REFERENCE

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

CONDITIONAL EXPECTATIONS

IK QUANTUM PROBABILITY THEORY

S. Twareque Ali

and

Gerard G. Emch

1974 MIRAMARE-TRIESTE

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL'PHYSICS

COVARIANT CONDITIONAL EXPECTATIONS IN QUANTUM PROBABILITY THEORY *

S. Twareque Ali

• International Centre for Theoretical Physics, Trieste, Italy,

and

Gerard G. Emch

Departments of Mathematics and of Physics and Astronomy,University of Rochester, Rochester, NY, USA.

MIRAMARE - TRIESTEJuly

* • To be submitted for publication.

wwm

This paper investigates the

extension of the concept of conditional

expectation to a non-commutative

probability theory, subject to some

group covariance requirements.

Over the last twenty years various extensions of the concept of

conditional expectation have been explored [3,7»H,12,13,l6]. The line of

investigation, to which the present paper belongs, deals with the possible

generalizations of the idea of conditional expectation from classical probability

theory [8] to non-commutative probability theories, in which the classical

algebra J^_ (X,y) of stochastic variables is replaced by a non-abelian von

Neumann algebra. Such a generalization, apart from its intrinsic mathematical

interest, is useful in discussions of the measurement process in quantum

mechanics [l,3,*0> an(l in the theory of coarse-graining in statistical mechanics.

[1?]. In this paper, we adopt Davies and Levis1 [3] definition of a conditional

expectation, as the dual to the concept of a measurement. We next subject a

measurement to a condition of covariance under a locally compact symmetry

group. The problem which then concerns us is that of expressing covariant

measurements in terms of certain operator-functions, these latter providing a

better basis for making contact with the theory of observables in quantum

mechanics [l]. We succeed in getting such a description for two classes of

measurements, which actually appear to be sufficient for all cases of

physical interest. We wish to point out that, in contrast to Davies'work [If],

we found it useful to stress here the measure theoretic aspects of a conditional

expectation and a measurement, A description of our results can be found at the

end of Sec. I.

I. PRELIMINARY DEFINITIONS AND FORMULATION OF THE PROBLEM

Throughout this paper X shall denote a separable, locally compact,

Hausdorff space, K(x) the space of all continuous complex-valued functions on

X which have compact supports (equipped with the standard inductive topology), and

W a separable Hilbert space, J£$¥), &($) and '&($) shall denote, respectively,

the Banach spaces of all bounded operators (on $f) under the operator norm, of

compact operators under the operator norm, and of trace-class operators under the

trace norm. It is well known [15] that the Banach space duality relations

hold between these spaces. We shall denote the positive cones of these spaces

(which in fact generate the spaces) by K(x) itfW) , etc.

-2-

Definition 1.1

A measurement on X is a bilinear map,

which is continuous and satisfies:

2) if f is a sequence of functions in K(X) such that f / I ,n -1

pointwise, then tr[£(fn,f)]y*tr f, V* f € cT'&O*. where 'tr

denotes the trace operation on «y W) •

Definition 1.2

A conditional expectation is a "bilinear map,

£*-.which is dual to a measurement through the relation

?)i(1.1)

As shown by Davies [k], Definition 1.2 generalizes the concept of a

conditional expectation of Umegaki [l6], which in turn generalizes, to operator

algebras, the concept of conditional expectation in classical probability theory

(in the sense of S.-T.C. Moy [ll]. Further, the measurement g is also a

generalization of von Neumann's expression for the "collapse of a wave packet"

in the course of a measurement [3].

Let G be a locally compact group, and suppose that for each g<£G there

exists a positive, norm-preserving automorphism of Zf&O onto itself. We shall

assume that this automorphism is unitarily implementable, so that we have a

weakly (hence strongly) continuous mapping g n- U of G into the set of unitary

operators of£f. The space X on which the measurement § is defined will be

assumed to be a transitive space in the sense of Mackey [9]» with respect to the

action of G (assumed to be acting on the right). Hence X will be

identifiable with the quotient G/H, of G by some closed subgroup H. In

particular, we shall take G to be a semi-direct product of the form:

G = H® X

where X itself is a commutative symmetry group of the system. Let g[f]

(= U *f>U ), g[f] and [x]g denote the transforms of f£j°(^0, f£K(x) and

respectively, under g

-3-

Definition 1.3

The measurement £ is said to be 'covariant under the action of the group

G if </-g£G, f£K(x) and

= r'l?(f,?Le])l.The conditional expectation g* is said to be covariant if it is dual to a

covariant measurement S

For feK(X), le t a(f) € Jf(#) be defined by

the resulting map a: f £ K(x) «• a.{t)e£&f} is positive and thus defines a

positive-operator-valued measure on X, which we denote by

a: EeB(x) w- a(E)eo£$0+, where B(X) is the set of all Borel sets of X. We

have, in particular, a(x) = I, the identity element in £&f).

Definition l.k

The normalized, positive-operator-valued (POV) measure f a(f), or

equivalently E » a(E), defined through Eq.. (1.3) is said to be the observable

determined by the measurement g .

The observable E <•*• a(E) determined by a covariant measurement satisfies

the relation

d 1 $ > d^)

[E]g~ being the translate of the set EeB(X) through g" .

Definition 1.5

The pair E •+ a(E), g *+ U satisfying Eq. {l.k) is said to form a

generalized system of imprimitivity. (in case a(E) is projection valued, this

reduces to a transitive system of imprimitivity in the sense of Mackey [<?])•

In the next section we obtain (Theorem 2.1) a general integral representation

for covariant measurements in terms of a Cf&P) space valued density function

x •+ I"}* (x). Next we find (Theorem 3.1) the conditions under which this density

may be written in terms of a certain bounded operator-valued function x *• T ,

namely Tl Cx) » T (f). This case covers all situations of physical interest

(including those treated in [l]), except the.case of a covariant observable with

a purely non-atomic projection-valued measure. This last case, however, is taken

-k-

care of by the results of the final section of the paper, where we analyse

a situation (Theorem U.l) in which' T. may "be unbounded. This allows us tox . . . .

answer the question as to which covariant measurements determine o"bservables

such as are projection-valued measures(Theorem k.2).

II. A CHARACTERIZATION THEOREM FOR COVARIANT MEASUREMENTS

Theorem 2.1

Let g be a measurement on a locally compact Hausdorff space X

S, : K(X)

$f being a separable Hilbert space, and let £ satisfy the group covariance

condition

where G is a locally compact group of the form

G = H®X,

with H a closed subgroup of G and X itself a commutative group. Further,

let there exist a strongly continuous representation g *+ U of G (implementing

the positive norm continuous automorphisms of &/{#) induced by G) by unitary

operators on $f.

Then, for each P, ^ is completely defined by a function.

which is positive if f €JW) , and which is measurable with respect to the Haar

measure y on X. £ is further given by the integral relation:

S(f,f) = f(x)Pr(x.)/ACc(.?t), (2.1)Jx /

vfeK(X) and f*etfkff), the convergence of the integral being in the weak topology

of tftyf). For all xeX and fcfffi), I£(x) satisfies the group covariance

condition

? u ? f u ; . (2.2)

Proof

The proof of this theorem will be given in four stages:

(a) The measure <?(f, f), for fixed f € 3"&) , will be written in terms of its

absolute value \B^\ as

-5-

f "being some \8J - jaeaaurable function with, values in

(b) For any geG and f£K(X) i t wi l l he shown tha t

and also that the map g »•»- ]<f rp-||(f), for fixed feK(x) and fG?°C#0, is

continuous.

(c) The results of (t>) will "be used to prove that the Haar measure u on X

dominates the measure | £p\ in (a).

(d) Using (c), the desired form for £f(f,p) will be obtained by writing first

Ifpl * Yf • V ,

where yp is some y-measurable numerical function, and then setting

V? (x) = f.f(x) YfU).

We proceed now with the detailed proof.

(a) For a fixed f£&($), ^(f,f) (with f£K(X)) . defines a vector-valued measure

on X, with values in v $f)* i.e.,

S(f, f) - |f(f) = \ fM &*&*). .(2.3)

We shall prove that.this measure is majorizable for the norm of [f&)t in the

sense of Bourbaki [2], and that \£_f\ exists. Indeed, from the definition of

the absolute value of a measure.

1£P| - Sup lA

where Ao^Cf) = tr[A^(f) ], for all feK(X). Since g is a continuous

bilinear map on the normed spaces K(x) and tf&f), for every compact subset

V ^ x it follows [5] that g restricted to K(V) is bounded by some number

. Hence, for all f£K(V)+ ,

f\\KCV)

-6-

which proves our contention.

Thus, £« is a measure on X with, values in 3"&0 , the strong dual of

the separable normed space &(#*;. Further, £=is majorizable for the norm of

V&f), and has as its absolute value the bounded positive measure |£p|> Hence

there exists [2] a function

/f :X—> ?(&),.measurable with respect to \£f\

a n^ satisfying:

(2) for all f£a7'<^0, | jf. » (x) j |y^\ = 1, for almost all x (with respect to

\if\) 'We may write then

so that Eq. (2.3) may be rewritten as

(b) Consider next the measure |£ rp-ij for any geG. We shall prove that

for all

Indeed, we have, for any

5^pIIAI;<J

p

u ? |trfA9-J[5(/',/[p])J]|, in virtue of Eq. (1.2),

Henc e,

Also, since by Eq. (1.2), g[S(g[f],f )] = (f, g[f])» therefore, we can

similarly show that

, +for all feKlXj , whence Eq. (2.5) follows.

-7-

To prove t h e c o n t i n u i t y of t h e map g **• \g- , , | ( f ) , for f ixed fcK(x)

and fe TS ), we prove first tne continuity in the norm topology of Hffi), of

the map g<+ g[A] = U AU *, for a l l k£&&). (Since G is a group, i t i sDO

enough to prove continuity in a neighbourhood of the identity element e of G)

We have

AU/ - Ai

Hit 3

Now, <gW being an ideal in the C* - algebra

compact operator. Hence, there exists some vector

, U AU * -A i s also a6 S

£ in c£f~ for which

< U\\im II (u;* - 1 )\im I (u? - r)A ?„0, as

because of the strong continuity of g •+ U . In fact, the continuity ofo

g H- u AU * is also seen from this proof to be uniform in the unit ball ofS g. This result now implies the continuity, in the trace norm topology of,

, of the map g H- g[f] = U*fU . Indeed,o o

\\u4A'ii A 11 <

J

-+•0, as g -+• e .

The continuity of g H- \g . J(f) is now immediate, for

SIf o ItS J

hr User,p)]i'£ KCX)

U\I<J UfUf

— i| o l\ yj- Hi II K(yj (I Ug Ug j3 JJo'/-'j , where V is some compact

region containing the support of f

- • 0 , as g -+• e .

(c) We shall prove next that the right invariant Haar measure ]i on X

dominates the positive measure \So\-

This will be achieved by first showing that u dominates the measures

] for y-almost all xeZ [lO]. To do this, let EeB(x). Then the set

E1 of all (x,y)€X x X such that xy' £ E is also a Borel set, and for all

x and y, Y-PI (x»y) - X-^i^y" )• (X denoting, as usual, the characteristic

function). Applying the Fubini theorem to x^ v e

fX

x

The left-hand side, because of the invariance of ]i is equal to u(E) |g|aj (X),

" ) = x l ( y x ~ )and since Xgtxy" ) = x

E-l(yx~ ) - X[E-l]x^y)» the right-hand side is equal to

Thus,

Hence, y(E) = 0 implies |£j([E~ ]x) = 0, for all x, except perhaps on—1

a set of y-measure zero. But 1J(E) = 0 iff u(E ) = 0, so that (in virtue ofEq. (2.5) above) u(E) = 0 implies \g r -. | (E) = 0, for y-almost all x,

whence our assertion follows.

The continuity of i£xrp]i i n ^ (as proved in (t>) above) now shows

that if'{x } is a sequence converging to e, then |g rplKE^ converges ton —X L1J

n

|gp|(E). But since U has support on the whole of X (being the invariant

Haar measure), therefore, ]g rpJ(E) = 0 for p-almost all x implies that

every neighbourhood of e has at least one point x1 for which

\8 trp-il(E) = 0. Let us choose a sequence of neighbourhoods ' fr } of e

:« lit Ji> V I-

such that V C V for all n, and further s.o that every neighbourhood Vn+i n • • • • •' ' . . e

of e contains at least one V . In each. V . let us choose a point x

such that |g r -||CE) = 0 . Then, taking the limit of n •*• », we immediately

get the result that u(E) * 0 implies |£P|(E) - 0, proving that u dominates

pi *

(d) Using the result in (c) above we have

(2.6)

for all feK(X), where Yp is a positive numerical function, measurable with

respect to u . Let us now define x "•*• n,(x) by setting

where f_p is defined as in Eq. (2.1+). Then it is straightforward to check

that rj, satisfies the stated conditions as a measurable function of x in

*fffi)i positive if f itself is positive. Also obvious is the covariance

property of FP as spelled out in Eq. (2.2). Hence finally, combining Eqs.

(2.k)t (2.6) and (2.7) we get the stated relation

The function x ^ ( x ) obtained above is measurable in the sense that

for each AzXQfr), the numerical function x '-> tr[Al^(x)] is measurable.

Definition 2.1

A measurable function x -»-Pp(x), of X into 3"W) t will be called the

density of the measurement g , for the state p and with respect to the base y

(assumed positive), if and only if it satisfies Eq. (2,l). It will further be

called,covariant if it also satisfies Eq. (2.2).

The density x*l"f (x), as defined through Eq. (2.7) is determined

completely, except at most on a set of u-measure zero, which could perhaps

depend on f . The next question thus is to find conditions under which this

null set is independent of f , for then we could use the linearity of Z (f »f)

in p. to define a linear operator T on tfffl) in the manner

(2.8)

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on a suitable domain in $&).. The'remainder of this paper is devoted to

answering this question in two cases of physical interest.

By its very construction, the function x **• T will have the properties;

(a) For almost all xfX (with respect to u), the operator T :,/(#) -*• r/G£Q

is positive and linear. It is "bounded whenever it is defined on the whole of

(•fa) If Cftf&f), '$&))* denotes the set of all positive linear maps on

hounded or otherwise, then the (positive-operator-valued) function

x£X «• T £$£f&), ifffl)) is measurable in the sense that for each f belonging

to the intersection of the domains of the operators T ( x£X, except perhaps

on a set of u-measure zero), and each A€jpi°), the numerical map

x «• tr[AT (f)] is measurable.

(c) For all P in the intersection of the domains of the operators T (p_

almost all xeX),

(2.9)

and, if £ is also covariant,

Tt.3f (f)

(d) For all P which are in the intersection of the domains of the operators

•p (vi-almost all

Definition 2.2

A measurable function x •+ T of x into (7(tf$0, ZfW)) will be

called the operator function inducing the measurement g , with respect to the .

base y (assumed positive), if and only if it satisfies Eq.. (2.9) (and will

further be called covariant if it also satisfies Eq. (2.10)) on a domain which

is dense in

III. MAJORIZABLE COVARIANT MEASUREMENTS

Lemma 3.1

Let X - ^ P P ( X ) be the density, with respect to p, of the covariant

measurement <f . Then, if for any xeX, Ff>(x) satisfies

(3.1)

for all fe/89, where \\r\\ is'a positive .number, then ||r|| is

independent of x and the'relation (3.1)is satisfied for all

Proof

Since £ is covariant, Tf>(x) satisfies Eg.(2.2) and hence, for any

xfcX

afrom which the result follows

Definition 3.1

Let i^fjid) be the density (with respect to y) for the covariant

measurement 8 • Then, Pe{x) will be called mat1orizable if, and only if it

satisfies the relation (3.1) at some, hence all, points x€X and

Lemma 3.2

If x *+ Vf> (x) is majorizable, then as a function of x it is

continuous in the trace norm topology of $"(&),

Proof

Let o denote the origin of the Abelian group X. We have, in virtue

of Eq. (2.2),

r? c*) - rP (o) i t m = I v* r^w (o;ux - rP to

<

-12-

because of the linearity, in f> of £(jf,f). Thus using. (3.1)

But since x U py * is continuous (cf. part ("b) in the proof of Theorem 2.1)

we get at once that

as x •+• o ^

Lemma 3.3

Let the covariant measurement £ be induced by the positive operator

valued function x '•+• T . Then, if T is a bounded operator at any point x,

the function x ••*• T is uniformly bounded for all x<?X.

The proof is an immediate consequence of the relation (2.8) and Lemma 3.1

Theorem 3.1

A measurement & on the locally compact Hausdorff space X, vhich is

covariant with respect to the action of the group G = H(£)X (assumed unitarily

implementable as before), is induced by a unique, (up to a set of y-measure •

zero) bounded, positive-valued function x ** T , with respect to the Haar

measure u on X, if its density function is majorizable.

Conversely, if w X ^ T ^ W i J($))* is a measurable function

such that

(a) I T ^ C ) ! ^ < I! T ff ftp H ^ , for all f<7WO andsome xeX, ||T11 being a positive number;

tr[ I f (x) T (f) u(dx)] / tr P , where f is a sequence in K(x)

+such that f SI, pointwise and

(b)

^ ^ t f p } t for all gcG and

duces a covariant measu

density function x^Pp(x) .

Proof

If x1-*- Pp(x) is majorizable then it is continuous (Lemma 3-2), and

hence as a function of x (for each fcVW it is completely determined at all

then x •* T induces a covariant measurement having a majorizable•x

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iE1""

points of X. Therefore, Eq,. (.2,8) jaay he used to define the operator

function x •-»• T > which "by the stated condition would he uniformly bounded.

Uniqueness in the sense stated is obvious.

Conversely, let x •+ T x be, given as stated. Then clearly it is

uniformly "bounded (cf. Lemma 3.1). Let f£K(x) and f€$$f). Then, for all5

fCx) t

'Pl«w J< °o

Thus the integral converges in the weak topology of tfffl). It is then easy to

see that

'Xis a continuous bilinear map and hence, in virtue of conditions (b) and (c)

defines a covariant measurement in the manner stated!

IV. PURE MEASUREMENTS

To proceed further, we need the concepts of a modified and a pure

measurement.

Following Definition (l.h) of the observable determined by a measurement,

we remarked that the observable determined by a covariant measurement satisfies

the generalized imprimitivity condition:

Suppose that there exists on the Borel sets of X a projection valued measure

E H- p(E) which also satisfies the relation

In that case* by the imprimitivity theorem of Mackey [9] the Hilbert space

is isomorphic to the space L^(Xfu) of all functions §: X -»• 3i which satisfy

vhere ?<5 is a Hirbert space carrying a unitary representation of the subgroup

H of G. Further, on L-^CX, y) the operator P(E) is given "by :A J^

where X I s the cha rac te r i s t i c function of the set E.

Definition k.l

A measurement K(x) X $"($) •*• Jffl) t covariant with respect to the

representation g >-*• \J£J&$) of G-, is said to "be a modified measurement if thereg

exists a projection valued measure E * P(E) on $f (E£B(X)) such thatEM- P(E) and g >•+- U form a protective system of imprimitivity on H, in the

sense of Mackey.

To define a pure measurement ve need a few more results. These will "be

stated in Lemmata k.l to J+.3

With the Hilbert space appearing in Mackey's imprimitivity theorem

as stated above, let L 1 _. (x, \i) denote the Banach space of all Borel functions

r_ : X -*-c7*(?<0, for which the norm defined by

^ x ) ' (U.3)

is finite, where \\ • • • \\ ft^A <3-enotes the trace norm in tf&O . Two functions

in L [frtA^* y) are identified if they differ on a set of y-measure zero.

Let o ^ y ^ C X , y) denote the corresponding space where such functions are not

identified, and let K J « A ( X ) be the subset of functions in L^^lX, y) which

are continuous and have compact supports. K^/^tX) is dense [2] in

fr/ovx (X, y) with respect to the semi-norm defined in (U.3). Also, let X be

the isometric,positive linear map

X:

which on elements in J*C^)+ of the form £<g>£, for $€.&, is defined by

One has then

= [XCU,- 1 5 -

for a l l g € Q and ji-alaoat a l l xeX.

Lemma U.I

The mapping X; ffffi'•*• L^-^CX, u) defined above is invariant under the

unitaries of ' {P(E)}" (the von Neumann algebra in JtW) generated by the

projection operators P(E) defined in (1*.2) above), and conversely, any operator

in J£&) which leaves A invariant is a unitary operator in ' {P(E)}''' Explicitly,

for all iaft \{Ul ®%t&) - \ti®l) if and only If U is a unitary operator

in 'fr(E)}"

The proof is easy and ve omit itj|

Lemma \,2.

Let g "be a measurement whose observable is a projection valued measure

E H - P ( E ) . Then, for all feK(X) and fetf&O, Z satisfies <f(f,f=) =

? i?>Uf>lf) for all unitary UC <P.(E)>".

Proof

If g* i s the conditional expectation dual to £ , then

tr

I being the identity element in JCffl), and where we have set (by the usual

method of extending a measure)

Thus,

^ * C E , I ) = P(E) • (U.6)

If now A is a positive operator in JfC#), ve have,

S*CEf]F , lAl! - A ) > o ,

for any E, F€B(X), since l!Ally(j ) x ^ A* Thus,

o < ff*CFnF, A ) < /JA// (fCEn'F, i) = |Af p( EnLet F' denote the complement of F in X. Then,

0 < £*(E.C) F , A) P ( F ' ) < IAJJPCE O F ) P ( F ' ) = O ,

Thus,

f*(EnF, A) p e r ) ' = o, •

-16-

so that,

S*(tr\r,A) = e*CEf\F, A ) [ P ( F ) •«• P(r'j]

= <?*(EnF, A) POO.

Similarly,

^(EAF, A) = P(F) (EOF, A),

so that,

e*(Ef]F, A)P(F) = P(F) £""CEnF, A ) .

(U.8)Replacing F by F' in ( .7) we get

£*Uf\F' , A) P(F) ' = P(F) g*(E OF', A) = 0 .

(U.9)

Adding (U.8) and C^.9) we have finally,

S*tE, A) P(F) . - P(F) £* (E , A) .(U.10.)

Hence, for a l l AeJ&ffl and a l l E€B(X), £*(E,A) commutes with every element

in 'fc(E)}11. If ^ f rCE)}" i s unitary, then clearly

&* g*(E, A) U = ff*(E, A),

so that, for all f / W )

O ( , ) f] = tr

i.e. ,

tr [A

from which the desired result follows

-IT-

Lemma k. 3

Let the measurement g he induced by'an operator function x *+ T , with

respect to the base u. Then, for Zl£ j£(ffl unitary, £ satisfies

Sit, Itffr) = <?(*,(=>). for all feK(X) and pc/OP)-, if and only if Tx

satisfies T^fft*) - T^Cf) for u almost all x£X and all f in the common

dense domain the operators T .

The proof is trivial§

In general a modified measurement need not "be invariant under the

unitaries of ' {p(E)>" in the sense that g(,t,Kftt*) need not "be the same as

We shall, therefore, introduce the concept of a pure measurement.

Definition U.2

A modified measurement § will be said to be pure if and only if it

satisfies the condition <f(f,p) = £(f£/f>U*), for all unitary U£ {P(E)}".

Remark

A covariant measurement whose observable is a projection valued measure

is necessarily modified and in fact, in virtue of Lemma ^.2, is also pure.

We state next a lemma, concerning the domains JD of the operators T ,

which will prove useful in the sequel.

Lemma ktk

Let x *-+ T induce a measurement g which is covariant. Assume that

JD (dense inji?') is invariant under the action of the unitary operators

U , V*x£X, i.e.4 Uv^oto Ux " ^ ^ » ^ € X , Then £) J^isx. for all xeX,

except perhaps on a set of y-measure zero. The same result is true if J©

is replaced by Jf) for any x€X,XJ

Proof

Remark

The space K^/^Cx) is invariant under the operators U ,

Consider now a pure measurement £ which has a density x **• I~f>(x) with

respect to the base u« As a result of Lemma ^.1 we see that for li-almost

all x£X the map

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may be considered, as one.acting in the jnanner

I —* (f(JC} * " * y^ '

(We use the same symbol I"1 to avoid notational pedantry.) Consider next the

restriction of this latter map to K«./<^\(X), and suppose that for some x e X

there exists a positive Borel measure y on X for which

f_£ K.j-,g,Ax). If £" is also covariant, then, in virtue of Eq,. {k.5) it is

clear that if the relation (1+.12) holds for some point x then a similaro

relation holds for all points xeX. Further, the measures y may all "be

obtained from u in the manner:

ff>Tf€K(X).

[A remark on a technical point is in order here. Let IT be the

canonical surjection

If K^,^v(x) denotes the image, under IT, of K^^^CX) in L^y^(X> y), then

clearly £-,,-,» (x) is dense in L^/^\(X, y). For an element f_" in

, which is an equivalence class, we may choose as its representative the

corresponding element t_ in Ky,_g, (X_]_. Clearly, this choice is both linear

and positive, and it is in this sense that the restriction to Ky^(X) of the

map (U.ll) is to be understood. However, as is easily seen, such a choice, in

a simultaneously positive and linear fashion, cannot be extended to the entire

spaces L J ^ J ( X , |i) and JfJ-^tX, u).]

Definition U.3

Let x •+ Tf> (x) be the density (with respect to u) of the pure,

covariant measurement <f . Then I >{x) • will be called relatively ma.iorizable

if and only if it satisfies the relation (U.12) for some, hence all, points

x€X and all f 6 ^ 0 such that

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Remark

If for a pure covariant measurement £ , the density Fp(x) is majorizable,

then it is also relatively majorizable.

Theorem U.I

A pure measurement S , on the locally compact Hausdorff space X, which

is covariant with respect to the action of the group G = H©X (assumed

unitarily implementable as "before), ia induced by a unique, (up to a set of

y-measure zero) positive-operator-valued function i * T , with respect to the

Haar measure \x on X, if its density function is relatively majorizable. For

y-almost all x€X, T is defined on the domain (dense in ^(550)

Conversely, if x€ X •*• T €&&{&)), /(#") ) + is a measurable function

(defined on domain <&), such that

(a) T (?/pa») » T (f), V^unitary ^^{P(E)}M a n d / f ^ ^ ;

< J jAfCTO^y tZxUxO, 4-?£$ and some

x£X, where u is a positive Borel measure on X;

(c) tr [ ff, (x) T xCf) itCeix)j^f^ where fQ is a sequence in K(X)+

such that f /• 1, pointwise, and

all S^G and fe^; then

x •+ T induces a pure, covariant measurement having a relatively

majorizahle density function xw-

Proof

We first show that for a pure, covariant measurement x ** f"p(x) is a

continuous function of x for f£$. Indeed, by definition*

- r f co

(cf. Remark following Lemma

X• 0, as x ->• o ,

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in virtue of the uniform continuity of the functions x U p) (JC) on compact

sets.

Thus, for all PC*@, Ccf. Lemma h.h) we may define an operator T

through Eq. C2.8). Also, since & is dense in c/C ) Ccf. Remark following

Lemma 1+.*+), x •+ T would indeed induce £ .

To prove the converse, we note that •</• f£^ t € K(V) for some compact

set VdX and

tr [ A /

Now, the function x •* (Xp)(x) is continuous and has compact support, so that

the integral within the square brackets converges uniformly with respect to x.

Hence, interchanging the orders of the two integrations,

:+ [A

for some positive number [j£| . Thus the expression

defines a continuous bilinear map on the dense domain x) and can therefore be

extended to the whole of J*£#).

The rest of the proof is straightforwardg

Let x »•»• T induce a pure, covariant measurement. Then, as a result of

Lemma U.3, T may be considered as a positive, linear operator (assumingx

Vf> (x) to be relatively majorizable):

(X)

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which, in virtue of {k. 12) satisfies, f_

Thus, T fulfills the conditions for being a dominated positive linear

transformation [6] between K^ij(x) and 'ifffi). Hence, there exists, for

p-almost all x€X, a unique positive measure T » with values in £(<7p£

(the set of all bounded linear transformations from &(&) to Jffl)), such that

1*in the domain jP='{f|Xf £ Kj-/^(x)}. Also, |T^[. < y , so that if

x I^(x) is majorizable then clearly JT [. <, u . .

An interesting particular case occurs when T is dominated, in the above

sense, by a J -measure concentrated at x, i.e.,

for all ^ f K j ^ ( X ) , In this case there again exists a positive measure

with values in £(&{%), &(%)), such that,for all

and IT I . < 5 . But since the 5-measure "is extremal, this latter

condition implies that ]T | is actually equivalent to £ , so that

-x J-x °x

where C is a function (measurable with respect to ):•*• x x

£» : x —*for y-almost a l l x. Further, i t i s c lear that <f (x ' ) need be specif ied, as

a function of x 1 , a t only x * x ' . We shal l assume therefore tha t £ (x1) -

^x for all x eX, where <px - £ ( & r t ^ +

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We may now give an equivalent characterization for a covariant

measurement which, has a projection valued observable.

'Lemma k.5

A covariant measurement § has a projection valued measure as its

observable if and only if it is induced by an operator-function x T

defined on the (dense) domain JD~' {f\\f C Krf*^Cx)) and satisfying:

GO T^Cf) = UXp)(x)}, / f £.0 and u-almost all x£X, where (fx is a,

bounded, positive, linear operator from /(5Q to /^) such that

(c) t r f j /^(x) TxCf)yu(^pc J / 1 t r / 5 , -where f is a sequence io K(X}such that f n ^l i point-wise and

(d) Tf«3, (f) = U?*TX(U^ f.U/)!^, /^eG and ^ £ . 0 .

Proof'

Let the operator function x •* T satisfy conditions (a) - (d) on <©.

Then, in virtue of (a), T (E/fM») « Tx(f), \f f € <& and all^e{p(E)}" which are

unitary. Also,

A € ^ ^ +*"</

where <£* * is the bounded, positive, linear map from $$f) to £{&) which is

dual to £ . Thus,

lT*(p)|i 1 I tx* I! fX

where f| *|J is the natural operator norm of <£ . Hence, x'+T fulfills

the conditions of Theorem k,l and, therefore induces a pure, covariant

measurement with a relatively majorizable density. To prove that this measure-

ment £ has a projection valued measure as its observable, we note that v

and E€B(X) , .

C A

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so that in virtue of condition . Cb},

J<X £, Up)>

where . < , "> denotes the canonical "bilinear form between the Banach space

QO1 QO

L y^:)(x» V) a n d i t s d u a l !• f£?e) x» " •fche *-algebra of all bounded functions

from X to %ty) which are y-measurable. x_ is the natural image of Xt.

in I* W ) ^ > Vi)« Let X denote the linear map

to which X is dual. Then, X(xE) = P(E) and hence,

< X E , U f ) > = t v [ A 3 CX £ ) f ] ,

so that, indeed,

tr f^(E,f)J = tr fP(E)p] ,and by continuity,

Conversely, let 8 determine a projection valued observable. Then, by

the remark following Definition 4.2, it is pure, so that Yf{x) s T*\p(x), ^r

fe XW). Also, J fc

On the other hand,

so that ,

•*"• • *" * fa.15)

-2U-

Considering'the restriction of Eq. 0+.15) to. %) and noting that, "by Eq. (2.7),

» fp(x) = YfU), it Is easy to see that in this case

Thus T~P(X) is relatively majorizable and, as noted in the discussion following

Theorem U.I, ve may define an operator-function x H- T which will induce g

and also have properties (a), (c) and (d). Further, in virtue of Eq. (1+.15),

Therefore, since f> is arbitrary in <?©, <£* satisfies condition

The results of this lemma and the discussion following Theorem k.l can

now "be collected into the following theorem:

Theorem k.2

Every pure, covariant measurement § , whose density function x ••>• I f(x)

is relatively majorizable, is uniquely determined for each x (up to a set of

y-measure zero) by a positive vector-valued measure T on X, with values in

, W ) ) , such that for all f in the (dense) domain *©-' {f

x

If n>(x) is also majorizable, then JT |. 5U (for u-almost all xeX).

Every covariant measurement £ which determines a projection-valued

•observable is necessarily pure and lias a relatively majorizable density. For

it [T | is equivalent to the measure 5 with unit mass placed at x, and

where £" is a bounded, positive, linear map (for y-almost all x6X) from

to J"(3?), satisfying

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for every $€$(&} . Further, in this case,'the group covariance of £ under

G * HSX implies that, for all x£X,

where o is the identity element of the commutative group X.

As a final remark, ve may mention that the covariance requirement on our

conditional expectations, while physically motivated, can sometimes "be lifted

at the cost of rather involved technicalities, which however do not pertain to

the simpler purposes of this paper.

ACKNOWLEDGMENTS

The final draft of this paper was completed while one of the authors

(GGE) was at the Ecole Polytechnique Federale de Lausanne. He wishes to

thank Professor Ph. Choquard for hospitality at Lausanne. The other author

(STA) is grateful to Professor Abdus Salam, the International Atomic Energy Agency

and UNESCO for hospitality at the International Centre for Theoretical

Physics, Trieste,

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1. S.T. All and G.G. Emch,, Fuzzy. Observables in'Quantum Mechanics, J. Math.

Phya. 12, 176-182 .(.lPTM.

2..N. Bourbaki, 'Element s de Mathematiques, Livre VI, Integration, Chapitre 6,

Hermann, Paris (1959).

3. E.B. Davies and J.T. Lewis, An Operational Approach to Quantum Probability,

Commun. Math. Phys. 1J_, 239 - 260 (.1970).

h. E.B. Davies,'On the Repeated Measurement of Continuous Qbservables in Quantum

Mechanics, J. Funct. Anal., £, 3l8 - 3U6 (1970).

5. See, for example, J. Dieudonne, Foundations of Modern Analysis, Chapter V,

Section 7» Academic Press, New York-London {1969).

6. See, for example, N. Dinculeanu, Vector Measures, Chapter III, Section 19,

International Series of Monographs in Pure and Applied Mathematics,

I.JT. Sneddon and M. Stark, Ed., Pergamon Press, (New York (1967).

7- S. Gudder and J.-P. Marchand, Non-commutative Probability on von Neuman

Algebras, J. Math. Phys. 13, 799 - 806 (1972).

8. See, for example, P.R. Halmos, Measure Theory, Chapter IX, Section h3,

The University Series in Higher Mathematics, M.H. Stone, Ed., Van Kostrand

Reinhold Company, New York (1950).

9. G.W. Mackey, Imprimitivity for Representations of Locally Compact Groups I,

Proc. Natl. Acad, Sci. U.S.A., 3J> 537 - 5^5 (19^9).

10. G.W. Mackey, On a Theorem of Stone and von Neumann, Duke Math. J. l6,

313 - 326 (19^9), cf. especially Lemma 3.3

11. S.-T.C. Moy, Characterizations of Conditional Expectation as a Transformation

on Function Spaces, Pacific J. Math. k_, Vf - 6k (195U).

12. M. Nakamura and T. Turumaru, Expectations in an Operator Alpebra, Tohoku

Math. J. 6_, 182 - 188 (195*0.

13. M. Nakamura and H. Umegaki, On von Neumann's Theory of Measurements in

•Quantum Statistics, Math. Japon., £, 151 - 157 (1962).

lU. I. Namioka, Partially Ordered Linear TopoloRical Spaces, Memoirs Am. Math.

Soc. 2h_, (1957).

15. See, for example, S. Sakai, C*-algebras and W*~alge"bras, Ergebnisse der

Mathematik und ihre Grenzgebiete, Band 60, Springer-Verlag (1961).

16. H. Umegaki,'Conditional Expectation'in an Operator Algebra, TShoku

Math.J. 6_, 177—I8I (195IO.' '

17. J.C Wolfe and G.G. Emch, C*-algebraic Formalism for'Coarse-Braining I, II

and III. Rochester preprints (1973).

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