(note that the hypothesis that the states are pure is not

used in the first part of their proof)). Hence and cOg* are not

disjoint. I

We now investigate some properties of states on the C*-algebra

A

. Before presenting the main results we prove some lemmas.

(10.3) Lemma. Every element A of a;A may be written

uniquely in the form A = X + Y for some X €

A Î

and some Y € L(P). Define a mapping 7t :

A

— ^ lT(P) by tc (A) = Y whenever A = X + Y

with X Ê and Y L(P). Then (L^ , % ) is a representation of

A

.

Proof. If T € AM^nlTfP) then T commutes with every

Ut

so

^ Ut TUt = T .

Hence

A t n l“(P) ~ {o}

and the uniqueness of the expression for each

A

now follows. By (9.18) is an ideal in

A

and it is now

easily verified that iz is a representation. I

- (10.4) Lemma. Let f,gcL^ If <f!Ep(R)g> = 0 for all Borel sets R

then there are disjoint Borel sets S, and in with

Ep(S| )f = f and Ep(S;i)g = g.

Proof. Since and

E

q _{are unitarily equivalent we may}

prove the result with

B

q

,

replacing

Ep

. Suppose

- {x6 IR” :

u(x) > o }

then

= 4,«- = o

so =

o

(almost everywhere). A similar argument shows

(almost everywhere) so

u = o

(almost everywhere). Repeating this

procedure with

u

replaced by the imaginary part of f we deduce

that f g = O' (almost everywhere). Hence f W = o or = o for

almost all ac e Let S, »

{=c; f(^)

t* o } and 5% - 3

o}

then £Q(5,)f = f , (Sa)g = 5 and Si o Sj has measure zero. The result

follows from this. *

(10.5) Definition. For each unit vector f in we denote

by the state on

A

defined by

a^TCA) = JÏ%<(4flAUtf> (VA <44). (This is a state by (9.2).)

(10.6) Theorem. Let f and g be unit vectors in l\/R'^)

then the states and on

A

are disjoint if and only if

there are disjoint Borel sets S, and Sj, with

Ep(S, )f = f and Ep (S_2)g = g.

Proof. Let tv be the representation of A defined in

(10.3) then it follows from the definition of A t that <v^(A) = <fi7T(A)f)>

and OJj(A) = < 3 I

for all A ^ A , Note that ir(A) - A whenever A & L*(P) and Tr(A) = o if A e A^.

<-F I ir(£p(R)) 3 > = < F l E p ( R )5> o

for some R and it follows that co* and

ccÇ

are not disjoint (4.3). Conversely suppose £/»(s,)f = and £/>(Sa)^ = 5 where

S,

and

Sx

are disjoint. Since 7t(a54) =

Lt^(P)

it follows that the range of £p/S, ) is invariant under Tc(/\H) and we shall denote by tt, the

corresponding subrepresentation of tt. Similarly let

Ttx^

be the

subrepresentation corresponding to the range of £"p (Sa). Let & be

the von Neumann algebra generated by TT(^$4) then

£, =

lF(P)

= <8^ and hence

Ep(.s,)

and are equal to their own

central supports in . Since S, and 5% are disjoint, )

and

Ep(Sx)

are orthogonal and it follows that and

Fx

are disjoint (Dixmier (1977) pll5; see also p375 for further details of central supports).

To' complete the proof it is now sufficient to show that the canonical cyclic representations associated with

cop

and oj* are unitarily equivalent to subrepresentations of tt, and

Fx

. Let M be the closure of the linear manifold {%(A)f ; A€/>s4 j = {Af ■

Then

M

is invariant under it

(A)

and

M

is contained in the range of

Ep(St) ,

Let

iZo

be the corresponding subrepresentation of

x,

then

co“ (A) = <(f 1 7Ttf(A)-r ]>

and f is clearly a cyclic vector for

~Fo

• Hence tTo is unitarily equivalent to the canonical cyclic representation associated with (Bratteli and Robinson (1979) p56), A similar argument shows

that the canonical cyclic representation associated with is

(10,7) Corollary. Let f be a unit vector in L^ and let S I be a Borel set with

Ep(S)f 0 ^ Ep(S)f

then

=* 11

£p(S)f

Jl cOg +

1} £p(st-f Ij

where 3 — llEp(s)f 11"* £p(s)f and L = 'Ep(S) f . Thus each cj™ is a mixture of two disjoint states.

Proof. By the theorem

co^

and are disjoint and if

A

then A ^ so •

(uTCA) = *2^ <f|ufAu+f>

Jt L < Ep(s)f j Ut A Ut Ep(s)f > + < E p (s /f I u^A Ut £p(s)^f >

/l£p(s)f 11^ cu” (A)

+ IIEp(S)^ftcJ>uiA)

I States of the form (J* (f e L^) may be generalised to give states which are limits of states described by density operators.

(10.8) Theorem. Let W be a density operator on L*" , then

the formula

C U ( A ) = titc o Ty ( W U ^A U t )

defines a state

co

on

A .

. If A = x+Y where X ^

A t

and Ye 1*(P) then

w ( A ) = Ty(>AJy)

In particular

cJ>

vanishes on oW* ,

Proof. Let A s ^ then A = X+Y for some Xe

A t

and some ye 1*(P). Then for every f€

t

we have

Let ; y £ M } be an orthonormal basis for

it

consisting of

eigenvectors of W and for each

f-eN

let ay be the eigenvalue »

associated with fy . Since IE #y =

I

it follows that

%

(if K-ff I

exists and also that

a, (2 2 L < fK u fx l4 f» > ) = o

since the above sum is uniform by the Weierstrass M-test (Apostol (1974) p223). Hence

GO ( A ) = Z <( fy I W uf AUt >

S a , < f. |Y f,> - Tf ( w y )

so

Ct

3 is well defined and co (A) = l>{wy). Clearly w is linear

and co(z) = = I.

Finally since is an ideal in

A

(9.18) we have

CO (A^A) = w ( x * x + x * y + y * x + y * y ) Ty (wy*y)

^ o

so CO is a state on ^4. I

(10.9) Definition. For each density operator W on L^ we

denote by co" the state on

A

defined in (10.8), A state

co

on xfl will be called normal if there is a density operator W on L^ with

(10.10) Corollary. If W is a density operator on L then CcC is not a normal state on

A

.

Proof. By way of contradiction suppose there is a density

In document An algebraic formulation of asmptotically separable quantum mechanics (Page 92-97)