On the Geometry of State Spaces

FUNCTIONAL ANALYSIS: SURVEYS AND RECENT RESULTS I 1 K.-D. Bierstedt, B. Fuchssteiner (eds.) 0 North-Holland Publishing Company, 1980

ON THE GEOMETRY OF STATE SPACES Erik M. Alfqen LJniversity of Oslo Blindern, O s l o 3 , Norway

The theme of this lecture is a geometric characterization of those compact convex sets which are (up to affine and topoiogical isomorphism) the state spaces of C*-algebras. Recall that a state of a C*-algebra p

of norm one which. is positive in that

is a linear functional p(a*a) 2 0 for all a E 01,

and recall also tliat the states of the particular C*-algebra Mn (of nxn-matrices) are given by the positive semi-definite nxn-matrices

of trace one

- the "density matrices". -

(This relates to the

"mixed states" of quantum theory d n d explains the terminology.) Generally, the w*-compact convex set of states of a C*-algebrd OL. is called the state space of M . The self-adjoint elements of a C*-algebra

(which represent "hounded observables" in physicdl applications) can be faithfully represented (wit11 preservation of linear structure, ordering and norm) Ot

as the continuous affine functions on the state space.

Thus, the

geometry of the state space determines the self-adjoint par-t as a partially ordered, normed linear space (over X). In fact, it can also he shown to determine the "squaring operation" a + a 2 , and

msa

hence the Jordan product

a oh

defined by

a o h = $(ab+ba).

(These

results are all due to Kadison.) Now, the first task is to determine the state spaces of the JB-algebras ("Jordan-Banach" a1 gebras) , which generalize the selfT h i s has been achieved in [ 3 1 . There are four axioms: the first two pertain to the facial structure and

-adjoint part of C*-algebras.

serve to provide spectral theory dnd functional calculus; t h e third gives the splitting of states into atomic and non-atomic parts (generalizing the corresponding splitting of measures in the abelian casek the fourth is a crucial geometric condition called the Hilbert ball property. It requires that the face B(p,n) generated by two distinct extreme points

p

and

c~ shall be affinely isomorphic to the

closed unit ball of some real Hilhert space.

(Any finite or infinite

54

E.M. ALFSEN Tur'nirig t o I

dirneiisiori i s p o : : r , i i > l + -).

wc n o t c ttiat tfie f i r s t

ic::,

t w o a x i o m s r ' e l a - i e t o t l l r rncai;urc-.inent p ~ o r ~ s( s "yes-no"

and p r o p o s i t i o n a l c a l c u l u s ) , w h i l e

"symnictry o f t r ' a n s i t i c J r i p r o b a b i l i t i e s " .

(See [ 3 ] for detaj.ls. 1

I t r e m a i n : ; t o r I - i d i - d c t e r i , : e tlic c 1 a t e i,p

o+

n t : ; d i f f i c u l t i c > ; < of^

hirid d u e t o t h c j l a c k o f u n i q u f : i i

riebi


the

w e m u s t f i r i d the r i g h t

aricj

g e o m e t r i c a x i o m s which

A

i s a , J o t > d a r ~Iiomoniorphisrn

PI ; m o r e o v e r , cp

.

iinto

a type I

rei~r'esei-itation iE

i to be a dcr1-e

ii,

a-weakly d e n s e i n

A

or

cp

br>ief d e t o u r i n t o the

,I

A r c p r e s e n t a t i o n o f a ,JR-JI ge-

r e p r - e : ; e n t a t i o r i t h e o r y of JB-alget.i';ir;.

(A " , J B W - a l g

i i ri

"

t r i v i a l c e n t e r ' a n d cont a i . n s r n i n j m , j l i d e m p o t e n t s . JBW-algehr'a

il

M

i s t h e w * - . t o p o l o g y 01-

PI

JRbJ-faclor cp(A)

js

J B - a l g e b r l a which i,; a l s o

a dual Uariach : ; p a c e ,
of

Theim is

make t h i s cl-ioice p o s s i b l e . B e f o r e w e approach t h i s p r o t ) l e i n , we m a k e

tr,i

d

multipticativc s t r ' u c t u r ' e

dc-l-erminetl by i t : ; s t a t e :)>ace a l o n e .

C * - a l g e b r , a i s rimt

a c l i o i c e t o be made, will

of C*-alF;ei)raq ilmong

Or r i l l J!i-cilecl

II!F r ' t d l - t : SPdCES

meilsuremeiiLs

l a s t one i s c o n n e c t e d w i - t l i t h e

tiit.

'I

i f it liar;

?'lie "17-weak

topoiogy"

a:; a d u d 1 tianach s p a c e

Note I l i a t the n o t i o n o f a r e p r ' e -

with u n i q u e p r c - d u a l , r f . [ 3 ] , [ C ] . )

s e n t a t i o n i s sti1.l r d t l i c r c o r i c r e i e , e\iert i n t11i:s new c o n t e x t , since tkiere arc only

d

f e w , wclI.-knowr~, [pr'olotyl!e? of t y p e I

They ar'e t h e J B W - d g e b r ~ d s o f all l ~ c ~ u i n i l s~ ed l f - a d j o i n t

t h e Cayley numbers.

of a 1 I s e l i - a ( I j o i n t

PI:

t3

p

of a J B - a l g : c h r a

entatiori

r a l way a densc-. r c p r

cp

P

:A

M

-f

3 x 3 - m a t r i c , e s over'

.

( S e e [ 5 I for. d e i - a i 1 s

To cacti pur-e s t a t e

operators on a

space, the s p i n f a c t o r s , and

r e a l , c o m p l e x and quaternionic H i 1 b i . r . t t l i e e x c e p t i o n a l algc

JRW-factors.

I,

A

corresponds i n a riatu-

( u n i q u e u p to J o r d a n equi-

v a l e n c e ) , a n d each d e r l c c r e p r e s e n t d t i o n can be o h t a i n e d i n this w a y . (Specifically,

M~ =

c ( o ) ~ * *

" c e n t r a l s u p p o r t " ot

0

,

cf.

and

cp(a) P

[3],[S].)

=

c(p)a

of c o m p l e x t y p e t f ~ l dle n s e repri-:;t.rltatiol?-~!~~,ritatioiis tpp

f o r some c o m p l e x H j l h e r t s p a c e

HD

.

,

wtiere

c ( p )

istlie

W e sdy t h a t a JB-algebr'd i s

are i n t o

MP = B(Hp)

(JH-algebras of r e a l , q u a t e r n i -

o n i c , s p i n - f a c t o r > , and t o t a l l y e x c e p t i o n a l t y p e a r e d e f i n e d analogC l e a r l y , the s e l f - a d j o i n t p a r t o f a C * - a l g e b r a i s a J B - a l g e bra o f c o m p l e x t y p e , s i n c e t h e d e n s e r e p r e s e n t a t i o n s r e d u c e to

ously.)

the c u s t o m a r y G N S - r e p r e s e n t a t i o n s

false.

i n t h i s case.

(pp

But t h e converse i s

An e x a m p l e t o t h i s e f f e c t i s t h e J B - a l g e b r a

A

o f all c o n -

t i n u o u s f u n c t i o n s frlom t h e u n i t c i r c l e i n t o t h e s e l f - a d j o i n t c o m p l e x 2xZ-mat.rices

such t h a t

f(-z) = f ( z ) ~( t = t r a n s p o s e ) . The d e n s e r e z of the unit circle,

p r e s e n t a t i o n s a r e t h e e v a l u a t i o n s at p o i n t s

so

A

i s o f complex t y p e .

Assuming

A

t o be t h e s e l f - a d j o i n t

part

ON THE GEOMETRY OF STATE SPACES

55 f,g E

of a C*-algebra, wc consider tihe -two constant functions

A

de-

fined by

By linear algebra,

and no others.

i(fg-gf)

can take the two values

By continuity, the same sign must hold throughout.

But this contradicts the definition of

A

.

An important step towards the final characterization is the follA JB-algebra A with state space K is of complex type iff B(p,o) is of dimension 3 or 1 f o r every pair p , u of owing result:

-

-

distinct extreme points ( dim F3(p , G ) = 3 'pp 'p, ). The example above shows that tliis dimension condition for. the balls B(p,o) a C*-algebra.

will not suffice to make A the self-adjoint part of In view of this example, it is perhaps not so surpris-

ing that there is an additional condition of dlgebraic topological nature which is needed: It must be possible to make a simultaneous in such a way that "neighbourorientation of all the 3-balls R ( p , o ) ing balls are oriented the same way". To make this statement precise, we consider the state space

K

of a JR-algebra A of complex type. First we define the topological space (D of all parametrized facial halls of K This is the set

.

of all injective affirie maps 3-ball and from

E3

(p:E3

(p(E3)

is a face of K (with usual topology) to

shall eliminate the parameters:

a

by writing

when

(p-Q

$-locpE0(3)

.

when

K

+

E3

where

is the standard

, provided with the weak topology K

(with w*-topology).

Then we

We define an equivalence relation on

(p(E3) = $ ( E 3 )

, or what is equivalent,

The quotient, which we denote by

@ / 3 ( 3 ) , is

called the space of all facial balls. In a similar way we define a stronger relation on 6 by writing C P ~ $when $ - l o ( p E SO(3) The new quotient, which we denote by ' / / s 0 ( 3 ) , is called the space of

.

all oriented facial balls.

Clearly

'/S0(3)

-t

'/0(3)

and this bundle can be shown to be locally trivial.

K

is a Z2-bundle, By definition,

is orientable if it is globally trivial. Now we are in the position to state the main result.

f o r details.)

(See [ 4 1

E.M. ALFSEN

56 Theorem.

Let

K

b e the s t a t i f spacci o f a J R - a l g e b r a

s a t i s f i e s t h e a f o r e m e n t i o n e d geornetiTica1 r e q u i r e m e n t s o f

i s t h e s e l f - a d j o i n t p a r t of a C*-algebra

K

the balls

(i)

K

a r e of d i m e n s i o n

K

[ 3 ] ; then

if a n d o n l y if 3

or

1

f o r a l l pair..;

o f d i s t iinct e x t r e m e p o i n t s ,

p ,a

(ii)

B(p,o)

(i.e.

is orientahle.

N o t e a l s o t h a t t h e o r i c n t e d s t d t e space ( i . e . t h e c o m p a c t c o n v e x

set

K t o g e t h e r w i t h a p r e s c r i h e d contiriuou::

c r o s s - s e c t i o n of t h e

b u n d l e ) , is a d u a l o b j e c t from which w e c a n r e c a p t u r e t h e C * - a l g s h r a .

References 1.

E.M.

A l f s e n a n d F.W. S h u l t z , N o n - c o m m u t a t i v e

spectral theory for

a f f i n e f u n c t i o n s p a c e s on convex s c t s , M e r n o i r s A.M.S. 172 ( 1 9 7 6 ) .

2.

E.M.

A l f s e n a n d F.W. S h u l t z , N o n - c o m m u t a t i v e

J o r d a n a l g e b r a s , I’roc 3.

E.M.

.

London M a t h . S O C

.

s p e c t r a l theory a n d

(To a p p e a r )

A l f s e n a n d F.W. S h u l t z , S t a t c sp;ices o f J o r 7 d a n a l g e b r a s ,

Acta M a t h . 1 4 0 (1978), 1 5 5 - 1 9 0 .

4.

5.

E . M . A l f s e n , F.W. Shultz a n d H . H a n c h e - O l s e n , C*-algebras. (To appear)

E.M.

Alfserl,

S t a t e s p a c e s of

F.W. S h u l t z a n d E . S t $ r r n e r , A G e l f a n d - N e u m a r k , A d v a n c e s in M a t h . 2 8 ( 1 9 7 8 ) , 1 1 - 5 6 .

t h e o r e m for J o r d a n a l g e b r a s

6.

F.W. S h u l t z , O n normed J o r d a n a l g e b r a s w h i c h a r e B a n a c h d u a l spaces

,

J . Functional Anal.

31 (1979), 3 6 0 - 3 7 6 .