Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics

30

an

infinite dimensional

Hilbert

space, U(H)

be

the

group

of

unitary

oper-

ators

on

H.

If

H is finite dimensional, U(H) is a

compact

Lie

group

and

it

is well known

that

any

compact

Lie

group

G

can

be

realized as a

subgroup

of

some U(n). Therefore

it

is

natural

to

think

of

a

subgroup

G

of

U(H)

as

the

generalizetion

of

the

Lie

subgroup

of

U(n)

and

discuss

their

"Lie

algebra". Since

most

infinite dimensional

unitary

representation

are

not

norm

continuous,

the

appropriate

topology

to

be

introduced

in

G c U(H)

would

be

the

strong

topology.

Then,

in view

of

Stone

theorem,

one

may

try

to

define

its

Lie

algebra

as a set 9 =

{X;

X*

=

-X,

e

tX

E

G,

\it} of

all (possibly

unbounded)

skew-adjoint

operators,

which

make

stongly

con-

tinous

one-parmeter

groups

of

G C U(H). However,

this

does

not

work

well

due

to

the

domain

problem

of

unbounded

operators:

even

though

X*

=

-X,

y*

=

-Yare

densely defined

on

H,

the

domain

of

the

sum

X + Y

and

Lie

bracket

XY

- Y X often fails

to

be

dense.

Even

worse,

it

is

possible

that

dom(X)

ndom(Y)

=

{O}.

In

addition

to

this,

there

is

another

problem:

what

kind

of

topology

should

we

introduce

in

u(H)?

Since

it

is

well known

that

the

sequence

{An}

of skew-adjoint

operators

on

Hand

skwe-adjoint

operator

A,

s-lim(An

+

1)-1

=

(A

+

1)-1

{?

e

tAn

----+

etA

for all t E R

n

Therefore,

"strong

resolvent topology"

may

be

the

appropriate

one. How-

ever,

there

is

no

knowing

wheter

s-lim(A

n

+

Bn

+ 1)-1 = (A + B + 1)-1,

n

holds

true

even if

An

and

Bn

converges respectively

to

A

and

B

in

the

strong

resolvent sense.

Taking

these

into

consideration, we see

that

G is

too

big

to

have a

natural

Lie algebra,

in

general. Is

there

kind

of

subgroups

G

that

have a nice Lie

algebra

beyond

finite dimensional ones? We an-

swered

to

this

question. More preciesely, let

9J1

be

a finite von

Neumann

algbera

on

H.

We showed

that

for

any

strongly

closed

subgroup

G

of

U(9J1)

,

the

unitary

group

of

9J1,

there

exists a Lie algebra, which is

complete

with

respect

to

the

strong

resolvent topology

and

studied

their

properties.

The

discussion is

based

on

the

Murray-von

Neumann's

result

stating

that

the

set

9J1

of all densely defined closed

operators

affiliated

with

9J1

has a

nat-

ural

*

-algebra

structure.

Furthermore,

we

determined

the

category

of

the

*-algebra

of

unbounded

operators

that

can

be

represented

as

9J1

for some

finite von

Neumann

algebra

9J1.

31

Notes.

After

finishing

this

work,

the

authors

were informed from

Pro-

fessor Daniel

Beltita

that

recently he

had

written

a

paper

whose

subject

was closely

ours

3.

2.

Murray-von

Neumann's

result

In

this

section, we review

the

fundamental

results

obtained

by

Murray-von

Neumann

6.

For

the

details

about

operator

algebras

and

operator

theory,

see Refs. 11, 15.

Let

1-{

be

a

Hilbert

space

with

an

inner

product

(C

1]), which is linear

with

respect

to

1].

We

denote

the

algebra

of

all

bounded

operators

on

1-{

by

'l3(1-{).

Let

9J1

be

a von

Neumann

algebra

on

1-{.

9J1'

is

the

commutant

of

9J1.

The

group

of

all

unitary

operators

in

9J1

is

denoted

by

U(9J1).

The

lattice

of

all projections in

9J1

is

denoted

by

P(9J1).

Next,

we recall

the

notion

of

an

affiliated

operator.

The

domain

of

a

linear

operator

T on 1-{ is

written

as

dom(T)

and

the

range

of

it

is

written

as

ran(T).

If

T is a closable

operator,

we

write

T for

the

closure of

T.

Definition

2.1.

A densely defined closable

operator

T

on

1-{ is

said

to

be

affiliated

with

a von

Neumann

algebra

9J1

if for

any

u E U(9J1') ,

uTu*

= T

holds.

If

T is affiliated

with

9J1,

so is

T.

The

set

of

all densely defined

closed

operators

affiliated

with

9J1

is

denoted

by

9J1.

Each

element in

9J1

is

called

an

affiliated operator.

In

general,

9J1

is

not

a *

-algebra

under

these

operations.

This

is

the

reason

for

the

difficulty

of

constructing

Lie

theory

in

infinite dimensions. However,

Murray

and

von

Neumann

proved,

in

the

pioneering

paper

6,

that

for a finite

von

Neumann

algebra

9J1,

9J1

does

constitute

a *-algebra

of

unbounded

operators.

That

is,

Theorem

2.1.

Murray-von Neumann 6 For

an

arbitrary finite

von

Neu-

mann

algebra

9J1,

the

set

9J1

forms

a *-algebra

of

unbounded operators,

where the algebraic operations are defined

b'll'

(X,Y)

f----+

X +

Y,

(X,

Y)

f----+

XY,

(a,X)

f----+

aX,

Xf----+X*.

This

theorem

is

the

starting

point

of

our

study.

aaX

equals

aX

when

a #

O.

However,

dom(O·

X)

= 7t #

dom(X).

32

Remark

2.1.

In

6,

Murray-von

Neumann

proved

Theorem

2.1 for count-

ably decomposable case.

It

can

be

generalized

to

arbitrary

finite von Neu-

mann

algebra. For

the

proof, see

1.

The

converse

of

Theorem

2.1 is also

true.

Namely, if

9J1

is a *-algebra,

then

9J1

must

be

of

finte type.

Theorem

2.2.

1 Let

9J1

be

a von

Neumann

algebra acting on a Hilbert

space H.

Assume

that, for all

A,

B E

9J1

, the domains

dom(A

+

B)

and

dom(AB)

are

dense

in

H.

If

the set

9J1

forms a *-algebra with respect to

the

sum

A +

B,

the scalar multiplication

o;A

(0;

E C), the multiplication

AB

and the involution A *, then

9J1

is a finite von

Neumann

algebra.

3.

Lie

Group-Lie

Algebra

Correspondences

In

this

section

we

state

and

prove

the

main

result

of

this

paper.

As ex-

plained

in

the

introduction,

Lie

theory

for

U(H)

is a difficult issue.

What

one

has

to

resolve for discussing

the

Lie group-Lie algebra correspondence

is a

domain

problem

of

the

generators

of

one

parameter

subgroups

of

G c

U(H).

The

second

to

be

discussed is a continuity of

the

Lie algebraic

operations. However

we

can

show

that,

for

any

strongly closed

subgroup

G

of

unitary

group

U(9J1)

of

some finite von

Neumann

algebra

9J1,

there

exists canonically a complete topological Lie algebra. Since

there

are

con-

tinuously

many

non-isomorphic finite von

Neumann

algebras

on

H,

there

are

also varieties

of

such groups.

3.1. Topological

properties

of

5)J1

We first endow

9J1

with

two topologies, called

the

strong

resolvent topol-

ogy

and

the

T-measure topolgoy.

The

former is

operator

theoretical one

and

the

latter

is

an

operator

algebraic one.

The

combined use

of

them

is

indispensable for

our

purpose.

3.1.1. Strong Resolvent Topology

First

of

all, we define

the

topology called

the

strong

resolvent topology

on

the

suitable

subset

of

densely defined closed operators.

Let

H

be

a Hilbert

space.

Definition

3.1.

We call a densely defined closed

operator

A

on

H belongs

to

the

resolvent class

fl~(1i)

if A satisfies

the

following two conditions:

33

(RC.1)

there

exist self-adjoint

operators

X

and

Y

on

H such

that

the

intersection

dom(X)

n

dom(Y)

is a core

of

X

and

Y,

(RC.2)

A=X+iY,

A*=X-iY.

Note

that

(RC.1) implies

dom(X)ndom(Y)

is dense, so X

+iY

and

X

-iY

are

closable.

Thus

X +

iY

and

X -

iY

are

always defined.

Furthermore,

we have

~(A

+

A*)

=

~(X

+

iY

+ X -

iY)

:J Xldom(X)ndom(Y).

Since

A+A*

is closable

and

by (RC.1),

we

get

~A

+

A*

:J

X.

As X is self-

adjoint,

X

has

no

non-trivial

symmetric

extension, we have

~A

+

A*

=

X.

Therefore, X is uniquely

determined.

As

same

as

the

above, Y is also

unique

and

-fiA

-

A*

=

Y.

We

denote

1---.---------.--,-

Re(A)

:=

X =

-A

+

A*,

2

1

---.---------.--,-

1m

(A)

:=

Y =

2iA

-

A*.

Also

note

that

bounded

operators

and

(possibility

unbounded)

normal

op-

erators

belong

to

!Jf!,#?(H).

Now

we

endow

!Jf!,#?(H)

with

the

strong resolvent

topology

(SRT for

short),

the

weakest topology for which

the

following

map-

pings

{

!Jf!'#?(H)

:3

A f------>

{Re(A)

- i}

-1

E

(SB(H),

SOT)

!Jf!,#?(H)

:3

A f------> {1m(A) - i}

-1

E

(SB(H),

SOT)

are

continuous.

Thus

a

net

{A,}",

in

!Jf!,#?(H)

converges

to

A E

!Jf!,#?(H)

with

respect

to

the

strong

resolvent topology if

and

only if

{Re(A",)-i}-l~

----;

{Re(A)-i}-l~,

{1m(A",)-i}-l~

----;

{1m(A)-i}-l~,

for each

~

E H.

This

topology is well-studied

in

the

field of

unbounded

operator

theory

and

suitable

for

the

operator

theoretical

study. We

denote

the

system

of

open

sets

of

the

strong

resolvent topology

by

OSRT.

Let

9J1

be

a finite

von

Neumann

algebra

on

a

Hilbert

space

H.

We

can

show

that

9J1

is a closed

subset

of

the

resolvent class

!Jf!'#?

(H).

3.1.2. T-Measure Topology

Let

9J1

be

a

count

ably

decomposable finite von

Neumann

algebra

acting

on

a

Hilbert

space

H.

Fix

a faithful

normal

tracial

state

T

on

9J1.

The

T-measure

topology

(MT

for

short)

on

9J1

is

the

linear topology whose

fundamental

34

system

of

neighborhoods

at

0 is given

by

N(

15)'=

{A

9J1.

there

exists a

projection

p E 9J1}

c,.

E,

such

that

IIApl1

< c,

T(PJ..)

<

15

'

where c

and

15

run

over all

strictly

positive real

numbers.

It

is

known

that

9J1

is a

complete

topological *

-algebra

with

respect

to

this

topology

9.

We

denote

the

system

of

open

sets

with

respect

to

the

T-measure

topology

by

Or.

Note

that

the

T-measure

topology

satisfies

the

first

count

ability

axiom.

Remark

3.1.

In

this

context,

the

operators

in

9J1

are

also called

T-

measurable operators

4.

Thus

there

are

two topologies

on

9J1,

the

strong

resolvent

topology

and

the

T-measure topology.

It

seems

that

these

two topologies

are

quite

dif-

ferent. However,

in

fact,

they

coincide

on

9J1,

i.e.,

Lemma

3.1.

Let

9J1

be

a countably decomposable finite von

Neumann

al-

gebra acting on a Hilbert space

H.

Then

the strong resolvent topology and

the T-measure topology coincide

on

9J1.

In

particular,

9J1

forms a complete

topological *-algebra with respect to the

strong resolvent topology. Moreover

the T-measure topology is independent

of

the choice

of

a faithful

normal

tracial state

T.

3.2.

Main

Results

Definition

3.2.

For a

strongly

closed

subgroup

G

of

U(9J1),

the

set

fJ

= Lie(G)

:=

{A

; A* =

-A

on

H,

etA

E

G,

for all t

E~}

is called

the

Lie

algebra

of

G.

The

complexification

fJlC

of

fJ

is defined

by

fJlC

:=

{A

+

iB

;

A,

BE

fJ}

.

If

G =

U(9J1),

we

sometimes

write

fJ

as u(9J1).

Remark

3.2.

In

general,

the

strong

limit

of

unitary

operators

is

not

nec-

essarily

unitary.

It

is

known

that

U(9J1)

is

strongly

closed

in

'l3(H) if

and

only if

9J1

is a finite von

Neumann

algebra.

At

first sight,

it

is

not

clear

whether

we

can

define algebraic

operations

on

fJ.

However,

Lemma

3.2.

Under the above notations,

fJ

C

9J1

holds.

35

Therefore

the

sum

A + B

and

the

Lie

bracket

AB

-

BA

are

well-defined

operations

in

9J1,

but

it

is

not

clear

whether

they

belong

to

9 again.

The

following

Lemma

3.4

guarantees

the

validity

of

the

name

"Lie

algebra".

The

former

part

of

the

proof

is

based

on

the

two

lemmata

established

by

Trotter-Kato

and

E. Nelson, which

are

of

importance

int

their

own.

Lemma

3.3.

Trotter-Kato, Nelson 8 Let

A,

B

be

skew-adjoint operators on

a Hilbert space

H.

(1)

If

A + B is essentially skew-adjoint on

dom(A)

n

dom(B),

then

it

holds that

for

all t E R

(2)

If

(AB

-

BA)

is essentially skew-adjoint on

dom(A

2

)

n

dom(AB)

n

dom(BA)

n

dom(B2),

then

it

holds that

for all t

> 0, where

[A,

B]

:=

AB

-

BA.

Lemma

3.4.

Let

G

be

a strongly closed subgroup

of

U(9J1).

Then 9 is a

real Lie algebra with the Lie bracket [X,

Y]

:=

XY

-

YX.

Based

on

the

above

preliminaries, we

can

prove

the

following

main

result.

Theorem

3.1.

Let

G

be

a strongly closed subgroup

of

the unitary group

U(9J1)

of

a finite von

Neumann

algebra

9J1.

Then 9 is a complete topological

real

Lie

algebra with respect to the strong resolvent topology. Moreover, 9c

is a complete topological Lie * -algebra.

Remark

3.3.

It

is

easy

to

see

that

for G =

U(9J1),

its

Lie

algebra

u(9J1)

is

equal

to

{A

E

9J1;

A * =

-A}

and

the

exponential

map

exp

:

u(9J1)

->

U(9J1)

is

continuous

and

surjective.

Proposition

3.1.

Let

9J1

1

,

9J1

2

be

finite von

Neumann

algebras on Hilbert

spaces

H

1

,

H2

respectively.

Let

G

i

be

a strongly closed subgroup

of

U(9J1

i

)

(i =

1,2).

For any strongly continuous group

homomorphism

36

rp

: G

1

----t G

2

,

there exists a unique SRT-continuous Lie algebra homomor-

phism

<P:

Lie(Gd

----t Lie(G

2

)

such that

rp(e

A

)

=

e<l>(A)

for all A E

Lie(Gd.

In

particular,

if

G

1

is isomorphic to G

2

as

a topological group, then Lie(

Gd

and Lie( G

2

)

are isomorphic as a topological Lie algebra.

As above, G

has

finite dimensional characters.

On

the

other

hand,

it

also

has

an

infinite dimensional character.

Proposition

3.2.

Let

9J1

be

a finite von

Neumann

algebra, then the fol-

lowing are equivalent.

(1) The exponential map

exp:

u(9J1):1 X

f-+

exp(X)

E

U(9J1)

is

locally injective. Namely, the restriction

of

the

map

onto some

SRT-neighborhood

of

0 E

9J1

is injective.

(2)

9J1

is finite dimensional.

Remark

3.4.

Lie( G) is

not

always locally convex, whereas

most

of

infinite

dimensional Lie theories, by

contrast,

assume local convexity. Indeed,

u(9J1)

is locally convex if

and

only if

9J1

is atomic.

Next,

we

characterize closed *-subalgebras

of

9J1.

Proposition

3.3.

Let

9J1

be

a finite

non

Neumann

algebra on a Hilbert

space

H,

!!l

be

a SRT-closed *-subalgebra

of

9J1

with

hi.

Then there exists

a unique von

Neumann

subalgebra l)1

of

9J1

such

that!!l

= l)1.

4.

Categorical

Characterization

of

9J1

In

this

section we consider some categorical

aspects

of

the

*

-algebra

9J1.

Especially, we

determine

when a

*-algebra!!l

of

unbounded

operators

on

a

Hilbert space

H

turns

out

to

be

of

the

form

9J1,

without

any

reference

to

von

Neumann

algebraic

structure

in advance. For

this

purpose,

we

define

the

category

fRng

of

unbounded

operator

algebras

and

compare

this

category

with

the

category

fvN

of

finite von

Neumann

algebras

and

show

that

both

of

them

have

natural

tensor

category

structures.

Furthermore,

we

will see

that

they

are

isomorphic as a

tensor

category,

in

spite

of

the

fact

that

the

object

in

fRng

is

not

locally convex

in

general while

the

one

in

fvN

is

a

Banach

space. To begin with, let us

introduce

the

structure

of

tensor

category into

fRng.

First,

it

is well known

that

the

usual

tensor

product

(9J1

1

,9J1

2

)

f-+

9J1

1

09J1

2

of

von

Neumann

algebras

and

the

tensor

product

of

a-weakly continuous homomorphisms

(¢1

1 ¢2)

f-+

¢1

0

¢2

makes

the

37

category

of

finite von

Neumann

algebras a

tensor

category. Therefore

we

define:

Definition

4.1.

The

category

fvN

is a category whose

objects

consist

of

pairs

(9J1,

'H)

of

a finite von

Neumann

algebra

9J1

acting

on

a Hilbert space

'H

and

whose morphisms

are

a-weakly continuous

unital

*-homomorphisms.

The

unit

object

is

(Cle,

C).

The

tensor

functor is

the

usual

tensor

product

functor of von

Neumann

algebras.

The

definition of left

and

right

unit

constraint

functors might

be

obvious.

If

we

are

to

characterize

the

objects

in

fRng,

we

must

settle

some sub-

tleties

due

to

the

fact

that

we

cannot

use von

Neumann

algebraic

structure

from

the

outset.

However,

this

difficulty

can

be

overcome

thanks

to

the

notion

of

the

strong

resolvent topology

and

the

resolvent class whose

definitions

are

independent

of

von

Neumann

algebras (See §3). We define

fRng

as follows.

Definition

4.2.

The

category

fRng

is a category whose

objects

(ge,

'H)

consist of a SRT-closed subset

ge

of

the

resolvent class

ge'-(?('H)

on

a Hilbert

space

'H

with

the

following properties:

(1) X + Y

and

XY

are

closable for all

X,

Y E

ge.

(2)

X +

Y,

aX,

XY

and

X*

again belong

to

ge

for all

X,

Y E

ge

and

a E

C.

(3)

ge

forms a *-algebra

with

respect

to

the

sum

X +

Y,

the

scalar

multiplication

aX,

the

multiplication

XY

and

the

involution

X*.

(4)

lH

E

ge.

The

morphism

set between (gel, 'Hd

and

(ge2,

'H

2

)

consists of SRT-

continuous

unital

*-homomorphisms from gel

to

ge2.

Remark

4.1.

From

the

definition

of

fRng,

it

is

not

clear whether, for each

objects

in

fRng,

its

algebraic

operations

are

continuous

or

not. However,

the

shows

that

ge

is a complete topological * -algebra.

Lemma

4.1.

Let

(ge,

'H)

be

an object

in

fRng.

Then there exists a unique

finite von

Neumann

algebra

9J1

on

'H

such that

ge

=

9J1.

Furthermore,

9J1

=

ge

n

Q3('H)

holds.

Note

that

for each finite von

Neumann

algebra

9J1

on a Hilbert space

'H,

(9J1,

'H)

is

an

object

in

fRng.

The

main

result

of

this

section is

the

38

Theorem

4.1.

The category

fRng

is a tensor category. Moreover,

fRng

and

fvN

are isomorphic as a tensor category.

To prove

this

theorem,

we

need

many

lemmata.

For

the

details, see

Ref

1.

Acknowledgement

The

authors

would like

to

express

their

sincere

thanks

to

Professor

Masanori

Ohya

and

Professor

Noboru

Watanabe

and

all

the

organizers

of

QBIC2010

for

their

kind

invitation

to

the

conference

and

giving

them

the

precious

opportunity

to

have a

talk

there.

They

are

grateful

to

Professor Daniel

Beltita

for informing

them

of his

important

paper

3

and

for his

kind

cor-

respondences

and

discussions

with

them.

They

also

thank

to

Ms.

Mari

Kuroda

for

her

warm

help

and

correspondences. H.A

and

1.0

also

thank

to

Mr. Ryo

Harada,

Mr. Takahiro Hasebe, Mr.

Kazuya

Okamura

and

Mr.

Hayato

Saigo for

their

useful

comments

and

discussions

during

the

seminar.

They

also

thank

to

Professor Asao

Arai

at

Hokkaido university

for

the

fruitful discussions, insightful

comments

and

encouragements,

Pro-

fessor

Konrad

Schmudgen

at

Universitat

Leipzig for informing

them

of

the

paper12

and

Mr.

Yutaka

Shikano

at

MIT

for his professional advice

about

LaTeX.

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1.

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Ando

and

Y.

Matsuzawa,

Lie

Group-Lie

Algebra

Correspondences

of

Unitary

Groups

in

Finite

von

Neumann

Algebras,

submitted.

http://arxiv.org/abs/1005.4850

2.

H.

Ando,

Y.

Matsuzawa

and

I.

Ojima,

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