Orthogonally scattered measures on non-Boolean semi-rings

Citation for published version (APA):

Kruszynski, P. (1985). Orthogonally scattered measures on non-Boolean semi-rings. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8507). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1985

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne

Take down policy

If you believe that this document breaches copyright please contact us at:

openaccess@tue.nl

providing details and we will investigate your claim.

to200S

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum 1985-07 April 1985

ORTHOGONALLY SCATTERED MEASURES ON NON-BOOLEAN SEMI-RINGS

by

Pawel Kruszynski

University of Technology

Department of Mathematics and Computing Science PO Box 513, 5600 MB Eindhoven

by

Pawet Kruszynski

Eindhoven University of Technology

Department of Mathematics and Computing SCience, P.O. Box 513, 5600 MB Eindhoven

The Netherlands

Summary

The definition of a countably additive orthogonally scattered measure on an orthogonal semi-ring is given. It is proved that there is no non-trivial c.a.o.s. measures mapping the lattice of orthogonal projections in a separable Hilbert space into a finite dimensional Hilbert space. Global properties of families of c.a.o.s. measures are investigated in connection with inductive limits of families of Hilbert spaces.

.. 2

-INTRODUCTION

Orthogonally scattered measures appeared in our theory of inductive-projective limits of Hilbert spaces in a natural way([EGK] ,

[EK] ,[ M ] ). We recall shortly the idea of our construction.

Let ~ be a generating family of operators i.e. a family of bounded selfadjoint operators in a Hilbert space H with the following properties: i) V a E 6l ii)

V

a,b

E

~ ab

=

ba iii) iv) V a,b E ~ 3 c E ~ V aE~3bE~ a .;;;; c and b';;;; c

a! .;;;;

b.

Let aH denote the Hilbert space consisting of vectors ah, where h E H, with the scalar product (ah I af)a := (r(a)h I r(a)f )H' where r(a) is the right (left) support of a E ~.

s~ will denote the inductive limit of the family of Hilbert spaces { aH } aE~ i. e.

s

=

U aH

aE~

Let E be the joint resolution of identity of the family

i.e. a projection valued countably additive measure defined on the a-ring

*

*

of Borel subsets of the spectrum A of the W -algebra W ( ~ ) generated by ~.

Let

r.

denote the semi-ring of Borel subsets of A defined as follows:

AEr. . if and only if A is a Borel subset of A and there exists a E ~

such that for some positive number c E( A ) .;;;; c' a.

r.

is called a family of ~-bounded subsets of A • If I ~ ~

We define a completely additive orthogonally scattered measure on E (c.a.o.s.m.) as a function v:E ~ H with the properties:

i) _{i f { ~ex }exEI} C E , ~ex

n ~a

=

0

for ex ,.,.

a

and ex~I~exEE then

1: V ( ~ )

₌

_{V (altI~ex ), where the series converges in norm in H.}

cxEI ex

ii) i f _{~]' ~2EE} and _{~ln ~2}

₌

₀

then ( V (~1 ) I V ( ~2 »H = O.

(cf. [11] ) •

A

c.a.o.s. measure ~ is said to be generated by a spectral measure

For a given family ~ c B(H) a c.a.o.s. measure ~ is called

~-bounded if it is generated by E and V a E ~ the mapping

E 3 ~ ~ a V ( ~ ) E H is a c.a.o.s. measure such that

sup II a ~ ( ~ ) II < <XI The set of ~ - bounded c.a.o.s. measures is

~EE

denoted by T~ and we endow it with the locally convex topology given by the following family of seminorms:

T~ V ~ II l.l II = sup II a l.l ( ~ ) II •

a ~EE

In [EK] we proved that the following duality holds:

S61

=

T~ •

Moreover we introduce a natural embedding j: S~ c T~ putting: j(s)( ~ ) = aE( ~ )h, where s E S is such that s

=

ah E aH. Note that then h E r(a)H is unique.

In this way we can describe the inductive limit s~ and its strong dual

S61

in terms of c.a.o.s. measures on the semi-ring E.

We can extend our definition of a c.a.o.s. measure l.l defined

on "characteristic functions" on r: onto an "integral" defined on the "family of functions" ~ by:

~ 3a ~ V (a) E H where l.l (a) is the unique vector in H such that

-

4-The function ~ :~ ~ H has interesting properties: if r(a) ~ r(b) for a,b E ~ then ~ (a) ~ ~ (b) , and i f a ~ b then II ~ (a)II ~ II ~ (b) II •

. In this way we have represented the order structure of the family ~

or, equivalently, the order type of the inductive limit S~ in terms of c.a.o.s. measures.

It is an easy observation that we can originally define c.a.o.s. measures on the family of projections H

= {

E( a

)1

aEL } instead of

defining them on L ,by:

::: 3 e ~ ~ (e)

=

~

(

a ) , where e = E ( a ) for some aEL

Suppose now that a family of operators ~ fulfils conditions i), iii) and iv), i.e. it is not commutative. Then we can define an inductive limit S~ of the family of Hilbert spaces {aH }aE~ ,taking as before

S~

=

a~~ aH with an adequate topology. This time however there is no joint resolution of identity for ~ and our previous idea of the representa-tion of S~ as a space of c.a.o.s. measures is not applicable directly.

This leads to the following idea: define orthogonally scattered measures on non-distributive (non-commutative) order structures, for instance

such as a lattice of projections of a non-commutative von Neumann algebra of operators in a Hilbert space.

The present paper is devoted to the including existence problem.

1. EXISTENCE OF C.A.O. S. MEASURES ON GENERAL DIRECTED SETS

We consider here certain particular type of directed sets which is modelled after the order structure of sets of orthogonal projections onto

subspaces of a Hilbert space.

1. 1. Definition

A set £ is called an orthogonal semi-ring if:

i) £ is a partially ordered directed set with respect to a relation ~

such that every finite family of elements of £ has its least upper bound.

£ contains the minimal element O.

ii) There is given an orthogonality relation ~ c £ x £ , such that:

1. a ~ b ... b ~ a

2 . a ~ b and a ~ b ... a = 0 •

3. if a,e E £ have the property that for each b E £ a ~ b

implies c ~ b, then c ~ a. 4. If a ~ band c ~ a then c ~ b.

5. If a ~ b then there exists (unique) c E £ such that

c ~ a and a v c = b.

6. (Weak modularity) If a ~ b , a ~ c , a v b = a V c and

b ~ c then b

=

c •

7. If a ~ b, a ~ c, b ~ d and a v c

=

b v d then d ~ c.

Although the conditions 1 - 7 of ii) are not independent it seems useful to display all of them at once.

6

-) .2. Definition

Let H be a Hilbert space and

t

be an orthogonal semi-ring. Then a function ].I: I -+ H is called a countably additive orthogonally scattered

measure on I (c.a.o.s.m.) if:

ii) _{For each countable family {a}

i } i EN c I , a. 1. 1. a. J i

:f

j and

iltN ai E I implies ].I (

~N

ail

=

~~(ai)

, where the series

converges in the Hilbert space norm.

For our purposes it is enough to consider the orthogonal semi-ring tp consisting of projections onto closed subspaces of a Hilbert space H.

Then for i f

e₃

=

e

l v e2, we take the orthogonal projection onto the closed subspace

elH v e_{2H and we put e} 1 1. e₂ i f e

1e2

=

O.

An example of a c.a.o.s. measure on t is given by p

I 3 e -+ II (e) := ex , where x E H.

P

Now let In denote the lattice of all orthogonal projections onto subspaces of the Euclidean space Rn with the above orthogonality relation.

1. 3. Pro12osition

I f m,n EN, n> m, n ~ 3 and ].I: I -+ Rm is a c.a.o.s. measure

n

with values in the real Hilbert space Rm then _lJ

=

O.

Proof:

We will use the following result

1.4. Lemma ( [G] ) Let p (1)

=

p : I -+ n [ 0, I] and if e 1. f then

be a function with the properties:

p(e + f)

=

p(e) + p(f). Assume n ~ 3.

Then there exists a density matrix W such that p(p)

=

Tr(Wp) for all pEt •

In particular it follows that the function p is continuous with respect to the natural parametrization of projections in Rn.

Observe that if 11: £ .... R m

n is a c.a.o.s.m. then its components

satisfy the assumptions of the above Lemma (up to normalization). It also can be applied to the scalar measure £3e .... llll(e)1I 2 •

n

Thus the measure 11 is continuous with respect to the natural parametrization of projections in Rn.

We prove the proposition by induction. Consider at first 11: £ .... R I .

n

Suppose that there exists an one-dimensional projectio.n pE £ , such

n

that I1(P)

f

O. Then for any qE £ , such that q ~ p, l1(q)

=

O. Let eE £

n n

~

be anyone-dimensional projection. Put f

=

P A (e v p) f O. Then e is;;; p + f

and l1(f)

=

O. We have:

l1(e) + I1(P + f - e )

=

I1(P).

Because l1(e) 11 (p + f - e)

=

0 so either l1(e)

=

0 or l1(e)

=

I1(P). This is a contradiction to the continuity of 11. Hence 11

=

O.

Let n

=

3, m

=

2.

For any triple {Pl,P2,P3} of mutually orthogonal one-dimensional projections we can assume that I1(P3)

=

O. We are going to show that if I1(PI) f 0 then I1(P2)

=

O.

Assume the contrary, i.e. I1(PI) and I1(P2) non-zero. Let e ~ PI + P3 be an one-dimensional projection. Then I1(PI + P3 - e) + l1(e)

=

I1(PI).

11 (p I) But

either l1(e)

=

0 or l1(e)

=

I1(Pl) which contradicts the continuity of 11 restricted to the projections e ~ PI + P3.

8

-Thus only one of the values p(p) or P(P2) can be non-zero. It follows that for any triple of mutually orthogonal one-dimensional projections in R3 at most one of them has non-zero measure.

h d d· · 1 . . . R3

Assume t at p an q are one- ~mens~ona proJect~ons ln such that p(p) ~ 0 and p(q) ~ 0 (hence p

t

q).

Let p'

=

pL A (p v q) , so p(p')

=

O. We have: p(p v q)

=

p(p + pI)

=

p(p)

and similarly p(p v q) .p(q)

=

p(p).

It means that p: £3

~

RI • Thus p

=

O. Let m = n - I.

Let us assume now that the only c.a.o.s. measure _{p: £ n-)} ~ R n-2

if p:£ ~ R n-I then is p

=

O. We will show that from this follows that

n

p

=

o.

Let {PI,P2, ••• ,Pn} be a collection of mutually orthogonal one-dimensional projections in £n and let p: £n

~

Rn-1 be a c.a.o.s.m. Suppose that p

~

O. We can assume that p(p)

=

0 and that there is such an index i €{l,2, ••• ,n-l}

n 0

that _{P(Pi ) ~} O. The restriction of p to the lattice of projections

0

n-2

£ n-) fill { q € £n I q<; p.L } has its values in R • Indeed: for each 1

q <; p.L

0

we have p(q) L \.l(Pi )

.

thus , by the assumption, \.ll£n-)

=

O.

1

,

0 0

It means that for an'arbitrary collection {PI ,P2"" ,Pn} there may by only one index i

o€ {1,2, ••• ,n} for which P(Pi) ~

o.

It follows that for any

o

one -dimensional projection q€ £n such that \.l(q) ~ 0, we have \.l(qL)

=

O. Let p,q € £ be a couple of one-dimensional projections. Then again

n \.l(p v q)

=

\.l(p)

=

\.l(q) and as before 1.5. Remark I p: £ ~ R • Hence p

=

O. n

The case p: £2 ~ R 1 obviously admits non-zero c.a.o.s. measures.

o

On the other hand the example £n 3 e ~ v(e)

=

e x € Rn , for some x € Rn shows that the assumption n > m is essential.

1. 6. Corollary

Let H be a separable Hilbert space (dim H ~ 3). Let ~ be a c.a.o.s.m. on the lattice £H of projections onto closed subspaces of H with values in a Hilbert space K. Let dim H

>

dim(lin.span.{ ~(E)I E E£H})' Then ~

=

O.

1.7. Corollary

Let ~I£ ~£ be a mapping which preserves the orthogonality relation n m

and which is additive on mutually orthogonal. projections. If n > m and n;2 then ~

₌

o.

In particular,

if

n 1M ~

men M mxm is a Jordan homomorphism

then n

=

0 for n > m.

Proof:

It is enough to notice that for any vector xE Cm the map

£ 3e ~ ~(e) x E Cm

n is a c.a.o.s. measure. Hence ~(. ) ==

o.

- 10

-2. REPRESENTATION OF A C.A.O.S. MEASURE

We extend here some of our results of [EK] onto the case of non-commutative domains of definition of c.a.o.s. measures. We exploit here the notion

of the bi-orthogonality relation ( i.e. compatibility) between two c.a.o.s. measures.

2.1. Definition

Let l be an orthogonal semi-ring. Let ~,v:l ~ H be a couple of c.a.o.s.measures with values in a Hilbert space H. Then {~,v} is called a bi-orthogonal couple ([M]) or a compatible couple of c.a.o.s. measures if for every a,bEl from a ~ b follows that ~(a)~ v(b).

Example

Let l

=

lH ,x,y

e:

H. Then couple of c.a.o.s. measures.

V(E) • Ex, v(E)

=

Ey is a bi-orthogonal

We say that a family of c.a.o.s. measures is bi-orthogonal if every pair of members of it is a bi-orthogonal couple. Every bi-orthogonal family of c.a.o.s. measures can be extended to a maximal one with respect to the set--inclusion relation.

Let e be a maximal bi-orthogonal family of c.a.o.s. measures on l. Denote: G(e)

=

{~(e)

I

~

e:

e }

for e

e:

l .

2.2. Theorem

Let e be a maximal bi-orthogonal family of c.a.o.s. measures on an orthogonal semi-ring l. Then:

ii) I f e,f €.£ and e ;;r: f _{then See)} ~ S(f).

iii) If e,f E

£

and e i f then See) i e(f) • iv) The set S

=

U See) is a linear subspace of H.

e€.£

Proof:

At first we notice that iii) easily follows from the definition of See). i) Let

~,v E

S,

~,e E

c

J• Then

~~

+

ev

is a c.a.o.s.m. compatible with each element of S • Thus by the maximality of S it belongs to S.

Let ~(e) denote the orthogonal projection onto the closure of See) in H. Then for e i f in virtue of iii) we have 0Ce) i GCf) and ~(e) i ~(f).

Moreover we have then 0(e V f)

=

0(e)

e

em-

,Le. /fI(e

v

f) = ~(e) fE> q;(f). Because

for n

r

m

q;(e) ~ 1 so for any family and ~Le E £ nt:.N n there exists e ) n == such that e i e n m l: q; (e ), nEN n

where the series converges strongly in B(H). Thus for any x E H the mapping:

£ :3 e -+ <P (e)x E H is a c.a.o.s.m. By the construction this measure is

compatible with 0 hence it belongs to 0

Now take x E 0Ce) • Then /fI(e)x == xE 0(e). Thus 0(e)

=

G(e).

This proves i).

ii) It is easy to notice that for e ~ f we have ~(e) ~ !p(f). Thus 0(e) ~ 0(f).

iv) The linearity of S follows from ii).

o

2.3. Remark

In distinction to the "collDllutative" case it may happen that the linear manifold S is not dense in H. For instance let £ == £ 3 and H

=

C4• Let S be a maximal bi-orthogonal family of c.a.o.s. measures containing all measures

4 3 3 4 •

of the form: £ 3 :3 e -+ e x E C where xE C and we embed C C C l.n

12

-Then we have £3 all 4x4 matrices of the form

where P is a 3x3 hermitian idempotent matrix i.e. a projection in C3• So C3 c S.

LetJ.l Ee •

vector x E C3

Then taking anyone-dimensional , we have :

P~

J.l (P )

=

O. Thus

x x

where _e I C3 . C4 . .

4 ~ 1n 1S a un1t vector.

projection P _x onto a unit j.t{P ) = y{P)x

e

A(P) e

4,

x x x

However for any triple of mutually orthogonal one-dimensional projections in C3 we have J.l (P .) ~ J.l (P . )

1 J , i

f

j, in particular

Thus there is only one index for which the number

A(P.) A (P.)

=

6 .. A(P.).

1 J 1J 1

A(P.) is different from O.

1

Notice that the map £3 3 Pi ~ A(P

i) e4 can be extended to a c.a.o.s. measure on £3 with values in C1, hence A - 0, i.e. S

=

C3

f

C4•

2.4. Corollary

For any maximal bi-orthogonal family of c.a.o.s. measures on £ there exists a map, which preserves the order and orthogonality relations

<t>: £ B(H)P, with values in projections in H, such that:

if e ~ f then <t>(e v f)

=

~(e) + <t>(f) and for every J.lE0 and each e E £ <t>(e) J.l (e) = J.l(e).

Let us consider now a particular case in which £

=

AP , i.e. £ is the set of all projections of a given W* -algebra A of operators acting in a separable Hilbert space H •

2.5. Theorem

Let £ = AP and let j.I be a bounded c.a.o.s. measure ll:£ ... H.

Then there exists a linear positive map

""

w:

A

...

B(H) and a vector x € H such that for each e € £ II (e) == N He) x.

Proof:

Let e be a maximal biorthogonal family of c.a.o.s. measures containing ll' Let ~(e) be the orthogonal projection onto the space e(e). Then by Corollary 2.4.

we obtain the complete additivity of ;p on families of mutually orthogonal

projections in AP• For e ~ f we have ;p(e) ~ ~(f) and for e ..;; f ;p (e)";; ;p (f) • Moreover for e";; f ;p (e) II (f)

=

;p (e) ( II (f - e) + II (e»

=

II (e) •

Now consider the net {ll(e)} e € £. Because it is bounded in norm it has cluster points. Let x € H be a cluster point of it. Then for every

e

>0, z € H and e € £ there exists f €£ such that f;> e and

I (

;p (e) z I II (f) - x)

I

< f, ,

Le.

e

>

I

(z I ;p (e) II (f) W (e) x)

I

==

I

(z I II (e) -

w

(e)x)

I

Since

e

is arbitrary we have q, (e)x

=

II (e) •

Because ;p : £ ... B(H)P is a normal map it can be extended by the spectral theorem to a positive map ~ defined on the whole algebra A. Linearity

follows from the generalization of Gleason theorem: there exists a linear positive (normal) functional w on A such that II II (e) II 2

=

w (e)

=

= II ;p (e)x II 2

=

_(x I

w

(e)x) (cf. [K j)

By the definition ~ is positive.

o

2.6. Corollary

The map II :AP ... H can be extended to a linear map ll: A... H with the property jl (a) == ~ (a)x. This map coincides with the "integral"

- 14

-CONCLUDING REMARKS

This is easy to observe that our description of c.a.o.s. measures can be closely related to the eigen-packet theory ( [M] ). Hence it would be desirable to describe the global properties of the dual of the

inductive limit S~for a non-commutative generating family ~ in terms of c.a.o.s. measures defined on the lattice of projections of W*(~).

Also a factorization of unbounded c.a.o.s. measures in the sense of Theorem 2.5. seems interesting.

At last we pose two technical problems , strictly connected with the present paper:

Problems

1. Find conditions under which the map ~ described in Theorem 2.5. is a Jordan homomorphism.

2. Extend the technics used in Corollary 1.7. onto the case of von Neumann factors - i.e. an easy proof of the existence or non-existence of

BIBLIOGRAPHY

[EGK] Eijndhoven,S.J.L.van~ Graaf J.de, Kruszynski P., 'Dual Systems of Inductive-Projective Limits of Hilbert spaces originating from Self-adjoint Operators', Memorandum 84-09~ July J984, Eindhoven

University of Technology,to appear in The Preceedings of KNAW, Sept.1985 [EK] Eijndhoven, S.J.L. van, Kruszynski P., 'Some Trivial Remarks on

Orthogonally Scattered Measures and Related Gelfand Triples',

Memorandum J984-1], December ]984, Eindhoven University of Technology, [K] Kruszynski P.,'Extensions of Gleason Theorem', Proceedings Quantum

Probability and Applications to the Quantum Theory of Irreversible Processes, Lecture Notes in Mathematics

lOSS,

Springer-Verlag 1984 [G] Gleason A.M., "Measures on the Closed Subspaces of Hilbert Space",

J.Math.Mech. ~(]957) 885-893,

[M] Masani P., "An outline of vector graphs and conditional Banach

spaces" in Linear Spaces and Approximation" , Birkhauser, Basel ]978, pp.71-89,

- " - "Remarks on eigenpackets of self-adjoint operators", pp.415-441 in Hilbert Space Operators and Operator Algebras, Proceedings, Tihany 1970, North Holland, Amsterdam 1972