E

⊗

σ

H

by dening

h

ξ

1

⊗

h

1

, ξ

2

⊗

h

2

i

=

h

h

1

, σ

(

h

ξ

1

, ξ

2

i

)

h

2

i

. Thus,

H

is viewed as a orrespondenefrom

M

to

C

via

σ

and

E

⊗

σ

H

isjust thetensor produtof

E

and

H

as

W

∗

-orrespondenes.

Note alsothat,given anoperator

X

∈ L

(

E

)

andan operator

S

∈

σ

(

M

)

′

, the map

ξ

⊗

h

7→

Xξ

⊗

Sh

denes a bounded operator on

E

⊗

σ

H

denoted by

X

⊗

S

. Therepresentationof

L

(

E

)

that resultswhenone lets

S

=

I

,isalled

the representation of

L

(

E

)

indued by

σ

and is often denoted by

σ

E

. The omposition,

σ

E

◦

ϕ

isarepresentationof

M

whihweshallalsosayisindued by

σ

,but weshallusually denoteitby

ϕ

(

·

)

⊗

I

.

Observe that if

E

is a

W

∗

-orrespondene over a von Neumann algebra

M

, then we may form the tensor powers

E

⊗n

,

n

≥

0

, where

E

⊗

0

is simply

M

viewed as the identity orrespondene over

M

, and we may form the

W

∗

-diret sum ofthe tensor powers,

F

(

E

) :=

E

⊗

0

⊕

E

⊗

1

⊕

E

⊗

2

⊕ · · ·

to obtain a

W

∗

-orrespondeneover

M

alledthe(full)Fokspae over

E

. Theations of

M

onthe left and right of

F

(

E

)

are the diagonal ations and, when it is onvenient to do so, we make expliit the left ation by writing

ϕ

∞

for it. That is,for

a

∈

M

,

ϕ

∞(

a

) :=

diag

{

a, ϕ

(

a

)

, ϕ

(2)

(

a

)

, ϕ

(3)

(

a

)

,

· · · }

, wherefor all

n

,

ϕ

(

n

)

(

a

)(

ξ

1

⊗

ξ

2

⊗ · · ·

ξ

n) = (

ϕ

(

a

)

ξ

1

)

⊗

ξ

2

⊗ · · ·

ξ

n

,

ξ

1

⊗

ξ

2

⊗ · · ·

ξ

n

∈

E

⊗n

. The tensor algebra over

E

, denoted

T

+

(

E

)

, is dened to be thenorm-losed subalgebra of

L

(

F

(

E

))

generated by

ϕ

∞(

M

)

and the reation operators

T

ξ

,

ξ

∈

E

,dened bytheformula

T

ξ

η

=

ξ

⊗

η

,

η

∈ F

(

E

)

. Wereferthereaderto

[28℄forthebasifats about

T

+

(

E

)

.

Definition 2.3

([31 ℄)Givena

W

∗

-orrespondene

E

overthe von Neumann algebra

M

,theultraweaklosureofthetensoralgebraof

E

,

T

+

(

E

)

,in

L

(

F

(

E

))

, isalledthe HardyAlgebraof

E

,andisdenoted

H

∞

(

E

)

.

Example 2.4

If

M

=

E

=

C,

then

F

(

E

)

an be identied with

ℓ

2

(

Z

+

)

or,

through the Fouriertransform,

H

2

(

T

)

. Thetensor algebrathen isisomorphi to the dis algebra

A

(

D

)

viewed asmultipliation operatorson

H

2

(

T

)

andthe Hardy algebraisrealizedasthe lassial Hardyalgebra

H

∞

(

T

)

.

Example 2.5

If

M

=

C

and

E

=

C

n

, then

F

(

E

)

an be identied with the spae

l

2

(

F

+

n

)

, where

F

+

algebrathen iswhat Popesu refersto as the non ommutative dis algebra

A

n

andtheHardyalgebraisits

w

∗

-losure. ItwasstudiedbyPopesu[32 ℄and by DavidsonandPittswho denoteditby

L

n

[17 ℄.

Weneedtoreviewsomebasifatsabouttherepresentationtheoryof

H

∞

(

E

)

andof

T

+

(

E

)

. See[28,31℄formoredetails.

Definition 2.6

Let

E

bea

W

∗

-orrespondeneover avon Neumannalgebra

M

. Then:

1. Aompletelyontrativeovariantrepresentationof

E

onaHilbertspae

H

isapair

(

T, σ

)

,where

(a)

σ

isanormal

∗

-representationof

M

in

B

(

H

)

.

(b)

T

is a linear, ompletely ontrative map from

E

to

B

(

H

)

that is ontinuousinthe

σ

-topologyof [6 ℄on

E

andthe ultraweaktopology on

B

(

H

)

.

()

T

is a bimodule map in the sense that

T

(

SξR

) =

σ

(

S

)

T

(

ξ

)

σ

(

R

)

,

ξ

∈

E

,and

S, R

∈

M

.

2. Aompletelyontrativeovariantrepresentation

(

T, σ

)

of

E

in

B

(

H

)

is alled isometri inase

T

(

ξ

)

∗

T

(

η

) =

σ

(

h

ξ, η

i

)

(1)

forall

ξ, η

∈

E

.

Itshould benotedthat theoperatorspaestrutureon

E

towhihDenition 2.6 refers is that whih

E

inherits when viewed as a subspae of its linking algebra.

As weshowedin [28, Lemmas3.43.6℄andin [31℄, ifaompletely ontrative ovariant representation,

(

T, σ

)

, of

E

in

B

(

H

)

is given, then it determines a ontration

T

˜

:

E

⊗

σ

H

→

H

dened by the formula

T

˜

(

η

⊗

h

) :=

T

(

η

)

h

,

η

⊗

h

∈

E

⊗

σ

H

. Theoperator

T

˜

intertwines therepresentation

σ

on

H

and

theinduedrepresentation

σ

E

◦

ϕ

=

ϕ

(

·

)

⊗

I

H

on

E

⊗

σ

H

;i.e.

˜

T

(

ϕ

(

·

)

⊗

I

) =

σ

(

·

) ˜

T .

(2)

Infatwehavethefollowinglemmafrom[31,Lemma 2.16℄.

The importane of theompletely ontrative ovariantrepresentationsof

E

(or, equivalently, theintertwining ontrations

T

˜

asabove)is that theyyield allompletelyontrativerepresentationsofthetensoralgebra. Morepreisely, wehavethefollowing.

Theorem 2.8

Let

E

bea

W

∗

-orrespondeneoveravonNeumannalgebra

M

. Toevery ompletely ontrative ovariantrepresentation,

(

T, σ

)

,of

E

there is a unique ompletely ontrative representation

ρ

of the tensor algebra

T

+

(

E

)

that satises

ρ

(

T

ξ) =

T

(

ξ

)

ξ

∈

E

and

ρ

(

ϕ

∞(

a

)) =

σ

(

a

)

a

∈

M.

The map

(

T, σ

)

7→

ρ

isabijetion between theset ofall ompletelyontrative ovariant representations of

E

and all ompletely ontrative (algebra) repre-sentationsof

T

+

(

E

)

whose restritions to

ϕ

∞(

M

)

are ontinuouswith respet tothe ultraweak topologyon

L

(

F

(

E

))

.

Definition 2.9

If

(

T, σ

)

is aompletely ontrativeovariant representation of a

W

∗

-orrespondene

E

overavonNeumannalgebra

M

,weall the repre-sentation

ρ

of

T

+

(

E

)

desribed in Theorem 2.8 the integrated form of

(

T, σ

)

andwrite

ρ

=

σ

×

T

.

Remark 2.10

Oneoftheprinipaldiulties onefaesindealingwith

T

+

(

E

)

and

H

∞

(

E

)

istodeidewhenthe integratedform,

σ

×

T

,ofaompletely on-trative ovariant representation

(

T, σ

)

extends from

T

+

(

E

)

to

H

∞

(

E

)

. This problemarises alreadyinthesimplestsituation,vis. when

M

=

C

=

E

. Inthis setting,

T

is given by asingle ontration operator on aHilbert spae,

T

+

(

E

)

is the dis algebra and

H

∞

(

E

)

is the spae of boundedanalyti funtions onthe dis. Therepresentation

σ

×

T

extendsfromthe disalgebrato

H

∞

(

E

)

preisely whenthere isnosingularparttothe spetral measureofthe minimal unitary dilation of

T

. We arenot aware of aomparableresult inourgeneral ontext but we have some suient onditions. One of them is given in the following lemma. Itisnotaneessaryonditioningeneral.

Lemma 2.11

[31 , Corollary 2.14℄ If

k

T

˜

k

<

1

then

σ

×

T

extends to a ultra-weakly ontinuousrepresentation of

H

∞

(

E

)

.

In[31℄weintroduedandstudiedtheoneptsofdualityandofpointevaluation (forelementsof

H

∞

(

E

)

). These playaentralroleinouranalysishere.

Definition 2.12

Let

E

bea

W

∗

-orrespondeneoveravonNeumannalgebra

M

andlet

σ

:

M

→

B

(

H

)

beafaithfulnormalrepresentationof

M

onaHilbert

spae

H

. Then the

σ

-dual of

E

,denoted

E

σ

,isdenedtobe

An important feature of the dual

E

σ

is that it is a

W

∗

-orrespondene, but overthe ommutant of

σ

(

M

)

,

σ

(

M

)

′

.

Proposition 2.13

With respettothe ationof

σ

(

M

)

′

andthe

σ

(

M

)

′

-valued innerprodutdenedasfollows,

E

σ

beomesa

W

∗

-orrespondeneover

σ

(

M

)

′

: For

Y

and

X

in

σ

(

M

)

′

,and

η

∈

E

σ

,

X

·

η

·

Y

:= (

I

⊗

X

)

ηY

,andfor

η

1

, η

2

∈

E

σ

,

h

η

1

, η

2

i

σ

(

M

)

′

:=

η

1

∗

η

2

.

Inthefollowingremarkweexplainwhatwemeanbyevaluatinganelementof

H

∞

(

E

)

atapointintheopenunit ballofthedual.

Remark 2.14

The importaneof this dual spae,

E

σ

,is that itislosely re-latedtotherepresentationsof

E

. Infat, theoperatorsin

E

σ

whosenormdoes not exeed

1

arepreisely the adjoints of the operators ofthe form

T

˜

for a o-variant pair

(

T, σ

)

. In partiular, every

η

in the openunitball of

E

σ

(written

D

(

E

σ

)

) gives rise to a ovariant pair

(

T, σ

)

(with

η

= ˜

T

∗

) suh that

σ

×

T

extendstoarepresentation of

H

∞

(

E

)

. Given

X

∈

H

∞

(

E

)

wean applythe representation assoiatedto

η

toit. The resultingoperator in

B

(

H

)

will bedenotedby

X

b

(

η

∗

)

. Thus

b

X

(

η

∗

) = (

σ

×

η

∗

)(

X

)

.

In this way, we view every element in the Hardy algebra as a

B

(

H

)

-valued funtion

b

X

:

D

(

E

σ

)

∗

→

B

(

H

)

onthe open unitballof

(

E

σ

)

∗

. Oneof ourprimaryobjetivesistounderstand the rangeofthe transform

X

→

X

b

,

X

∈

H

∞

(

E

)

.

Example 2.15

Suppose

M

=

E

=

C

and

σ

the representation of

C

on some Hilbertspae

H

. Thenitiseasytohekthat

E

σ

isisomorphito

B

(

H

)

. Fixan

X

∈

H

∞

(

E

)

. Aswementionedabove,thisHardyalgebraisthelassial

H

∞

(

T

)

andweanidentify

X

withafuntion

f

∈

H

∞

(

T

)

. Given

S

∈

D

(

E

σ

) =

B

(

H

)

, it is not hard to hek that

X

b

(

S

∗

)

, as dened above, is the operator

f

(

S

∗

)

denedthroughthe usualholomorphi funtionalalulus.

Example 2.16

In [17 ℄ Davidson and Pitts assoiate toevery element of the freesemigroupalgebra

L

n

(seeExample2.5)afuntionontheopenunitballof

C

n

. Thisisaspeialaseofouranalysiswhen

M

=

C,

E

=

C

n

and

σ

isaone dimensionalrepresentationof

C.

Inthis ase

σ

(

M

)

′

=

C

and

E

σ

=

C

n

. Note, however, thatourdenitionallowsustotake

σ

tobetherepresentationof

C

on an arbitraryHilbert spae

H

. If we doso, then

E

σ

isisomorphi to

B

(

H

)

(

n

)

Example 2.17

Partof the reentwork ofPopesuin [35℄maybe astin our framework. We will follow his notation. Fix aHilbert spae

K

, and let

E

be the olumn spae

B

(

K

)

n

. Take, also, aHilbert spae

H

and let

σ

:

B

(

K

)

→

B

(

K

⊗

H

)

be the representation whih sends

a

∈

B

(

K

)

to

a

⊗

I

H

. Then,

sinetheommutantof

σ

(

B

(

K

))

isnaturallyisomorphito

B

(

H

)

,itiseasyto see that

E

σ

isthe olumnspae over

B

(

H

)

,

B

(

H

)

n

. It follows that

D

(

E

σ

)

is the openunit ballin

B

(

H

)

n

. Afreeformalpowerseries withoeientsfrom

B

(

K

)

is aformal series

F

=

P

α∈

F

+

n

A

α

⊗

Z

α

where

F

+

n

isthe freesemigroup

on

n

generators,the

A

α

areelements of

B

(

K

)

andwhere

Z

α

is themonomial in nonommuting indeterminates

Z

1

,

Z

2

,...,

Z

n

determined by

α

. If

F

has radiusofonvergeneequalto

1

,thenonemay evaluate

F

atpointsof

D

(

E

σ

)

∗

togetafuntionon

D

(

E

σ

)

∗

withvaluesin

B

(

K

⊗

H

)

,vis.,

F

((

S

1

, S

2

,

· · ·

S

n)) =

P

α∈

F

+

n

A

α

⊗

S

α

. See[35 , Theorem 1.1℄. Infat, under additional restritions ontheoeients

A

α

,

F

maybeviewedasafuntion

X

in

H

∞

(

B

(

K

)

n

)

insuh awaythat

F

((

S

1

, S

2

,

· · ·

S

n)) =

X

b

(

S

1

, S

2

,

· · ·

S

n)

inthesensedenedin[31 ,p. 384℄ and disussed above in Remark 2.14. The spae that Popesu denotes by

H

∞

(

B

(

X

)

n

1

)

arises when

K

=

C,

and isnaturallyisometrially isomorphi to

L

n

[35 ,Theorem3.1℄. Wenotedinthe preeding examplethat

L

n

is

H

∞

(

C

n

)

. The point of [35 ℄, at least in part, is to study

H

∞

(

B

(

X

)

n

1

)

≃ L

n

=

H

∞

(

C

n

)

through all the representations

σ

of

C

on Hilbert spaes

H

, that is, through evaluating funtions in

H

∞

(

B

(

X

)

n

1

)

at points the unit ball of

B

(

H

)

n

for all possible

H

's. The spae

B

(

K

)

n

isMoritaequivalentto

C

n

inthesenseof [30 ℄, at leastwhen

dim(

K

)

<

∞

, and, in that asethe tensor algebras

T

+

(

B

(

K

)

n

)

and

T

+

(

C

n

)

are Morita equivalent in the sense desribed by [15℄. The tensor algebra

T

+

(

C

n

)

, in turn, is naturally isometrially isomorphi to Popesu's nonommutative dis algebra

A

n

[33 ℄. The analysis in [15 ℄ suggests a sense in whih

C

n

and

B

(

K

)

n

are Morita equivalent even when

dim(

K

) =

∞

, and thattogetherwith[30℄suggeststhat

H

∞

(

B

(

K

)

n

)

shouldbeMoritaequivalentto

H

∞

(

B

(

X

)

n

1

)

≃

H

∞

(

C

n

)

. Thiswouldsuggestanevenloseronnetionbetween Popesu's free power series, and all that goes with them, and the perspetive wehave takeninthis paper, whih,asweshall see, involvesgeneralizedShur funtionsandtransferfuntions. Theonnetionseemslikeapromisingavenue toexplore.

In[31℄ weexploited theperspetiveofviewing elementsof theHardyalgebra as

B

(

H

)

-valued funtions on the open unit ball of the dual orrespondene to prove a Nevanlinna-Pik type interpolation theorem. In order to state it we introdue some notation: For operators

B

1

and

B

2

in

B

(

H

)

, we write

Ad

(

B

1

, B

2

)

forthemapfrom

B

(

H

)

toitselfthatsends

S

to

B

1

SB

∗

2

. Also,given

elements

η

1

, η

2

in

D

(

E

σ

)

, we let

θ

η

1

,η

2

denote the map, from

σ

(

M

)

′

to itself that sends

a

to

h

η

1

, aη

2

i

. Thatis,

θ

η

1

,η

2

(

a

) :=

h

η

1

, aη

2

i

=

η

∗

1

aη

2

,

a

∈

σ

(

M

)

′

.

Theorem 2.18

([31 ,Theorem5.3℄)Let

E

bea

W

∗

-orrespondeneoveravon Neumannalgebra

M

andlet

σ

:

M

→

B

(

H

)

beafaithfulnormalrepresentation of

M

onaHilbertspae

H

. Fix

k

points

η

1

, . . . η

k

inthedisk

D

(

E

σ

)

2

k

operators

B

1

, . . . B

k

, C

1

, . . . C

k

in

B

(

H

)

. Then thereexistsan

X

in

H

∞

(

E

)

suhthat

k

X

k ≤

1

and

B

i

X

b

(

η

i

∗

) =

C

i

for

i

= 1

,

2

, . . . , k,

if and only if the map from

M

k(

σ

(

M

)

′

)

into

M

k

(

B

(

H

))

denedbythe

k

×

k

matrix

(

Ad

(

B

i

, B

j

)

−

Ad

(

C

i

, C

j

))

◦

(

id

−

θ

η

i

,η

j

)

−

1

(3)

isompletely positive.

That is,themap

T

,say,givenbythematrix(3) isomputedbytheformula

T

((

a

ij

)) = (

b

ij

)

,

where

b

ij

=

B

i((

id

−

θ

η

i

,η

j

)

−

1

(

a

ij)

B

∗

j

−

C

i((

id

−

θ

η

i

,η

j

)

−

1

(

a

ij

)

C

j

∗

and

(

id

−

θ

η

i

,η

j

)

−

1

(

a

ij

) =

a

ij

+

θ

η

i

,η

j

(

a

ij) +

θ

η

i

,η

j

(

θ

η

i

,η

j

(

a

ij)) +

· · ·

We lose this setion with two tehnial lemmas that will be needed in our analysis. Let

M

and

N

be

W

∗

-algebrasand let

E

be a

W

∗

-orrespondene from

M

to

N

. Given a

σ

-losed suborrespondene

E

0

of

E

we know that the orthogonal projetion

P

of

E

onto

E

0

is a right module map. (See [6, Consequenes1.8(ii)℄). Inthefollowinglemmaweshowthat

P

alsopreserves theleftation.

Lemma 2.19

Let

E

be a

W

∗

-orrespondene from the von Neumann algebra

M

to the von Neumann algebra

N

, and let

E

0

be a sub

W

∗

-orrespondene

E

0

of

E

that is losed in the

σ

-topology of [6 , Consequenes 1.8 (ii)℄. If

P

is the orthogonal projetion from

E

onto

E

0

, then

P

is abimodule map; i.e.,

P

(

aξb

) =

aP

(

ξ

)

b

for all

a

∈

M

and

b

∈

N

.

Proof.

Itsuestohekthat

P

(

eξ

) =

eP

(

ξ

)

forall

ξ

∈

E

andprojetions

e

∈

M

. For

ξ, η

∈

E

andaprojetion

e

∈

M

,wehave

k

eξ

+

f η

k

2

=

kh

eξ, eξ

i

+

h

f η, f η

ik ≤ kh

eξ, eξ

ik

+

kh

f η, f η

ik

=

k

eξ

k

2

+

k

f η

k

2

,

where

f

= 1

−

e

. So,forevery

λ

∈

R

wehave

(

λ

+ 1)

2

k

f P

(

eξ

)

k

2

=

k

f P

(

eξ

+

λf P

(

eξ

))

k

2

≤ k

eξ

+

λf P

(

eξ

)

k

2

≤ k

eξ

k

2

+

λ

2

k

f P

(

eξ

)

k

2

.

Hene,forevery

λ

∈

R,

and,thus,

(

I

−

e

)

P

(

eξ

) =

f P

(

eξ

) = 0

.

Replaing

e

by

f

=

I

−

e

weget

eP

((

I

−

e

)

ξ

) = 0

and,therefore,

P

(

eξ

) =

eP

(

eξ

) =

eP

(

ξ

)

.

Sine

M

isspanned byitsprojetions,wearedone.

Lemma 2.20

Let

E

bea

W

∗

-orrespondeneover

M

,let

σ

beafaithfulnormal representationof

M

ontheHilbertspae

E

,andlet

E

σ

bethe

σ

-dual orrespon-deneover

N

:=

σ

(

M

)

′

. Then

(i) The left ation of

N

on

E

σ

is faithful if and only if

E

is full (i.e. if and only if the ultraweakly losed ideal generated by the inner produts

h

ξ

1

, ξ

2

i

,

ξ

1

, ξ

2

∈

E

,isall of

M

).

(ii) Theleft ationof

M

on

E

isfaithfulifand onlyif

E

σ

isfull.

Proof.

We shall prove (i). Part (ii) then follows by duality (using [31, Theorem 3.6℄). Given

S

∈

N

,

Sη

= 0

for every

η

∈

E

σ

if and only if for all

η

∈

E

σ

and

g

∈ E

,

(

I

⊗

S

)

η

(

g

) = 0

. Sine the losed subspaespanned by the rangesof all

η

∈

E

σ

is allof

E

⊗

M

E

([31℄), this is equivalent to the equation

ξ

⊗

Sg

= 0

holdingforall

g

∈ E

and

ξ

∈

E

. Sine

h

ξ

⊗

Sg, ξ

⊗

Sg

i

=

h

g, S

∗

h

ξ, ξ

i

Sg

i

, we nd that

SE

σ

= 0

if and only if

σ

(

h

E, E

i

)

S

= 0

, where

h

E, E

i

isthe ultraweakly losedidealgenerated byall innerproduts. Ifthis

idealisallof

M

wendthattheequation

SE

σ

= 0

impliesthat

S

= 0

. Inthe other diretion,ifthisisnotthease, thenthisideal isoftheform

(

I

−

q

)

M

forsomeentralnonzeroprojetion

q

andthen

S

=

σ

(

q

)

isdierentfrom

0

but vanisheson

E

σ

.

3

Schur class operator functions and realization

Throughout this setion,

E

will be a xed

W

∗

-orrespondene over the von Neumannalgebra

M

and

σ

willbeafaithfulrepresentationof

M

onaHilbert spae

E

. We then form the

σ

-dual of

E

,

E

σ

, whih is aorrespondeneover

N

:=

σ

(

M

)

′

, and we write

D

(

E

σ

)

for its open unit ball. Further, we write

D

(

E

σ

)

∗

for

{

η

∗

|

η

∈

D

(

E

σ

)

}

.

The following denition is learly motivated by the ondition appearing in Theorem2.18andShur'stheorem fromlassialfuntion theory.

Definition 3.1

Let

Ω

be asubset of

D

(

E

σ

)

and let

Ω

∗

=

{

ω

∗

|

ω

∈

Ω

}

. A funtion

Z

: Ω

∗

→

B

(

E

)

will be alled a Shur lass operator funtion (with valuesin

B

(

E

)

)if,forevery

k

andeveryhoieofelements

η

1

, η

2

, . . . , η

k

in

Ω

, the map from

M

k(

N

)

to

M

k(

B

(

E

))

denedbythe

k

×

k

matrixof maps,

((

id

−

Ad

(

Z

(

η

i

∗

)

, Z

(

η

∗

j

)))

◦

(

id

−

θ

η

i

,η

j

)

−

1

)

,

Note that,when

M

=

E

=

B

(

E

)

and

σ

isthe identityrepresentationof

B

(

E

)

on

E

,

σ

(

M

)

′

is

C

I

E

,

E

σ

isisomorphito

C

and

D

(

E

σ

)

∗

anbeidentiedwith the open unit dis

D

of

C.

In this ase our denition reovers the lassial Shur lass funtions. More preisely, these funtions are usually dened as analyti funtions

Z

from anopensubset

Ω

of

D

into the losed unit ballof

B

(

E

)

butitisknownthatsuhfuntionsarepreiselythoseforwhihthePik

kernel

k

Z

(

z, w

) = (

I

−

Z

(

z

)

Z

(

w

)

∗

)(1

−

z

w

¯

)

−

1

is positivesemi-denite on

Ω

. The argument of [31, Remark 5.4℄ shows that the positivity of this kernelis equivalent, in ourase, to the onditionof Denition 3.1. This ondition, in turn, isthesameasassertingthatthekernel

k

Z(

ζ

∗

, ω

∗

) := (

id

−

Ad

(

Z

(

ζ

∗

)

, Z

(

ω

∗

))

◦

(

id

−

θ

ζ,ω

)

−

1

(4)

isaompletelypositivedenitekernelon

Ω

∗

in thesenseofDenition3.2.2of [14℄.

Forthesakeofompleteness,wereordthefatthat everyelementof

H

∞

(

E

)

ofnormat mostonegivesrisetoaShurlassoperatorfuntion.

Theorem 3.2

Let

E

bea

W

∗

-orrespondeneoveravonNeumannalgebra

M

and let

σ

be afaithful normal representation of

M

in

B

(

H

)

for some Hilbert spae

H

. If

X

isanelementof

H

∞

(

E

)

ofnormatmostone,thenthefuntion

η

∗

→

X

b

(

η

∗

)

dened in Remark 2.14 is a Shur lass operator funtion on

D

((

E

σ

))

∗

with valuesin

B

(

H

)

.

Proof.

Onesimplytakes

B

i

=

I

forall

i

and

C

i

=

X

b

(

η

∗

i

)

in Theorem2.18.

Theorem 3.3

Let

E

bea

W

∗

-orrespondeneoveravonNeumannalgebra

M

. Suppose also that

σ

a faithful normal representation of

M

on a Hilbert spae

E

and that

q

1

and

q

2

are projetions in

σ

(

M

)

. Finally, suppose that

Ω

is a

subset of

D

((

E

σ

))

and that

Z

is a Shur lass operator funtion on

Ω

∗

with values in

q

2

B

(

E

)

q

1

. Then thereisaHilbert spae

H

,anormalrepresentation

τ

of

N

:=

σ

(

M

)

′

on

H

and operators

A, B, C

and

D

fullling the following onditions:

(i) Theoperator

A

lies in

q

2

σ

(

M

)

q

1

.

(ii) Theoperators

C

,

B

,and

D

,areinthespaes

B

(

E

1

, E

σ

⊗

τ

H

)

,

B

(

H,

E

2

)

, and

B

(

H, E

σ

⊗

τ

H

)

,respetively,andeahintertwinestherepresentations

of

N

=

σ

(

M

)

′

on the relevant spaes (i.e. , for every

S

∈

N

,

CS

=

(

S

⊗

I

H

)

C

,

Bτ

(

S

) =

SB

and

Dτ

(

S

) = (

S

⊗

I

H)

D

).

(iii) Theoperator matrix

V

=

A

B

C

D

,

(5)

viewedas anoperator from

E

1

⊕

H

to

E

2

⊕

(

E

σ

⊗

τ

H

)

, isaoisometry,

(iv) For every

η

∗

in

Ω

∗

,

Z

(

η

∗

) =

A

+

B

(

I

−

L

∗

η

D

)

−

1

L

∗

η

C

(6)

where

L

η

:

H

→

E

σ

⊗

H

is dened by the formula

L

η

h

=

η

⊗

h

(so

L

∗

η

(

θ

⊗

h

) =

τ

(

h

η, θ

i

)

h

).

Remark 3.4

BeforegivingtheproofofTheorem3.3,wewanttonotethatthe resultbears astrong resemblane tostandardresultsin the literature. Weall speial attention to [1 , 2, 7, 9 , 10 , 11 , 12 , 13℄. Indeed, we reommend [7 ℄, whih is a survey that explains the general strategy for proving the theorem. What isnovel inourapproahisthe adaptation of the resultsin the literature toaommodateompletelypositive denitekernels.

Sinethe matrixin equation(5) andthefuntion inequation (6)arefamiliar onstrutsinmathematialsystemstheory,morepartiularlyfrom

H

∞

-ontrol theory(see,e.g.,[38℄),weadoptthefollowingterminology.

Definition 3.5

Let

E

bea

W

∗

-orrespondeneover avon Neumannalgebra

M

. Supposethat

σ

isafaithfulnormalrepresentation of

M

onaHilbertspae

E

and that

q

1

and

q

2

are projetions in

σ

(

M

)

. Then an operator matrix

V

=

A

B

C

D

,where the entries

A

,

B

,

C

,and

D

,satisfyonditions

(

i

)

and

(

ii

)

of Theorem 3.3for somenormal representation

τ

of

σ

(

M

)

′

on aHilbert spae

H

,isalleda systemmatrixprovided

V

isaoisometry (that isunitary,if

E

is full). If

V

is a system matrix, then the funtion

A

+

B

(

I

−

L

∗

η

D

)

−

1

L

∗

η

C

,

η

∗

∈

D

(

E

σ

)

∗

isalledthe transferfuntion determinedby

V

.

Proof.

Aswejustremarked,thehypothesisthat

Z

isaShurlassfuntion on

Ω

∗

meansthat thekernel

k

Z

in equation(4)isompletelypositivedenite inthesenseof[14℄. Consequently,wemayapplyTheorem3.2.3of[14℄,whihis alovelyextensionofKolmogorov'srepresentationtheoremforpositivedenite kernels,to ndan

N

-

B

(

E

)

W

∗

-orrespondene

F

andafuntion

ι

from

Ω

∗

to

F

suhthat

F

isspannedby

N ι

(Ω

∗

)

B

(

E

)

andsuhthatforevery

η

1

and

η

2

in

Ω

∗

andevery

a

∈

N

,

(

id

−

Ad

(

Z

(

η

∗

1

)

, Z

(

η

2

∗

)))

◦

(

id

−

θ

η

1

,η

2

)

−

1

(

a

) =

h

ι

(

η

1

)

, aι

(

η

2

)

i

.

Itfollowsthatforevery

b

∈

N

and every

η

1

, η

2

in

Ω

∗

,

b

−

Z

(

η

∗

1

)

bZ

(

η

∗

2

)

∗

=

h

ι

(

η

1

)

, bι

(

η

2

)

i − h

ι

(

η

1

)

,

h

η

1

, bη

2

i

ι

(

η

2

)

i

=

h

ι

(

η

1

)

, bι

(

η

2

)

i − h

η

1

⊗

ι

(

η

1

)

, bη

2

⊗

ι

(

η

2

)

i

.

Thus,

Set

G

1

:=

span

{

bZ

(

η

∗

)

∗

q

2

T

⊕

bι

(

η

)

q

2

T

|

b

∈

N, η

∈

Ω

∗

, T

∈

B

(

E

)

}

and

G

2

:=

span

{

bq

2

T

⊕

(

bη

⊗

ι

(

η

)

q

2

T

)

|

b

∈

N, η

∈

Ω

∗

, T

∈

B

(

E

)

}

.

Then

G

1

is a sub

N

-

B

(

E

)

W

∗

-orrespondene of

B

(

E

)

⊕

F

(where we use the assumption that

q

2

Z

(

η

∗

) =

q

2

Z

(

η

∗

)

q

1

) and

G

2

is a sub

N

-

B

(

E

)

W

∗

-orrespondeneof

B

(

E

)

⊕

(

E

σ

⊗

N

F

)

. (Thelosureinthedenitionsof

G

1

, G

2

is

inthe

σ

-topologyof[6℄. Itthenfollowsthat

G

1

and

G

2

are

W

∗

-orrespondenes [6,Consequenes1.8(i)℄). Dene

v

:

G

1

→

G

2

bytheequation

v

(

bZ

(

η

∗

)

∗

q

2

T

⊕

bι

(

η

)

q

2

T

) =

bq

2

T

⊕

(

bη

⊗

ι

(

η

)

q

2

T

)

.

It followsfrom (7)that

v

is anisometry. Itis alsolearthat itisabimodule map. Wewrite

P

i

fortheorthogonalprojetiononto

G

i

,

i

= 1

,

2

and

V

˜

forthe map

˜

V

:=

P

2

vP

1

:

q

1

B

(

E

)

⊕

F

→

q

2

B

(

E

)

⊕

(

E

σ

⊗

N

F

)

.

Then

V

˜

is apartial isometry and, sine

P

1

, v

and

P

2

are all bimodule maps (seeLemma 2.19),sois

V

˜

. Wewrite

V

˜

matriially:

˜

V

=

α β

γ

δ

,

where

α

:

q

1

B

(

E

)

→

q

2

B

(

E

)

,

β

:

F

→

q

2

B

(

E

)

,

γ

:

q

1

B

(

E

)

→

E

σ

⊗

F

and

δ

:

F

→

E

σ

⊗

F

and all these maps are bimodule maps. Let

H

0

be the Hilbert spae

F

⊗

B

(

E

)

E

and note that

B

(

E

)

⊗

B

(

E

)

E

is isomorphito

E

(and theisomorphismpreservestheleft

N

-ation). Tensoringontherightby

E

(over

B

(

E

)

)weobtainapartialisometry

V

0

:=

A

0

B

0

C

0

D

0

:

E

1

H

0

→

E

2

E

σ

⊗

H

0

.

Here

A

0

=

α

⊗

I

E

,

B

0

=

β

⊗

I

E

,

C

0

=

γ

⊗

I

E

and

D

0

=

δ

⊗

I

E

. Thesemaps are well dened beausethe maps

α, β, γ

and

δ

areright

B

(

E

)

-module maps. Sinethesemaps arealso left

N

-modulemaps,soare

A

0

, B

0

, C

0

and

D

0

. Bythedenitionof

V

0

,itsinitialspaeis

G

1

⊗ E

anditsnalspaeis

G

2

⊗ E

. Infat,

V

0

induesanequivalene ofthe representationsof

N

on

G

1

⊗ E

and on

G

2

⊗ E

.

Itwillbeonvenienttousethenotation

K

1

N

K

2

iftheHilbertspaes

K

1

and

K

2

arebothleft

N

-modulesandtherepresentationof

N

on

K

1

isequivalenttoa

Usingthisnotation,weanwrite

G

1

⊗ E ≃

N

G

2

⊗ E

. Form

M

2

:= (

E

2

⊕

(

E

σ

⊗

H

0

))

⊖

(

G

2

⊗ E

)

, whih isaleft

N

-module, andnote that

L

:=

F

(

E

σ

)

⊗ M

2

also is a left

N

-module, where the representation of

N

on

L

is the indued representation. Sine

L

=

F

(

E

σ

)

⊗ M

2

=

L

∞

n

=0

((

E

σ

)

⊗n

⊗

(

M

2

))

,itisevident that

(

E

σ

⊗

L

)

⊕ M

2

≃

N

L

. Indeed,theisomorphismsarejustthenaturalones that givetheassoiativityof thetensorprodutsinvolved. Thus,

E

2

⊕

(

E

σ

⊗

(

H

0

⊕

L

)) =

E

2

⊕

(

E

σ

⊗

H

0

)

⊕

(

E

σ

⊗

L

) =

G

2

⊗E ⊕M

2

⊕

E

σ

⊗

L

≃

N

G

2

⊗E ⊕

L

≃

N

G

1

⊗ E ⊕

L

N

E

1

⊕

(

H

0

⊕

L

)

. Consequently,weobtainaoisometrioperator

V

:

E

1

⊕

(

H

0

⊕

L

)

→ E

2

⊕

E

σ

⊗

(

H

0

⊕

L

)

thatintertwinestherepresentations

of

N

and extends

V

0

. Note that,if

V

0

were knownto beanisometry (sothat

G

2

⊗ E ≃

N

G

1

⊗ E

=

E

1

⊕

H

0

),thenwewouldhaveequivaleneaboveand

V

anbehosentobeunitary.

Assume that

E

isfull. We alsowrite

M

1

for

(

E

1

⊕

H

0

)

⊖

G

1

⊗ E

. Sine

E

is full,therepresentation

ρ

of

N

on

E

σ

⊗ E

isfaithful(Lemma2.20)anditfollows that everyrepresentationof

N

isquasiequivalenttoasubrepresentationof

ρ

. Write

E

∞

forthediretsumofinnitelymanyopiesof

E

. Then

E

σ

⊗E

∞

isthe

diretsumofinnitelymanyopiesof

E

σ

⊗E

and,thus,everyrepresentationof

N

isequivalenttoasubrepresentationoftherepresentationof

N

on

E

σ

⊗ E

∞

.

Inpartiular, weanwrite

M

1

⊕ E

∞

N

E

σ

⊗ E

∞

. Thus

E

1

⊕

(

H

0

⊕ E

∞) =

(

G

1

⊗ E

)

⊕ M

1

⊕ E

∞

N

E

2

⊕

(

E

σ

⊗

H

0

)

⊕

(

E

σ

⊗ E

∞) =

E

2

⊕

(

E

σ

⊗

(

H

0

⊕ E

∞))

.

So,replaing

H

0

by

H

0

⊕ E

∞

,weanreplae

V

0

byanisometryand,usingthe argumentjustpresented,weonludethattheresulting

V

isaunitaryoperator intertwiningtherepresentationsof

N

andextending

V

0

.

Sowelet

V

betheoisometryjust onstruted(andtreatitasunitarywhen

E

isfull). Writing

H

:=

H

0

⊕

L

,weanexpress

V

inthematriialform asin

part(iii) ofthestatementof thetheorem. Conditions (i)and(ii) thenfollow from thefat that

V

intertwinestheindiatedrepresentationsof

N

. It isleft to prove(iv).

Setting

b

=

T

=

I

inthedenitionof

v

aboveandwriting

v

inamatriialform weseethat

α β

γ

δ

Z

(

η

∗

)

∗

q

2

ι

(

η

)

q

2

=

q

2

η

⊗

ι

(

η

)

q

2

.

Tensoringby

I

E

ontherightandidentifying

B

(

E

)

⊗

B

(

E

)

E

with

E

asabove,we ndthat

A

0

B

0

C

0

D

0

Z

(

η

∗

)

∗

g

ι

(

η

)

⊗

g

=

g

η

⊗

(

ι

(

η

)

⊗

g

)

,

for

g

∈ E

2

. Sine

A, B, C

and

D

extend

A

0

, B

0

, C

0

and

D

0

respetively, we andropthesubsript

0

. Wealsousethefat that thematrixweobtainisa oisometry, andthus its adjointequals itsinverse onits range. We onlude

that

A

∗

C

∗

B

∗

D

∗

g

η

⊗

(

ι

(

η

)

⊗

g

)

=

Z

(

η

∗

)

∗

g

ι

(

η

)

⊗

g

.

(8)

Thus

ι

(

η

)

⊗

g

=

B

∗

g

+

D

∗

(

η

⊗

(

ι

(

η

)

⊗

g

)) =

B

∗

g

+

D

∗

L

η(

ι

(

η

)

⊗

g

)

and

Combiningthisequalitywiththeotherequationthatwegetfrom(8),wehave

Z

(

η

∗

)

∗

g

=

A

∗

g

+

C

∗

L

η

(

I

−

D

∗

L

η)

−

1

B

∗

g , g

∈ E

.

Takingadjointsyields(iv).

Thus,Theorem3.3assertsthateveryShurlassfuntiondeterminesasystem matrixwhose transferfuntion representsthefuntion. The systemmatrixis notuniqueingeneral,butastheproofofTheorem3.3shows,itarisesthrough aseries of naturalhoies. Of ourse,equation (6) suggeststhat everyShur lassfuntionrepresentsanelementin

H

∞

(

E

)

. Thisisindeedthease,asthe followingonverseshows.

Theorem 3.6

Let

E

be a

W

∗

-orrespondene over a

W

∗

-algebra

M

, and let

σ

be a faithful normal representation of

M

on a Hilbert spae

E

. If

V

=

A

B

C

D

is a system matrix determined by a normal representation

τ

of

N

:=

σ

(

M

)

′

on a Hilbert spae

H

, then there is an

X

∈

H

∞

(

E

)

,

k

X

k ≤

1

, suhthat

b

X

(

η

∗

) =

A

+

B

(

I

−

L

∗

η

D

)

−

1

L

∗

η

C

,

for all

η

∗

∈

D

(

E

σ

)

∗

and, onversely, every

X

∈

H

∞

(

E

)

,

k

X

k ≤

1

, may be representedinthis fashion for asuitablesystemmatrix

V

=

A

B

C

D

.

Proof.

Forevery

n

≥

0

wedeneanoperator

K

n

from

E

to

(

E

σ

)

⊗n

⊗ E

as follows. For

n

= 0

,weset

K

0

=

A

-anoperatorin

B

(

E

)

. For

n

= 1

,wedene

K

1

,mapping

E

to

E

σ

⊗ E

,tobe

(

I

1

⊗

B

)

C

,whereforall

k

≥

1

,

I

k

denotesthe identityoperatoron

(

E

σ

)

⊗k

. For

n

≥

2

,weset

K

n

:= (

I

n

⊗

B

)(

I

n−

1

⊗

D

)

· · ·

(

I

1

⊗

D

)

C.

Note,rst,thatitfollowsfrom thepropertiesof

A, B, C

and

D

that,forevery

n

≥

0

and every

a

∈

N

,

K

n

a

= (

ϕ

n(

a

)

⊗

I

E

)

K

n

where

ϕ

n

denes the left

multipliationon

(

E

σ

)

⊗n

. Thus,writing

ι

fortheidentityrepresentationof

N

on

E

,

K

n

lies inthe

ι

-dual of

(

E

σ

)

⊗n

whih, byTheorem 3.6and Lemma3.7 of [31℄, is isomorphi to

E

⊗n

. Hene, for every

n

≥

0

,

K

n

denes a unique element

ξ

n

in

E

⊗n

.

For every

n

≥

0

and

η

∈

E

σ

we shall write

L

n(

η

)

for the operator from

(

E

σ

)

⊗n

⊗ E

to

(

E

σ

)

⊗

(

n

+1)

⊗ E

given by tensoring on the left by

η

. Also note that, for

k

≥

1

and

n

≥

0

,

I

k

⊗

K

n

is an operator from

(

E

σ

)

⊗k

⊗ E

to

(

E

σ

)

⊗

(

k

+

n

)

⊗ E

. With thisnotation,itiseasytoseethat,forall

k

≥

1

and

n

≥

0

,

(

I

k

+1

⊗

K

n)

L

k(

η

) =

L

k

+

n

(

η

)(

I

k

⊗

K

n)

.

(9)

Note, too,thatweanwrite

andeveryoperatoron

F

(

E

σ

)

⊗E

anbewritteninamatriialformwithrespet to this deomposition(with indiesstarting at

0

). Forevery

m

,

0

≤

m

≤ ∞

, welet

S

m

betheoperatordenedbythematrixwhose

i, j

entryis

I

j

⊗

K

i−j

, if

0

≤

j

≤

i

≤

m

,andis

0

otherwise. (For

m

=

∞

,itisnotlearyetthat the matrixsoonstrutedrepresentsabounded operator,but thiswill beveried later).

Sofarwehavenotusedtheassumptionthat

V

isaoisometry. Butifwetake this into aount, form the produt

V V

∗

, and set it equalto

I

E⊕

(

E

σ

⊗H

)

, we ndthat

I

E

−

AA

∗

=

BB

∗

(10)

CC

∗

=

I

E

σ

⊗

τ

H

−

DD

∗

(11)

AC

∗

=

−

BD

∗

(12)

Welaimthat,for

1

≤

j

≤

i

≤

m

,thefollowingequationshold,

(

I

−

S

m

S

m

∗

)i,j

= (

I

i

⊗

B

)(

I

i−

1

⊗

D

)

· · ·

DD

∗

· · ·

(

I

j−

1

⊗

D

∗

)(

I

j

⊗

B

∗

);

(13)

that for

0

< i

≤

m

,

(

I

−

S

m

S

∗

m

)i,

0

= (

I

i

⊗

B

)(

I

i−

1

⊗

D

)

· · ·

DB

∗

,

(14)

andthat for

i

=

j

= 0

,

(

I

−

S

m

S

m

∗

)

0

,

0

=

BB

∗

.

(15)

Equation(15)followsimmediatelyfrom(10)sine

(

S

m)

0

,

0

=

A

. For

0

< i

≤

m

weompute

(

I

−

S

m

S

∗

m

)i,

0

=

−

(

S

m)i,

0

(

S

m)

∗

0

,

0

=

−

(

I

i

⊗

B

)(

I

i−

1

⊗

D

)

· · ·

(

I

1

⊗

D

)

CA

∗

= (

I

i

⊗

B

)(

I

i−

1

⊗

D

)

· · ·

(

I

1

⊗

D

)

DB

∗

where, in the last equalitywe used (12). It is left to prove(13). Letus write

R

i,j

for theleft hand sideof (13). (For

j

= 0

< i

we have

R

i,

0

= (

I

i

⊗

B

)(

I

i−

1

⊗

D

)

· · ·

DB

∗

and when bothare

0

,

R

0

,

0

=

BB

∗

). Wehave

K

0

K

∗

0

=

AA

∗

=

I

−

BB

∗

=

I

−

R

0

,

0

R

∗

0

,

0

. For

0 =

j < i

≤

m

wehave

K

i

K

∗

0

= (

I

i

⊗

B

)(

I

i−

1

⊗

D

)

· · ·

(

I

1

⊗

D

)

CA

∗

=

−

(

I

i

⊗

B

)(

I

i−

1

⊗

D

)

· · ·