Index for C*-subalgebras, Issues 422-424 issue 424

Number 424

Yasuo Watatani Index for C*-subalgebras

Memoirs of the American Mathematical Society Providence· Rhode Island • USA January 1990· Volume 83 . Number 424 (end of volume) . ISSN 0065-9266

TABLE OF CONTENTS

INTRODIJCTION ... CHAPTER

I.

1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1. 7. 1.8. 1.9. 1 10 1.11. CHAPTER

II. 2 2 2 2 3 2.4. 2.5. 2.6. 2 7 2.8. 2.9. 2.10.

1

Purely algebraic preliminaries Assum tions .......................................... . Conditional expectations ............................. . Morita equivalence .................................... Uniqueness of conditional expectations ................ Index under direct sums and tensor products ........... Dual conditional expectations ..........•..•........... Composition of conditional expectations ... ............ Coring structure and separable extensions ... ......... Hattori-Stalling rank and index ............•.......... Generalized Casimir elements ...................... Modular automorphisms................................. Index for C*-algebras Reduced C*-basic construction

Unreduced C*-basjc construct jon Values of index....................................... Index for finite dimensional C*-algebras .............. Index for factors.. . .. . ... . . . ...... . ........ . ...... ... Some constants ..................... . ..... ....... ...... Finite index .......................................... Index, freeness of group actions and covering spaces . . Some other examples . . .. . . . ........ . ................... Radon-Nikodym derivatives .. . ........ . ..... . ....... . ... 2 11 Reduct jon ............ 2.12. Minimizing index ...... . . .. . . ..... . ........... .. . .... . . CHAPTER

III 3 1

3.2. 3.3.

6 6 11 14 16 17 20 21 23 25 27

39

43 52 56 59 63 65 68 74 77 84

87

Transfer Induced and co-jnduced modules . . . . . . . ....... 91 Transfer maps in Tor and Ext ............. ......... .... 93 Transfer maps in the K-theory of C*-algebras .......... l03

REFERENCES. . .•.....•.................. , . , " " " ' , .•. , .. , ............ 114

iii

ABSTRACT In this paper we study the index. denoted by Index E. for conditional expectations E on C* - algebras. which is a generalization of the index for subfactors by Jones and Kosaki. Under a certain condition we show that the spectrum of Index E is contained in {4cos2~/n ; n=3.4.5 •...}V [4. ~1. The notion of the index is related with coring structure. separable extension. generalized Casimir elements. G-Galois basis. covering spaces. modular automorphisms and Nakayama automorphisms on Frobenius algebras. We introduce transfer maps in Tor. Ext and K-theory. We show that the transfer map is related with the multiplication map by Index E up to the evaluation by a trace.

Key Words and Phrases C* -algebra. index. conditional expectation. basic construction. K- theory. transfer. modular automorphism. Galois theory. covering space. coring structure. separable extension. Casimir element.

iv

To my wife Toshiko for everything

INTRODUCTION

Jones' index theory[36j for subfactors of type III is an exciting and epochmaking work. but

also

He createS a new world, not only for operator algebras many

for

unexpected

branches

of

statement(38] concerning the geometric, combinatrial and discrete

of Ill-factors has been developed (16],

Goodman,

nature

Chodaf14l,(15J.

by Bian-Nadal [2J, M.

de la Harpe and Jones [29J,

His

rnathematics[39].

Ocneanu[53],

Pimsner

and

Popa[58] and Wenz'1[76J.

On for

the other hand, Kosaki (44] defined index, denoted by

a

(normal

semifinite faithful) conditional

onto

a subfactor.

He used

Index E,

expectation

arbitrary

factor

the

Connes[21J

and the theory of operator valued weights by

E

spatial

of

an

theory

of

Haagerup[31J.

The case for factors of type I I I with the ergodic theory has been studied by Hamachi and Kosaki [32] ) (33] ) [34] and Loi [45 J and Riai [35 J • In

this

E

expectation transfer

in

Index E. [58J.

note on

we a

shall

investigate

C1'-algebra.

Index E

We shall

establish

a a

conditional link

between

the K-theory of C)':-algebras and the multiplication

Our definition of index is inspired by the

map

Pimsner-Popa

by

basis

Kosaki's formula "Index E "" E-l(l)" -and the Casimir

elements

for

Roughly speaking, index [M:N]

measures

the

semi-simple

Lie algebras.

maximal number of (disjoint) copies of N in M. our

for

Hence it is evident

that

index should coincide with the number' of sheets of a covering

in the case of a commutative C1'-algebra. exactly

Hattori-Stalling

index

definition

tilge.braic

formulation immediately implies that

index

almost

equivalent

Jones index is

Hence we shall start from

algebraic

is

of

rank.

For Ill-factors

to

unify

many

to the union of

space

related

a

purely

notions.

Our

the property of finite two

things:

the

coring

structure (a certain generalization of Hopf algebra) by Sweedler[68]

and

the notion of separable extension. Let

M be

a factor of type III on a Hilbert space

Hand

NC M

a

Received by the editor January 24, 1989. Received in revised form May 1989.

YASUO WATATANI

2

We denote by

subfae tor.

di~(H)

the coupling constant of M on H [48J.

Then Jones index [M:N] is defined by Jones[36] as [M:N]

if N' is finite.

= dimN(H)/dimM(H)

Surprisingly, Jones determined that the set of possible

values of the index [M: NJ for subfactors is exactl, the set

n ~ 3,4,5, ... }

{4cos 2 w/n

Pimsner

and

V [4,w].

only if there exists a Pimsner-Popa basis {ro , ... l (Le. L j mjE(m/~X) and some mor.e condition on orthonormality. (II)

is

l:j mjeNmj* "" 1

calculated

condition

(U)

by

if

shows that the Jones index [M:NJ is finite

Popa[S~J

the

Pimsner-Popa

reminds

us

of

the

=

,ffi

+

n 1

and

satisfying

}C M

x for x0M) In this case the Jones

basis

notion

as

~

[M:NJ

of

E

j

G-Galois

index

*.

The

basis

(cf.

mjm

j

[23J, [46Jand [54]).

be

Kosaki[44]

extended Jones' index theory for arbitrary factors. Let

an

with a subfactor N.

factor

We assume the

existence

(semifinite faithful) conditional expectation E : M-7'N.

of

M

normal

The set of

all

normal semifinite faithful operator-valued weights is denoted by P{M.N). Haagerup peN' ,M').

constructed

an order-reversing bijection

between

P(M,N)

and

Based on the spatial theory by Connes[21]. the above bijection

(denoted by E ~ E -1 ) is given by the equation of spatial derivatives: d(¢oE) Id~ • d$/d(~oE-1),

where ¢ and ware normal semifinite faithful weights on Nand M'.

Kosaki

defined the index. depending on E, by Index E • E- 1 (1).

And

he shows that the set of values of indices is the same set as

obtained. bas~s

Jones

If Index E is finite. then there exists a Pimsner-Popa

{mI' ...

~mn}CM

satisfying (U) in this case too.

type

Furthermore we can

show that for x€:N'. Hence we recover that Index E "" E-I(l) "'" Ej mjmji'. In

our

approach, the Pimsner-Popa basis is generalized to a

guasi-basis {(u1.vl)' ...• (un'vn )} satisfying

(PU) (,i uiE(vix)

=

x

= Ei E(xui)v i

·

notion

of

INDEX FOR C*-SUBALGEBRAS

3

Here the coefficients over N are not necessarily unique, so M~Nn genera1l • 7 We shall define Index E analogously by

Index E

~

E. u.v. 1.

1.

1..

On the other hand, PiIDsner and Pop a shows that if M:::>N are factors of

III

type

and

N'n M ~

~,

then

the

Connes-St¢'rmer

coincides with Jones index in the sense that H(MIN) PiIDsner ,and Popa, Kawakami and Yoshida

after

several

interesting

succeeds in

III

in

formulae.

is

It

relative

= In(M:NJ.

[41J, [42J, [78J

surprising

that

entropy

Following developed

Ocneanu[53J

classifying subfactors of the hyper finite factor M

the case of (M:N]< 4.

But we shall not

discuss

of

these

type

matters

further here. Goodman, de la

Neumann

algebras

Harpe 'and Jones[29j extended the index theory to

with finite dimensional centers.

Their

work,

von

though

very interesting, differs somewhat from ours in its approach. While

this note was being prepared, a preprint[6] by Baillet, Denize·

au and.Havet appeared.

They consider the index of conditional

expectation

of von Neumann algebras and some of their result are similar to ours. In chapter I a purely algebraic definition of index is introduced and several coring

consequences structure,

G-Galois

are

noted. The notion of index

separable extension,

basis,

generalized

modular automorphisms

and

is

related

Casimir

Nakayama

with

elements,

automorphisms on

Frobenius algebras. In

chapter II we consider a conditional expectation E : B-)A

C*-algebra B onto a C*-subalgebra A.

of

a

We introduce the notion of C*-basic

construction C*. The C*-algebra C* is exactly the algebra of "compact operators" on a Hilbert C*-module E, where 8. is the completion of

B with

the A-valued inner product

construction

of

Jones'

iterated

(xly)

basic

interesting result by Wenzl[tt1] to show that

=

E(x*y).

extensions,

By we

a

similar

can

use an

if Index E is contained in

A, then the spectrum of Index E is included in the set {4cos 2 Tf/n ; n"" 3,4,5, .. )

If any

V

(4,00[.

Index E is not contained in A, then the spectrum of index E can values

natural.

in [1,00[.

But to consider Index E as being in

A

is

take very

Because if the C'lLalgebra B is simple, then Index E is a scalar

YASUO WATATANI

4

and

hence

covering

in

A.

In the case

spaces.

of

commutative

C*-algebras,

Then that Index E is in A tells

that the

consider

conditional

expectation E is given by taking the mean on each fiber. If

C~Lalgebras

the

following

after

B and A are simple, then we can minimize

Riai's idea [35J.

Thus we can define [B:A],

independent of the choice of conditional expectations. pairs

B,:, A of simple

above

set

instead can

C*-alge~ras

of weak closure in

analyze

the

index

equivalence[63]. If the properties

which

is

Many examples

of

are constructed with the index

by considering AF-algebras obtained by

taking

Wenzl's construction [76].

owing to

index

Rieffel' s

work

is finite, then

two

in corrnnon, for example. Hyperfiniteness

index E

on

norm

in

the

closure

Furthermore (strong)

algebras [ZOJ)

we

Morita

share

many

fullness[58],

property T [Z].[ZZ],[40J. similarity problem[491. nuclearity, etc. In chapter III transfer maps are introduced in the theory of Tor, Ext

K.

and

We

multiplication

shall

show

that

the

transfer

map

is

related

to

the

map by Index E up to evaluation by a trace. Suppose under some condition, then Index E is an eigen-value

that of

a

certain integral matrix S which is constructed by using a transfer map. For

example

4cosZ~/5

=

let

e l .eZ.e 3 •... be Jones proj ections $Z, where $ is the golden number. Let

with

T

-1

C*(l,e Z ,e , ... ). Then there exists a conditional expectation 3 E : B-7 A such that E(xely) "" 1:xy for x,yE.A. If so. the above integral and

A

=

matrix KO(B)?i'-

corresponds to the multiplication by Index E

.2 on Z + $Z

c

IR

and Index E is the Perron-Frobenius root of S. This is an extended version of an o.nnouncement [75] in C. R.

Acknowledgement. I am greatly indebted to Prof. Vaughan Jones for many Prof.

suggestions

and new ideas on my research.

Special

thanks

go

Hideki Kosaki for his introducing me his theory and Y. Sekine

improving an inequality.

his to for

I would like to thank Y. Kato for the influence

of his singular methods on research.

I would like to thank Prof. H.

and

INDEX FOR C*-SUBALGEBRAS M.

Choda,

Suzuki,

M.

Enomoto, J. Fujii, M. Fujii, E. Kamei,

Masahiro

Kitamura,

I would

deep gratitude to Prof. Osamu Takenouchi for his

Finally it is my great pleasure

constant encouragement.

Prof.

K.

H. Takehana and T. Yoshida for their discussions.

to express my and

5

Nakamura,

a

pioneer

algebras in Japan, who predicted many

among

researchers

like

guidRnce to

of

H.

thank

operator

important tools, ideas and notions

at an early stage of the development of the research of operator algebras in Japan.

Added in Proof: After submission of this paper, I received (and was informed of) the following interesting preprints on the index:

(HaJ J.-F. Havet, Esperance conditionnelle minimale.

[HO] R. Herman and A. Ocneanu, Index theory and Galois theory for infinite index inclusions of factors. [Hi] F. Riai,. Minimum index for subfactors and entropy. [Jo] P. Jolissaint, Index for pairs of finite von Neumann algebras. [Lo] P. Loi, On the theory of index and type III factors. [Lo] R. Longo, Index of subfactors and statistics of quantum fields. [PP] M. Pimsner and S. Popa, Finite dimensional approximation of pairs of of algebras and obstructions for the index. [Po] S. Popa, Relative dimension, towers of projections and commuting squares of subfactors. [Tsl S. -K. Tsui, Index of faithful normal c'onditional

expectations.

CHAPTER I.

PURELY ALGEBRAIC PRELIMINARIES

1.1. Assumptions.

Let

k

Throughout

be

commuto...tive ring with an

a

identity

element

1.

this chapter, we consider a k-algebra B and a k-subalgebra

of B with a common identity element 1.

A

In this chapter, we consider only

purely algebraic aspects of conditional expectations E : B ~ A and

indices Index E as preliminaries for Chapter II

their

and III.

1.2. Conditional expectations

Definition 1.2.1. A conditional expectation E

is

B-1 A

an

onto

k-linear map satisfying E(ab) for

=

=

aE(b), E(ba)

a E A and b EB.

E(b)a

and E(a)

=

a

But when we consider operator algebras,

we

always

assume that E is positive, that is, E is a projection of norm one studied

J.

by

Tomiyama

[73J

We say E

E(bB)

o

implies b "'" 0, for

then

E is

non-degenerate

E(b*b) "'" 0

implies

inequality

'liE(x*y)1\

for

conditional

b:= 0,

bE B. if

is

non-degenerate If we consider

and only if

for

~ \iE(x*x)111/2I\E(Y*Y)lr/2 •

expectations

in operator

E(Bb) = 0

operator

E is

bE: B, because

if

we

or

algebras,

is.

faithful,

that

have

following

the

See also

algebras.

Umegaki[74j

Throughout

paper, we assume that conditional expectations are non-degenerate

this unless

stated otherwise.

Motivated

by Pimsner and Popa basis of subfactors in (58J. we

shall

introduce a notion of quasi-basis.

Definition

1.2.2.

A finite

family

{(u

l ' v l ), ... , (un' v n )} C B x.. B

called a guasi-basis for E if for x\:-B.

is

INDEX FOR C*-SUBALGEBRAS A

expectation

conditional

E : B-7A is of index-finite

exists a quasi-basis for E, Index E

Remark

Center

B

7

~

if

there

In this case we define the index of E by Li uiv i E: B

=

(1) If E is of index-finite type, then Index E is in fact

and

the

value

Index E does not

depend

on

the

choice

in

of

quasi-bases, cf. Proposition 1.2.8. (2)

Note

that

"coefficients" E(vix),

(or E(xu

i

»,

are

not necessarily

unique.

(3) If B and A 'are C*-algebras, then we can choose a quasi-basis {(ui,v ) i ; i

=

1, ... ,n} with vi = u i

(4) Sometimes advance.

we

But

do

not

the

non - degenera te.

*.

know whether E is

existence In

fact

cf. Lemma 2.1.7.

of a

non-degenerate

quasi-basis

guarantees

or

not

in

E

is

that

if

Similarly E(bB) = 0 implies that b "" O.

(5) We see that G-Galois basis (cf. [23J ,[46J,[54J) is also a quasi-basis for the trace map up to normalization

Example 1.2.3. group

algebras

B

Let G be a group and H a subgroup of G. ~

k[G]

{)"(h) ; h6 H} over k.

and A

~

k[H]

with

bases

{A(g)

Define a conditional expectation E :

E(Lg
a

quasi-basis

INDEX FOR C*-SUBALGEBRAS

15

Proposition 1.2.9.

Proof.

Since F is in HomA (BA,AA)' there exists a unique

F "" Eh by Proposition 1.3.6, where h "" Ei F(ui)v i ,

that

hE B

such

Since for

aEA

and x E:.- B E(hax)

ha

Eh(",-Xl = F(ax) = aF(x) = aE(hx) = E(ahx),

= ah because E is non-degenerate.

that E(h)

Hence h is in CBCA).

= F(l) = 1.

Suppose

that F is of index-finite type.

exists k in CBCA) such that E

F , k

E

is

>::i uiF(h

for x

E(hkx)

non-degenerate,

invertible.

By the same argument

Similarly

hk = 1.

kh

Conversely suppose that h is invertible. -1 -1 1: i lii E(hh

vix)

there

Then

E(x) = Fk(x) = E(kx)

Since

It is trivial

vix)

&i uiE(vix)

=

E: ~

B.

1.

Thus

h

is

is

of

expectation

of

Then x

and -1

Ei F(xui)h vi Hence {(ui,h -1 Vi)

=

i = l , ... ,n}

is

aquasi-basisforFand

F

Now it is clear that Index F "" Li uih -1 vi

index-finite type. and HF(x)

h-1hx ::: x.

HE (xh- 1 )

Q.E.D.

By the symmetric argument we have the following.

1.4.2.

Proposition index-finite another

type

(not

Let E : B -7 A be a

with

a

conditional

quasi-basis{(ui,v ) ; i""l, ... ,n}. i

necessarily

non-degenerate)

Take

conditional

expectation

1 such that F

More

F : B ~ A. Then we have the fo llowing. 0) There exists a unique h G:CB(A) with E(h) precisely h

=

Ei uiF(V i )·

(2) F is of index-finite type if and only if the above h is invertible. (3)

Suppose

(.euih

-1

that F is of index-finite type.

,vi); i

=

1, ... ,n} and Index F

Moreover HF(x) = HE(h-lx)

Finally

we

shall

:=

Ei uih

Then F has -1

a

quasi-basis

vi'

for xE: CE(A).

give

a

criterion

for

the

uniqueness

of

the

16

YASUO WATATANI

conditional expectation.

Corollary index- fini te

type.

another

Take

centralizer CB(A)C A, then F

Proof.

By

such that F

~

conditional

Q..

expectation

expectation

F : B -) A.

Since. h is in CB(A)C A,

E(hx)

~

hE(x)

If

of the

E.

~

Proposition 1..4.1. there exists hE:.CB(A)

Eh'

F(x)

E : B-'7 A be.

Let

1.4.3.

~

E(h)E(x)

~

for

XE;

with

E(h)

1

B

Q.E.D.

E(x).

1.5. Index under direct sums and tensor products Let E.:B. -"7A. be conditional expectations for i 1.

1,2.

1.

1.

Consider the

direct sum E 1 (j)E 2 ' B1@B2--7A1$A2 defined by (E $E )(b (j)b ) ~ E1 (b 1 )@E 2 (b ) 1 Z 1 Z Z

for bieBi'

Then EItt> E Z is a conditional expectation.

1.5.1.

Proposition

index-finite type. El~E2

LetE i

: Bi -7 Ai be cond~tional expectations

of

Then the. direct sum

: Bl~B2--7Al@A2 is also of index-finite type and Index(E 1G7E ) ~ (Index E )$(Index E 2 )1i- B B by F(x@y)

F is a B-B-bimodu1e map.

=

O':i uiv i ) "'" 1. for fEo is non-degenerate, there exists

(V l "" ,vnl C B in the sense 1,_ •. n)

B-lk.

fi~ld

Then

that

is a quasi-basis for E.

Therefore the

"modular"

automorphism is exactly the Nakayama ,automorphism in [52.]

Example 1.11.8. over IR.

Let k "" A "" Rand B "" Mn (!R) be the

Let h be an invertible element in B,

n by n

Define E ; B-)A

matrices by

INDEX FOR C*-SUBALGEBRAS

E(x) "'" tr(xh) for xGB. where tr is the normalized trace of Mn(JR). the "modular" automorphism eE on C (B) = B is inner and given A 1 SE(x) "" hxh- . In fact, for x, yE-B.

Tr(xyh)

Example 1.11.9. Let k

~

A

~

IR, Bi'2MZ(R) and C n

Define an injection Cn--yCn+l by x~x\&)l.

of

Let

C , n

h

h®, .. fl)h E-e

n

Define an automorphism an on Cn x E:C , n

E(xy) .

n by

~

B €,)· .. 0Bn. 1

Let B be the inductive

for an invertible

element

Define E : B--')IR by Ele

on M 0R). 2

satisfy the modular condition for

n

0:.

M2 (k) be the 2 by 2 matrix algebra OVer k and D We fix the standard matrix units: =

0)

o Each

1)

ell~(O

xED is represented

quasi-bases.

=

B~M2'

e2\~(O

0'

0

I

Let tr be

"" tr&:> ... Q)tr The"

A-bimodule maps with M2

h €: M2 (lR

h xh -1 n n

an(x)

Then O:n can be extended to an automorphisl'YJ a on B.

normalized trace

1

by a matrix

1Z X ) " L e,; "X'·i x 22 The balanced map G ~ El ~E2 of E1 and E2 is defined to be the

x'"

map of

ee'i

(XU x 21

A-bimo.

D to A, given by

1.11.10.

Lemma for E t Then

1

(t

=

1,2).

Let {(ui(t),vi(t» By adding

° if

; i "" 1, ... ,n } be a quasi t

necessary, we assume that n

{(ui(t)@est,viCt)@ets) ; s =- 1,2,

t "" 1,2,

Let

x "" L:k,r x kr ®e kr E--D.

Then we have that

[s,t [i (Ui(t)!Sest)G«viCt)