Paul S. Muhly

(1)

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

^be^a

W ^∗

-orrespondeneoveravonNeumannalge- bra

M

^and^let

H ^∞ (E)

^be^theâssoiated^Hardyâlgebra.Îf

σ

^is^a^faith-

ful normalrepresentationof

M

^on^a^Hilbert ^spae

H

^, ^then^one^may

form thedual orrespondene

E ^σ

^and^represent^elementsⁱⁿ

H ^∞ (E)

as

B(H )

^-valued^funtions^on^the^unit^ball

D (E ^σ ) ^∗

^. ^The^funtions^that

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)

ⁱⁿ^terms^of^speial^Möbiustransformationson

D (E ^σ )

^.

Partiularattentionisdevotedtothe

H ^∞

^-algebras^that^are^assoiated

tographs.

2000 Mathematis Subjet Classiation: 46E22, 46E50, 46G20,

46H15,46H25,46K50,46L08,46L89,

Keywords and Phrases: Hardy Algebras, Tensor Algebras, Shur

lassfuntions,

W ^∗

-orrespondene,nonommutativerealizationthe- ory,Möbiustransformations,freesemigroupalgebras,graphalgebras,

Nevanlinna-Pikinterpolation

1

SupportedinpartbygrantsfromtheNationalSieneFoundationand fromthe U.S.-

IsraelBinationalSieneFoundation.

2

SupportedinpartbytheU.S.-IsraelBinationalSieneFoundationandbytheFundfor

thePromotionofResearhattheTehnion.

(2)

1 Introduction

Let

M

^be^a

W ^∗

^-algebra^and^let

E

^be^a

W ^∗

-orrespondeneover

M

^. ^In^[31℄^we

builtanoperatoralgebrafromthisdatathatwealledtheHardyalgebraof

E

and whih wedenoted

H ^∞ (E)

^. ^If

M = E = C

-the omplexnumbers,then

H ^∞ (E)

îs^the^lassial^Hardyâlgebraônsistingôf âll^bounded ânalyti^fun-

tionsontheopen unit dis,

D

(seeExample 2.4below.) If

M = C

again,but

E = C ⁿ

, then

H ^∞ (E)

^is ^the^free ^semigroup^algebra

L n

^studied ^by ^Davidson

andPitts[17℄,Popesu[32℄andothers(seeExample2.5.) Oneoftheprinipal

disoveriesmadein [31℄,andthesoureofinspirationforthepresentpaper,is

thatattahedtoeahfaithfulnormalrepresentation

σ

^of

M

^thereîsâ^dualôr-

respondene

E ^σ

^, ^whih^is^a

W ^∗

-orrespondeneovertheommutantof

σ(M )

^,

σ(M ) ^′

^, ând ^theêlementsôf

H ^∞ (E)

^dene^funtions ôn^the ôpenûnit ^ballôf

E ^σ

^,

D (E ^σ )

^. ^Further,^the^valuedistributiontheoryofthesefuntionsturnsout 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

^is^the

1

-dimensional representationof

C

onitself,then

E ^σ

^is

C

again. Therepresentationof

H ^∞ (E)

ⁱⁿ^terms^of^fun-

tionson

D (E ^σ ) = D

isjusttheusualwaywethink of

H ^∞ (E)

^. ^In^this^setting,

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

σ

is arepresentationof

C

on aHilbert spae

H

^,

dim(H ) > 1

^, ^then

E ^σ

^may^be

identiedwith

B(H)

^and^then

D (E ^σ )

^beomes^the^spae^of^strit^ontrations

on

H

^, î.e., âll ^those ôperators ôf ^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 ^through^the

usual holomorphi funtional alulus. OurNevanlinna-Piktheorem givesa

solutiontoproblems suh asthis: given

k

^operators

T 1 , T 2 , . . . , T k

^all^of^norm

lessthan

1

^and

k

^operators,

A 1 , A 2 , . . . , A k

^,^determine^the^irumstanes^under

whihoneanndaboundedanalytifuntion

f

ôn^theôpenûnit^disôf^sup

norm at most

1

^suh ^that

f (T i ) = A i

^,

i = 1, 2, . . . , k

^(See ^[31, ^Theorem^6.1℄.)

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

thatlieinamultiplierspaestud- iedin detailbyArveson[5℄. Moreaurately,thefuntionalrepresentationof

H ^∞ (E) = L n

ⁱⁿ ^termsôf^these^funtions êxpresses^this^spaeâsâ^quotient ôf

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 ^be^salar multipliation onanauxiliaryHilbert spae.

Ourobjetiveinthepresentnoteisbasiallytwofold. First,wewishtoidentify

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those funtionson

D (E ^σ )

^that ârise^fromêvaluatingêlementsôf

H ^∞ (E)

^. ^F^or

this purpose, weintrodueafamilyof funtions on

D (E ^σ )

^that ^we^all^Shur

lassoperatorfuntions(seeDenition3.1). Roughlyspeaking,thesefuntions

aredened sothat aPik-likekernelthatonemayattahtoeahone isom-

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 ^points^of

D (E ^σ )

ândônversely, êah ^funtion ⁱⁿ

H ^∞ (E)

^has^a

representationintermsofatransferfuntion. Themeaningofthenotationwill

bemadepreisebelow,but weuseitheretohighlighttheonnetionbetween

our analysis and realization theory as it omes from mathematial systems

theory. Thepointto keepinmindis thatfuntionson

D (E ^σ )

^that^ome^from

elementsof

H ^∞ (E)

âre^not, â^priori, ânalytiⁱⁿ ânyôrdinary^senseând îtîs

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

⁾^at^points^of

D (E ^σ )

^. ^The^fat^that^eah^suh^funtion^may

berealizedasatransferfuntion exhibitsasurprisinglevelofanalytiitythat

isnotevidentin thedenition of

H ^∞ (E)

^.

Ourseondobjetiveisto onnettheusualholomorphipropertiesof

D (E ^σ )

with the automorphisms of

H ^∞ (E)

^. ^As ^a^spae,

D (E ^σ )

îs ^theûnit ^ball ôf â

J ^∗

^-triple^system. Consequently,everyholomorphiautomorphismof

D (E ^σ )

^is

theomposition ofaMöbiustransformation andalinearisometry [20℄. Eah

ofthese implementsanautomorphismofthealgebraofallbounded, omplex-

valued analyti funtions on

D (E ^σ )

^, ^but ⁱⁿ ôur ^setting ônly êrtain ôf ^them

implement automorphisms of

H ^∞ (E)

^- ^those ^for ^whih ^the ^Möbius ^part ^is

determinedbyaentralelementof

E ^σ

^(see^Theorem^4.21).^Our^proof^requires

thefatthattheevaluationoffuntionsin

H ^∞ (E)

^(of^norm^at^most

1

⁾^at^points

of

D (E ^σ )

âre^preisely^the^Shur^lassôperator^funtionsôn

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 )

^. În ^the ^last ^setion, ^we âpply ôur ^general ânalysis ôf ^the

automorphisms of

H ^∞ (E)

^to ^the ^speial ^ase ^of

H ^∞

^-algebras ^oming ^from

direted graphs.

Inonludingthisintrodution,wewanttonotethatapreprintofthepresent

paperwasposted on thearXiv on June 27, 2006. Reently, inspired in part

by ourpreprint, Ball,Biswas,Fang andter Horst [8℄ wereable to realizethe

Fokspaethat wedesribehereintermsofthetheoryofompletelypositive

denitekernelsadvaned byBarreto,Bhat,LiebsherandSkeide[14℄thatwe

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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 ^∗

^-modules^that^we^shall^use. ^Let

A

^be^a

C ^∗

^-algebra^and

E

^be^a^right^module^over

A

^endowed^with^abi-additive map

h·, ·i : E × E → A

^(referred ^to ^as^an

A

^-valued^inner ^produt) ^suh ^that,

for

ξ, η ∈ E

^and

a ∈ A

^,

hξ, ηai = hξ, ηia

^,

hξ, ηi ^∗ = hη, ξi

^, ^and

hξ, ξi ≥ 0

^,^with

hξ, ξi = 0

^only ^when

ξ = 0

^. ^Also,

E

îs âssumed ^to ^be ômpleteⁱⁿ ^the^norm

kξk := khξ, ξik ^1/2

^. ^We ^write

L(E)

^for ^the ^spae ^of ^ontinuous, adjointable,

A

^-module^maps^on

E

^. Îtîs^known^to^beâ

C ^∗

^-algebra. ^If

M

^is^a^von^Neumann

algebraandif

E

^is^a^Hilbert

C ^∗

^-module^over

M

^,^then

E

^is^said^to^be^self-dualⁱⁿ

aseeveryontinuous

M

^-module^map^from

E

^to

M

îs^given^byânînner^produt

withanelementof

E

^. ^Let

A

^and

B

^be

C ^∗

^-algebras. ^A

C ^∗

^-orresp^ondene^from

A

^to

B

^is ^a^Hilbert

C ^∗

^-module

E

^over

B

êndowed ^withâ ^struture ôf â^left

moduleover

A

^via^anondegenerate

∗

-homomorphism

ϕ : A → L(E)

^.

Whendealingwithaspei

C ^∗

-orrespondene,

E

^,^from^a

C ^∗

^-algebra

A

^to^a

C ^∗

^-algebra

B

^, ^it^will ^be ^onvenient^sometimes ^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 ^the^balaned^tensor ^produt,

E ⊗ B F

^is ^an

A, C

^-bimodule ^that

arriestheinner produtdenedbytheformula

hξ 1 ⊗ η 1 , ξ 2 ⊗ η 2 i E⊗ B F := hη 1 , ϕ(hξ 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

^be^von^Neumann^algebras^and^let

E

^be^a^Hilbert

C ^∗

^-module^over

N

^. ^Then

E

îsâl^le^dâ ^Hilbert

W ^∗

^-module^over

N

ⁱⁿ^ase

E

^is

self-dual. Themodule

E

îsâlledâ

W ^∗

-orrespondenefrom

M

^to

N

ⁱⁿ^ase

E

isaself-dual

C ^∗

-orrespondenefrom

M

^to

N

^suh^that^the

∗

-homomorphism

ϕ : M → L(E)

^, ^giving ^the ^left ^module ^struture ^on

E

^, ^is ^normal. ^If

M = N

weshallsay that

E

^is^a

W ^∗

^-orresp^ondene ^over

M

^.

Wenote thatif

E

^is^a^Hilbert

W ^∗

^-moduleôverâ^von^Neumannâlgebra,^then

L(E)

îs^notônlyâ

C ^∗

^-algebra,^butîs âlsoâ

W ^∗

^-algebra. ^Thus^it^makes^sense

to talkaboutnormalhomomorphismsinto

L(E)

^.

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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

hΨ(e), Ψ(f )i = σ(he, 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

^-topology^of

[6℄ in order to get a

W ^∗

-orrespondene. However, we will not distinguish notationallybetweenthe

C ^∗

^-tensor^produt^and^the

W ^∗

^-tensor^produt. ^Note

alsothatgivena

W ^∗

-orrespondene

E

^over

M

^and^a^Hilbert^spae

H

^equipped

with anormalrepresentation

σ

^of

M

^,^we^an^form ^the^Hilbert ^spae

E ⊗ σ H

by dening

hξ 1 ⊗ h 1 , ξ 2 ⊗ h 2 i = hh 1 , σ(hξ 1 , ξ 2 i)h 2 i

^. ^Thus,

H

îs ^viewed âs â

orrespondenefrom

M

^to

C

via

σ

^and

E ⊗ σ H

^is^just ^the^tensor ^produt^of

E

^and

H

^as

W ^∗

-orrespondenes.

Note alsothat,given anoperator

X ∈ L(E)

ândân ôperator

S ∈ σ(M ) ^′

^, ^the

map

ξ ⊗ h 7→ Xξ ⊗ Sh

^denes â ^bounded ôperator ôn

E ⊗ σ H

^denoted ^by

X ⊗ S

^. ^Therepresentationof

L(E)

^that ^results^when^one ^lets

S = I

^,^is^alled

the representation of

L(E)

^indued ^by

σ

ând îs ôften ^denoted ^by

σ ^E

^. ^The

omposition,

σ ^E ◦ ϕ

^is^arepresentationof

M

^whih^we^shallâlso^sayîsîndued

by

σ

^,^but ^we^shall^usually ^denote^it^by

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

^alled^the^(full)^Fok^spae ^over

E

^. ^The^ations

of

M

ôn^the ^left ând ^right ôf

F(E)

âre ^the ^diagonal âtions ând, ^when ît îs

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), · · · }

^, ^where^for ^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 ^the^norm-losed

subalgebra of

L(F(E))

^generated ^by

ϕ ∞ (M )

^and ^the ^reation ^operators

T ξ

^,

ξ ∈ E

^,^dened ^by^the^formula

T ξ η = ξ ⊗ η

^,

η ∈ F (E)

^. ^We^refer^the^reader^to

[28℄forthebasifats about

T + (E)

^.

Definition 2.3

^{([31 ℄)}^Given^a

W ^∗

^-orresp^ondene

E

^over^the ^von ^Neumann

algebra

M

^,^theûltraweak^losureôf^the^tensorâlgebraôf

E

^,

T + (E)

^,ⁱⁿ

L(F(E))

^,

isalledthe HardyAlgebraof

E

^,^and^is^denoted

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 )

^. ^The^tensor âlgebra^then îsîsomorphi

to the dis algebra

A( D )

^viewed ^asmultipliation operatorson

H ² ( T )

^and^the

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 ⁺ _n

is the free semigroup on

n

generators. The tensor

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algebrathen iswhat Popesu refersto as the non ommutative dis algebra

A n

ând^the^Hardyâlgebraîsîts

w ^∗

^-losure. ^It^was^studied^by^Popesu^{[32 ℄}^and

by DavidsonandPittswho denoteditby

L n

^{[17 ℄.}

Weneedtoreviewsomebasifatsabouttherepresentationtheoryof

H ^∞ (E)

andof

T + (E)

^. ^See^[28,^31℄^for^more^details.

Definition 2.6

^Let

E

^be^a

W ^∗

-orrespondeneover avon Neumannalgebra

M

^. ^Then:

1. Aompletelyontrativeovariantrepresentationof

E

^on^a^Hilbert^spae

H

^is^a^pair

(T, σ)

^,^wher^e

(a)

σ

^is^a^normal

∗

-representationof

M

ⁱⁿ

B(H)

^.

(b)

T

îs â ^lineâr, ômpletely ôntrative ^map ^from

E

^to

B(H )

^that ^is

ontinuousinthe

σ

^-topology^of ^{[6 ℄}^on

E

^and^the ^ultraweak^topology

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(hξ, ηi)

⁽¹⁾

forall

ξ, η ∈ E

^.

Itshould benotedthat theoperatorspaestrutureon

E

^to^whih^Denition

2.6 refers is that whih

E

înherits ^when ^viewed âs â ^subspae ôf îts ^linking

algebra.

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

ovariant representation,

(T, σ)

^, ^of

E

ⁱⁿ

B(H )

îs ^given, ^then ît ^determines â

ontration

T ˜ : E ⊗ σ H → H

^dened ^by ^the ^formula

T(η ˜ ⊗ h) := T (η)h

^,

η ⊗ h ∈ E ⊗ σ H

^. ^The^operator

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℄.

Lemma 2.7

^The^map

(T, σ) → T ˜

îsâ^bijetion ^betweenâllômpletelyôntra-

tiveovariantrepresentations

(T, σ)

^of

E

^on^the^Hilbert^spae

H

^and^ontrative

operators

T ˜ : E ⊗ σ H → H

^that^satisfy^equation^(2). ^Given^suh^a

T ˜

^satisfying

this equation,

T

^,^dened^by ^the ^formula

T(ξ)h := ˜ T (ξ ⊗ h)

^, ^together ^with

σ

^is

a ompletely ontrative ovariant representation of

E

^on

H

^. ^F^urther,

(T, σ)

isisometri ifandonly if

T ˜

îsân îsometry.

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The importane of theompletely ontrative ovariantrepresentationsof

E

(or, equivalently, theintertwining ontrations

T ˜

âsâbove)îs ^that ^they^yield

allompletelyontrativerepresentationsofthetensoralgebra. Morepreisely,

wehavethefollowing.

Theorem 2.8

^Let

E

^be^a

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

îsâ^bijetion ^between ^the^set ôfâll ômpletelyôntrative

ovariant representations of

E

ând âll ômpletely ôntrative ^(algebra) ^repre-

sentationsof

T + (E)

^whose restritions to

ϕ ∞ (M )

âre ôntinuous^with ^respêt

tothe ultraweak topologyon

L(F(E))

^.

Definition 2.9

^If

(T, σ)

îs âômpletely ôntrativeôvariant representation of a

W ^∗

-orrespondene

E

ôverâ^von^Neumannâlgebra

M

^,^we^all ^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

Ôneôf^the^prinipal^diulties ône^faesⁱⁿ^dealing^with

T + (E)

and

H ^∞ (E)

^is^to^deide^when^the ^integrated^form,

σ × T

^,ôfâômpletelyôn-

trative ovariant representation

(T, σ)

^extends ^from

T + (E)

^to

H ^∞ (E)

^. ^This

problemarises alreadyinthesimplestsituation,vis. when

M = C = E

^. ^In^this

setting,

T

îs ^given ^by â^single ôntration ôperator ôn â^Hilbert ^spae,

T + (E)

is the dis algebra and

H ^∞ (E)

îs ^the ^spae ôf ^boundedânalyti ^funtions

onthe dis. Therepresentation

σ × T

^extends^from^the ^dis^algebra^to

H ^∞ (E)

preisely whenthere isnosingularparttothe spetral measureofthe minimal

unitary dilation of

T

^. ^We âre^not âware ôf âômparable^result ⁱⁿôur^general

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 Tk ˜ < 1

^then

σ × T

êxtends ^to â ûltra-

weakly ontinuousrepresentation of

H ^∞ (E)

^.

In[31℄weintroduedandstudiedtheoneptsofdualityandofpointevaluation

(forelementsof

H ^∞ (E)

^). ^These ^playâêntral^roleⁱⁿôurânalysis^here.

Definition 2.12

^Let

E

^be^a

W ^∗

M

^and^let

σ : M → B(H )

^be^a^faithful^normalrepresentationof

M

^on^a^Hilbert

spae

H

^. ^Then ^the

σ

^-dual ^of

E

^,^denoted

E ^σ

^,^is^dened^to^be

{η ∈ B(H, E ⊗ σ H ) | ησ(a) = (ϕ(a) ⊗ I)η, a ∈ M }.

(8)

An important feature of the dual

E ^σ

îs ^that ît îs â

W ^∗

-orrespondene, but overthe ommutant of

σ(M )

^,

σ(M ) ^′

^.

Proposition 2.13

^With ^respet^to^the ^ation^of

σ(M ) ^′

^and^the

σ(M ) ^′

^-valued

innerprodutdenedasfollows,

E ^σ

^beomes^a

W ^∗

-orrespondeneover

σ(M ) ^′

^:

For

Y

^and

X

ⁱⁿ

σ(M ) ^′

^,^and

η ∈ E ^σ

^,

X ·η·Y := (I⊗X )ηY

^,^and^for

η 1 , η 2 ∈ E ^σ

^,

hη 1 , η 2 i σ(M) ^′ := η ₁ ^∗ η 2

^.

Inthefollowingremarkweexplainwhatwemeanbyevaluatinganelementof

H ^∞ (E)

âtâ^pointⁱⁿ^theôpenûnit ^ballôf^the^dual.

Remark 2.14

^The împortaneôf ^this ^dual ^spâe,

E ^σ

^,îs ^that îtîs^losely ^re-

latedtotherepresentationsof

E

^. ^In^fat, ^the^operatorsⁱⁿ

E ^σ

^whose^norm^does

not exeed

1

âre^preisely ^the âdjoints ôf ^the ôperators ôf^the ^form

T ˜

^for ^a^o-

variant pair

(T, σ)

^. ^In ^partiular, ^every

η

ⁱⁿ ^the ôpenûnit^ball ôf

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)

^we^an ^apply^the representation assoiatedto

η

^to^it. ^The

resultingoperator in

B(H)

^will ^be^denoted^by

X b (η ^∗ )

^. ^Thus

X(η b ^∗ ) = (σ × η ^∗ )(X ).

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

B(H )

^-valued

funtion

X b : D (E ^σ ) ^∗ → B(H)

onthe open unitballof

(E ^σ ) ^∗

^. Ôneôf ôur^primaryôbjetivesîs^toûnderstand

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

^. ^Thenîtîsêasy^to^hek^that

E ^σ

^is^isomorphi^to

B(H)

^. ^Fix^an

X ∈ H ^∞ (E)

^. Âs^we^mentionedâbove,^this^Hardyâlgebraîs^the^lassial

H ^∞ ( T )

andweanidentify

X

^with^a^funtion

f ∈ H ^∞ ( T )

^. ^Given

S ∈ D (E ^σ ) = B(H )

^,

it is not hard to hek that

X(S b ^∗ )

^, âs ^dened âbove, îs ^the ôperator

f (S ^∗ )

denedthroughthe usualholomorphi funtionalalulus.

Example 2.16

În ^{[17 ℄} ^Davidson ând ^Pitts âssoiate ^toêvery êlement ôf ^the

freesemigroupalgebra

L n

^(seeÊxample^2.5)â^funtionôn^theôpenûnit^ballôf

C ⁿ

. Thisisaspeialaseofouranalysiswhen

M = C

,

E = C ⁿ

and

σ

îsâône

dimensionalrepresentationof

C

. Inthis ase

σ(M ) ^′ = C

and

E ^σ = C ⁿ

. Note, however, thatourdenitionallowsustotake

σ

^to^be^therepresentationof

C

on an arbitraryHilbert spae

H

^. ^If ^we ^do^so, ^then

E ^σ

^is^isomorphi ^to

B(H ) ⁽ⁿ⁾

^,

the ntholumn spae over

B(H )

^, ândêlements ôf

L n

^dene ^funtions^on ^the

openunit ballof this spaeviewed asaorrespondene over

B(H)

^with ^values

in

B(H )

^. ^This îs ^the ^pêrspetive âdopted ^by Constantinesu and Johnson in [16 ℄. In the analysis of [17℄ it is possible that a non zero element of

L n

^will

give rise tothe zero funtion. Weshall show in Lemma 3.8 that, byhoosing

an appropriate

H

^we^an ^insure ^that^this ^does ^not^happen.

(9)

Example 2.17

^Partôf ^the ^reent^work ôf^Popesuⁱⁿ ^[35℄^may^be âstⁱⁿ ôur

framework. We will follow his notation. Fix aHilbert spae

K

^, ^and ^let

E

^be

the olumn spae

B(K) ⁿ

^. ^Tâke, âlso, â^Hilbert ^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))

^is^naturally^isomorphi^to

B(H )

^,îtîsêasy^to

see that

E ^σ

îs^the ôlumn^spae ôver

B(H )

^,

B(H) ⁿ

^. ^It ^follows ^that

D (E ^σ )

^is

the openunit ballin

B(H) ⁿ

^. ^A^free^formal^power^series ^with^oeients^from

B(K)

^is ^a^formal ^series

F = P

α∈ F ⁺ _n A α ⊗ Z α

^where

F ⁺

n

^is^the ^free^semigroup

on

n

generators,the

A α

âreêlements ôf

B(K)

^and^where

Z α

^is ^the^monomial

in nonommuting indeterminates

Z 1

^,

Z 2

^,^.^.^.^,

Z n

^determined ^by

α

^. ^If

F

^has

radiusofonvergeneequalto

1

^,^then^one^may ^evaluate

F

ât^pôintsôf

D (E ^σ ) ^∗

togetafuntionon

D (E ^σ ) ^∗

^with^valuesⁱⁿ

B(K⊗H)

^,^vis.,

F ((S 1 , S 2 , · · · S n )) = P

α∈ F ⁺ _n A α ⊗ S α

^. ^See^{[35 ,} ^Theorem ^1.1℄. În^fat, ûnder âdditional restritions ontheoeients

A α

^,

F

^may^be^viewed^as^a^funtion

X

ⁱⁿ

H ^∞ (B(K) ⁿ )

ⁱⁿ^suh

awaythat

F ((S 1 , S 2 , · · · S n )) = X(S b 1 , S 2 , · · · S n )

ⁱⁿ^the^sense^denedⁱⁿ^{[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 ,}^Theorem^3.1℄. ^We^notedⁱⁿ^the ^preeding ^example^that

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 )

ât ^points ^the ûnit ^ball ôf

B(H ) ⁿ

^for ^al^l

possible

H

^'s. ^The ^sp^ae

B(K) ⁿ

^is^Morita^equivalent^to

C ⁿ

inthesenseof [30 ℄, at leastwhen

dim(K) < ∞

^, ând, ⁱⁿ ^that âse^the ^tensor âlgebras

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

âre ^Morita êquivalent êven ^when

dim(K) = ∞

^, ^and

thattogetherwith[30℄suggeststhat

H ^∞ (B(K) ⁿ )

^should^be^Morita^equivalent^to

H ^∞ (B(X ) ⁿ ₁ ) ≃ H ^∞ ( C ⁿ )

^. ^This^would^suggestânêven^loserônnetion^between

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 ôn ^the ôpen ûnit ^ball ôf ^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 )

^for^the^map^from

B(H)

^to^itself^that^sends

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

hη 1 , aη 2 i

^. ^That^is,

θ η 1 ,η 2 (a) := hη 1 , aη 2 i = η ₁ ^∗ aη 2

^,

a ∈ σ(M ) ^′

^.

Theorem 2.18

^{([31 ,}^Theorem^5.3℄)^Let

E

^be^a

W ^∗

-orrespondeneoveravon Neumannalgebra

M

^and^let

σ : M → B(H )

^be^a^faithful^normalrepresentation of

M

^on^a^Hilbert^spae

H

^. ^Fix

k

^p^oints

η 1 , . . . η k

ⁱⁿ^the^disk

D (E ^σ )

^and^hoose

(10)

2k

^operators

B 1 , . . . B k , C 1 , . . . C k

ⁱⁿ

B(H )

^. ^Then ^there^exists^an

X

ⁱⁿ

H ^∞ (E)

suhthat

kX k ≤ 1

^and

B i X(η b _i ^∗ ) = C i

for

i = 1, 2, . . . , k,

îf ând ônly îf ^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^,^given^by^the^matrix⁽³⁾ ^is^omputed^by^the^formula

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 ^∗

^-algebras^and ^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

^also^preserves

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 ^∗

^-orresp^ondene

E 0

^of

E

^that ^is ^losed ⁱⁿ ^the

σ

^-topôlogy ôf ^{[6 ,} ^Conse^quenes ^1.8 ^(ii)℄. Îf

P

is the orthogonal projetion from

E

^onto

E 0

^, ^then

P

îs â^bimodule ^map; î.e.,

P (aξb) = aP (ξ)b

^for ^all

a ∈ M

^and

b ∈ N

^.

Proof.

^It^sues^to^hek^that

P(eξ) = eP (ξ)

^for^all

ξ ∈ E

^and^projetions

e ∈ M

^. ^For

ξ, η ∈ E

^and^a^projetion

e ∈ M

^,^we^have

keξ + f ηk ² = kheξ, eξi + hf η, f ηik ≤ kheξ, eξik + khf η, f ηik = keξk ² + kf ηk ² ,

where

f = 1 − e

^. ^So,^for^every

λ ∈ R

wehave

(λ + 1) ² kf P (eξ)k ² = kf P (eξ + λf P (eξ))k ² ≤ keξ + λf P (eξ)k ²

≤ keξk ² + λ ² kf P (eξ)k ² .

Hene,forevery

λ ∈ R

,

(2λ + 1)kf P (eξ)k ² ≤ keξk ²

(11)

and,thus,

(I − e)P (eξ) = f P (eξ) = 0.

Replaing

e

^by

f = I − e

^we^get

eP((I − e)ξ) = 0

^and,^therefore,

P(eξ) = eP (eξ) = eP (ξ).

Sine

M

^is^spanned ^by^itsprojetions,wearedone.

Lemma 2.20

^Let

E

^be^a

W ^∗

^-orresp^ondene^over

M

^,^let

σ

^be^a^faithful^normal

representationof

M

^on^the^Hilbert^spae

E

^,^and^let

E ^σ

^be^the

σ

^-dual^orrespon-

deneover

N := σ(M ) ^′

^. ^Then

(i) The left ation of

N

^on

E ^σ

îs ^faithful îf ând ônly îf

E

^is ^ful^l ^(i.e. ^if

and only if the ultraweakly losed ideal generated by the inner produts

hξ 1 , ξ 2 i

^,

ξ 1 , ξ 2 ∈ E

^,îsâll ôf

M

^).

(ii) Theleft ationof

M

^on

E

îs^faithfulîfând ônlyîf

E ^σ

^is^full.

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

îf ând ônly îf ^for

all

η ∈ E ^σ

^and

g ∈ E

^,

(I ⊗ S)η(g) = 0

^. ^Sine ^the ^losed ^subspae^spanned

by the rangesof all

η ∈ E ^σ

îs âllôf

E ⊗ M E

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

equation

ξ ⊗ Sg = 0

^holding^for^all

g ∈ E

^and

ξ ∈ E

^. ^Sine

hξ ⊗ Sg, ξ ⊗ Sgi = hg, S ^∗ hξ, ξiSgi

^, ^we ^nd ^that

SE ^σ = 0

îf ând ônly îf

σ(hE, Ei)S = 0

^, ^where

hE, Ei

^is^the ultraweakly losedidealgenerated byall innerproduts. Ifthis idealisallof

M

^we^nd^that^the^equation

SE ^σ = 0

^implies^that

S = 0

^. ^In^the

other diretion,ifthisisnotthease, thenthisideal isoftheform

(I − q)M

forsomeentralnonzeroprojetion

q

^and^then

S = σ(q)

^is^dierent^from

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

^will^be^a^faithfulrepresentationof

M

^on^a^Hilbert

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 îts ôpen ûnit ^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 ^a^subset ^of

D (E ^σ )

^and ^let

Ω ^∗ = {ω ^∗ | ω ∈ Ω}

^. ^A

funtion

Z : Ω ^∗ → B(E)

^will ^be âlled â ^Shur ^lass ôperator ^funtion ^(with

valuesin

B(E)

⁾^if,^for^every

k

ândêvery^hoieôfêlements

η 1 , η 2 , . . . , η k

ⁱⁿ

Ω

^,

the map from

M k (N)

^to

M k (B(E))

^dened^by^the

k × k

^matrix^of ^maps,

((id − Ad(Z(η _i ^∗ ), Z(η ^∗ _j ))) ◦ (id − θ η i ,η j ) ⁻¹ ),

isompletely positive.

(12)

Note that,when

M = E = B(E)

^and

σ

^is^the ^identityrepresentationof

B(E )

on

E

^,

σ(M ) ^′

^is

C I E

^,

E ^σ

^is^isomorphi^to

C

and

D (E ^σ ) ^∗

^an^be^identied^with

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 ^an^open^subset

Ω

^of

D

into the losed unit ballof

B(E)

^butîtîs^known^that^suh^funtionsâre^preisely^those^for^whih^the^Pik

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 ^the^sense^of^Denition^3.2.2^of

[14℄.

Forthesakeofompleteness,wereordthefatthat everyelementof

H ^∞ (E)

ofnormat mostonegivesrisetoaShurlassoperatorfuntion.

Theorem 3.2

^Let

E

^be^a

W ^∗

M

and let

σ

^be ^a^faithful ^normal representation of

M

ⁱⁿ

B(H)

^for ^some ^Hilbert

spae

H

^. ^If

X

îsânêlementôf

H ^∞ (E)

ôf^normât^mostône,^then^the^funtion

η ^∗ → X b (η ^∗ )

^dened ⁱⁿ ^Remark ^2.14 îs â ^Shur ^lass ôperator ^funtion ôn

D ((E ^σ )) ^∗

^with ^valuesⁱⁿ

B(H)

^.

Proof.

^One^simply^takes

B i = I

^for^all

i

^and

C i = X(η b ^∗ _i )

ⁱⁿ ^Theorem^2.18.

Theorem 3.3

^Let

E

^be^a

W ^∗

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

îs â ^Shur ^lass ôperator ^funtion ôn

Ω ^∗

^with

values in

q 2 B(E)q 1

^. ^Then ^there^is^a^Hilbert ^spae

H

^,^a^normalrepresentation

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

^,^areⁱⁿ^the^spaes

B(E 1 , E ^σ ⊗ τ H )

^,

B(H, E 2 )

^,

and

B(H, E ^σ ⊗ τ H)

^,^respêtively,ândêahintertwinestherepresentations 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 )

^, îsâôisometry,

whih isunitaryif

E

^is^full.

(13)

(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(hη, θi)h

^).

Remark 3.4

^Before^giving^the^proof^of^Theorem^3.3,^we^want^to^note^that^the

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

^be^a

W ^∗

-orrespondeneover avon Neumannalgebra

M

^. ^Suppose^that

σ

^is^a^faithful^normalrepresentation of

M

^on^a^Hilbert^spae

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

^,^satisfy^onditions

(i)

^and

(ii)

of Theorem 3.3for somenormal representation

τ

^of

σ(M ) ^′

^on ^a^Hilbert ^spae

H

^,îsâlledâ ^system^matrix^provided

V

îsâôisometry ^(that îsûnitary,îf

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 ) ^∗

^is^alled^the ^transfer^funtion ^determined^by

V

^.

Proof.

^As^we^just^remarked,^the^hypothesis^that

Z

^is^a^Shur^lass^funtion

on

Ω ^∗

^means^that ^the^kernel

k Z

ⁱⁿ êquation⁽⁴⁾îsômpletely^positive^denite

inthesenseof[14℄. Consequently,wemayapplyTheorem3.2.3of[14℄,whihis

alovelyextensionofKolmogorov'srepresentationtheoremforpositivedenite

kernels,to ndan

N

^-

B(E) W ^∗

-orrespondene

F

^and^a^funtion

ι

^from

Ω ^∗

^to

F

^suh^that

F

^is^spanned^by

N ι(Ω ^∗ )B(E)

^and^suh^that^for^every

η 1

^and

η 2

ⁱⁿ

Ω ^∗

^and^every

a ∈ N

^,

(id − Ad(Z (η ^∗ ₁ ), Z(η ₂ ^∗ ))) ◦ (id − θ η 1 ,η 2 ) ⁻¹ (a) = hι(η 1 ), aι(η 2 )i.

Itfollowsthatforevery

b ∈ N

^and ^every

η 1 , η 2

ⁱⁿ

Ω ^∗

^,

b − Z(η ^∗ ₁ )bZ(η ^∗ ₂ ) ^∗ = hι(η 1 ), bι(η 2 )i − hι(η 1 ), hη 1 , bη 2 iι(η 2 )i

= hι(η 1 ), bι(η 2 )i − hη 1 ⊗ ι(η 1 ), bη 2 ⊗ ι(η 2 )i.

Thus,

b + hη 1 ⊗ ι(η 1 ), bη 2 ⊗ ι(η 2 )i = hι(η 1 ), bι(η 2 )i + Z (η ^∗ ₁ )bZ(η ₂ ^∗ ) ^∗ .

⁽⁷⁾

(14)

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)

^. ^(The^losureⁱⁿ^the^denitions^of

G 1 , G 2

^is

inthe

σ

^-topology^of^[6℄. ^It^then^follows^that

G 1

^and

G 2

^are

W ^∗

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

v : G 1 → G 2

^by^the^equation

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

îs ânîsometry. Îtîs âlso^lear^that îtîsâ^bimodule

map. Wewrite

P i

^for^the^orthogonal^projetion^onto

G i

^,

i = 1, 2

^and

V ˜

^for^the

map

V ˜ := P 2 vP 1 : q 1 B(E ) ⊕ F → q 2 B(E) ⊕ (E ^σ ⊗ N F).

Then

V ˜

îs â^partial îsometry ând, ^sine

P 1 , v

^and

P 2

^are ^all ^bimodule ^maps

(seeLemma 2.19),sois

V ˜

^. ^We^write

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

ând âll ^these ^maps âre ^bimodule ^maps. ^Let

H 0

^be ^the

Hilbert spae

F ⊗ B(E) E

^and ^note ^that

B(E ) ⊗ B(E) E

^is ^isomorphi^to

E

^(and

theisomorphismpreservestheleft

N

^-ation). ^Tensoring^on^the^right^by

E

^(over

B(E)

⁾^weôbtainâ^partialîsometry

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

^. ^These^maps

are well dened beausethe maps

α, β, γ

^and

δ

^are^right

B(E)

^-module ^maps.

Sinethesemaps arealso left

N

^-module^maps,^so^are

A 0 , B 0 , C 0

^and

D 0

^.

Bythedenitionof

V 0

^,îtsînitial^spaeîs

G 1 ⊗ E

ândîts^nal^spaeîs

G 2 ⊗ E

^.

Infat,

V 0

înduesânêquivalene ôf^the representationsof

N

^on

G 1 ⊗ E

^and

on

G 2 ⊗ E

^.

Itwillbeonvenienttousethenotation

K 1 N K 2

^if^the^Hilbert^spaes

K 1

^and

K 2

^are^both^left

N

^-modules^and^therepresentationof

N

^on

K 1

îsêquivalent^toâ

subrepresentationoftherepresentationof

N

^on

K 2

^. ^This^means,^of^ourse,^that

thereisanisometryfrom

K 1

^into

K 2

^thatintertwinesthetworepresentations.

Ifthetworepresentationsareequivalentwewrite

K 1 ≃ N K 2

^.

Paul S. Muhly

Schur Class Operator Functions and Automorphisms of Hardy Algebras

Paul S. Muhly

and Baruch Solel

Received: June 12, 2007

Communicated by Joachim Cuntz

Abstract.

E

W ∗

M

H ∞ (E)

σ

M

H

E σ

H ∞ (E)

B(H )

D (E σ ) ∗

H ∞ (E)

D (E σ )

H ∞

W ∗

1 Introduction

M

W ∗

E

W ∗

M

E

H ∞ (E)

M = E = C

H ∞ (E)

D

M = C

E = C n

H ∞ (E)

L n

σ

M

E σ

W ∗

σ(M )

σ(M ) ′

H ∞ (E)

E σ

D (E σ )

M = E = C

σ

1

C

E σ

C

H ∞ (E)

D (E σ ) = D

H ∞ (E)

σ

C

H

dim(H ) > 1

E σ

B(H)

D (E σ )

H

1

f ∈ H ∞ (E)

T ∈ D (E σ )

f (T )

k

T 1 , T 2 , . . . , T k

1

k

A 1 , A 2 , . . . , A k

f

1

f (T i ) = A i

i = 1, 2, . . . , k

M = C

E = C n

σ

E σ

W ^∗

H ^∞ (E)

E ^σ

H ^∞ (E)

D (E ^σ ) ^∗

H ^∞ (E)

D (E ^σ )

H ^∞

W ^∗

W ^∗

W ^∗

H ^∞ (E)

H ^∞ (E)

E = C ⁿ

H ^∞ (E)

E ^σ

W ^∗

σ(M ) ^′

H ^∞ (E)

E ^σ

D (E ^σ )

E ^σ

H ^∞ (E)

D (E ^σ ) = D

H ^∞ (E)

E ^σ

D (E ^σ )

f ∈ H ^∞ (E)

T ∈ D (E ^σ )

E = C ⁿ

E ^σ

C ⁿ

D (E ^σ )

B ⁿ

H ^∞ (E) = L n

B ⁿ

H ^∞ (E) = L n

H ^∞ (E) = L n

E = C ⁿ

D (E ^σ )

H ^∞ (E)

D (E ^σ )

A + B(I − L ^∗ _η D) ⁻¹ L ^∗ _η C

H ^∞ (E)

D (E ^σ )

H ^∞ (E)

D (E ^σ )

H ^∞ (E)

H ^∞ (E)

D (E ^σ )

H ^∞ (E)

D (E ^σ )

H ^∞ (E)

D (E ^σ )

J ^∗

D (E ^σ )

D (E ^σ )

H ^∞ (E)

E ^σ

H ^∞ (E)

D (E ^σ )

D (E ^σ )

H ^∞ (E)

D (E ^σ )

H ^∞ (E)

H ^∞

C ^∗

C ^∗

hξ, ηi ^∗ = hη, ξi

kξk := khξ, ξik ^1/2

C ^∗

C ^∗