Scholarly article on topic 'Remarks on the order for quantum observables'

Remarks on the order for quantum observables Academic research paper on "Mathematics"

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Academic research paper on topic "Remarks on the order for quantum observables"

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VERSITA Slovaca

DOI: 10.2478/s12175-007-0048-x Math. Slovaca 57 (2007), No. 6, 589-600

REMARKS ON THE ORDER FOR QUANTUM OBSERVABLES

S. PULMANNOvA — E. VlNCEKOYA

(Communicated by Anatolij Dvurecenskij)

ABSTRACT. Relations between generalized effect algebras and the sets of classical and quantum observables endowed with an ordering recently introduced in [GUDDER, S.: An order for quantum observables, Math. Slovaca 56 (2006), 573-589] are studied. In the classical case, a generalized OMP, while in the quantum case a weak generalized OMP is obtained. Existence of infima for arbitrary sets and suprema for above bounded sets in the quantum case is shown. Compatibility in the sense of Mackey is characterized.

©2007 Mathematical Institute Slovak Academy of Sciences

1. Introduction

The set of bounded observables for a quantum system is usually represented by the set S(H) of bounded self-adjoint operators on a complex Hilbert space H. The traditional order for A,B E S(H) is defined by A < B if {Ax, x) < {Bx, x) for every x E H. In [9], this order is called numerical. Under this ordering, (S(H), <) becomes a partially ordered set (poset). A well-known theorem due to R. Kadison [14] shows that (S (H), <) is not a lattice, it is even an anti-lattice in the sense that A A B exists if and only if A < B or B < A, and A A B is the smaller of the two. If this ordering is applied to the self-adjoint elements S(A) of a von Neumann algebra A, then S(A) is a lattice if and only if A is abelian.

Another ordering, so-called spectral order, was introduced in [17], [8] as follows: let A,B E S(H), and let (PA)xeR, (P\ )agr be spectral families of A, B,

2000 Mathematics Subject Classification: Primary 81P10; Secondary 06A06, 47C15. Keywords: observable, self-adjoint operator, generalized effect algebra, (weak) generalized orthomodular poset, Riesz ideal, Mackey compatibility.

This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-032002, grant VEGA 2/3163/23 and Center of excellence SAS, CEPI I/2/2005.

Springer

respectively. We define A <s B if P^ < P ^ for every A £ R. Then <s extends < on projections, is coarser than the usual ordering, but agrees with it on abelian subalgebras, and turns S(H) into a boundedly complete lattice, that is, every bounded family in S(H) has a supremum and infimum.

In [9], a new order for quantum observables, represented by the set of bounded self-adjoint operators S(H) on a complex Hilbert space H, has been introduced. This new order is determined by assuming that A B if the proposition that A has a value in A implies that B has a value in A for every Borel set A not containing 0. It is called the logical order. In the commutative case, we may represent observables by fuzzy random variables, and study the new ordering on them. There are several characterizations of the ordering <, e.g., A B if and only if AB = A2. This shows that is the restriction of D r a z i n's order ([4]) from the set B(H) of all bounded operators on H, to the self-adjoint part S(H). Indeed, the Drazin order a <d b is introduced by the binary relation a* a = a*b = b*a and aa* = ab* = ba*. To the difference of both the traditional and the spectral order, the logical order is algebraic in the sense that a partial binary operation © can be introduced in S(H) such that A B if there is C such that A © C = B.

In the present paper, we will study the structure of classical and quantum observables with respect to the new ordering in more details. We will prove that in the classical case, the set M(A) of random variables on a probability space (Q, A,^) forms a generalized a-orthocomplete orthomodular poset (GOMP), which satisfies the Riesz decomposition properties. Nevertheless, its unitization does not form a Boolean algebra. We also show that the set of functions with finite support form a Riesz ideal. In the quantum case, the set S(H) forms a weak generalized orthocomplete orthomodular poset (WGOMP). Moreover, the infimum of any two elements of S(H) exists, while the supremum exists if and only if the two elements have a common upper bound. This extends the results of [9], where the structure of a generalized a-orthoalgebra has been shown for the classical, and a generalized orthoalgebra for the quantum case, and the existence of infima in S(H) has been proved only for the finite-dimensional H. More generally, we show that the infimum of an arbitrary family exists, and the supremum of an arbitrary above bounded family exists. We also find a characterization of the Mackey compatibility. Since Mackey compatibility is usually interpreted as simultaneous measurability, it turns out that two observables corresponding to self-adjoint operators A and B, are simultaneously measurable (with respect to (S(H), ©, 0)) if and only if AB = (A A B)2.

2. Generalized effect algebras

Effect algebras were introduced in [6] (see also [7] and [15] for alternative definitions) as an abstract generalization of the Hilbert space effects, that is, self-adjoint operators between the zero and identity operators (in the usual ordering), which play an important role in the theory of quantum mechanical measurements ([1]).

Another important example is the unit interval [0,1] of real numbers organized into an effect algebra by defining a ± b if a+b < 1, and then putting a©b = a+ b. We note that [0,1] is also a prototypical example of an MV-algebra, a structure introduced by C hang [2] as an algebraic base for many-valued logic. For the details about effect algebras and related structures see, e.g., [5].

Definition 2.1. An effect algebra (EA) is a system (E, ©, 0,1) consisting of a set E with two special elements 0,1 G E and with a partially defined binary operation © satisfying the following conditions for all p,q,r G E:

(E1) if p © q is defined, then q © p is defined and p © q = q © p (commutative law),

(E2) if q © r is defined and p © (q © r) is defined, then p © q and (p © q) © r are defined and p © (q © r) = (p © q) © r (associative law),

(E3) for every p G E there exists a unique q G E such that p © q is defined and p © q =1 (orthosupplement law),

(E4) if p © 1 is defined, then p = 0.

The element q in (E3) is denoted by p' and is called an orthosupplement of the element p.

We recall that an effect algebra E is an orthoalgebra (OA) if a ± a implies a = 0; E is an orthomodular poset (OMP) if whenever a, b, c are mutually orthogonal, then a ± b © c; E is an orthomodular lattice if it is a lattice ordered OMP.

Definition 2.2. ([11], [5]) A generalized effect algebra (GEA) is a system (P, ©, 0) satisfying conditions (E1) and (E2) and (E3') if a © b = a © c, then b = c (cancelation law), (E4') if a © b =0, then a = 0 = b (positivity), (E5) a © 0 = a for every a G P.

On an effect algebra or a generalized effect algebra E, we further define:

• a binary relation ± by

a ± b ^^ a © b exists,

• a dual partial binary operation © to the operation © by

c © a = b ^^ a ± b and a © b = c,

• and a binary relation < by

a < b ^^ 3c e E : c © a = b,

which is a partial order on E, where 0 is the least element and in the case of effect algebra, 1 is the greatest element. A generalized effect algebra becomes an effect algebra iff it contains a greatest element 1.

A GEA P is a generalized orthoalgebra (GOA) if a ± a implies a = 0; P is a weak generalized orthomodular poset (WGOMP) if a © b = a V b whenever a © b exists, and a ± (b © c) whenever a, b, c are mutually orthogonal; P is a generalized orthomodular poset (GOMP) if a © b = a V b whenever a ± b, and if b V c exists and a ± b, a ± c then a ± b V c. A lattice ordered GOMP is a generalized orthomodular lattice (GOML).

It is well known that every generalized effect algebra P can be embedded into a uniquely defined effect algebra E such that for every a e E, either a e P or its orthosupplement a' e P. In analogy with the theory of rings, we call this effect algebra E the unitization of the GEA P (for more details see [11], [5], [19], [18]).

Moreover, a GEA P is a generalized orthoalgebra iff its unitization E is an orthoalgebra; P is a WGOMP iff its unitization E is an orthomodular poset ([16]); P is a GOMP iff E is an orthomodular poset and the embedding of P into E preserves existing suprema ([16]); and finally P is a generalized orthomodular lattice iff its unitization E is an orthomodular lattice ([12]).

3. The commutative case

Classical commuting observables are represented by random variables on a probability space (Q, A, j), where A is usually thought of as the set of events for some statistical experiment. The set A can be organized into an effect algebra if we put A ± B if A n B = 0 and define the orthosum A © B = A U B whenever A ± B. Then (A, ©, 0, Q) is an effect algebra with A' = Ac(= Q \ A). In fact, A is a Boolean a-algebra.

We identify an event A e A with its characteristic function xa, which can be considered as a two-valued measurement with outcomes 0 and 1, or "no" and "yes". That is, for any w e Q, xa(^) gives the values 1 or 0 depending on whether w e A or w / A. We have j(A © B) = j(A) + j(B) whenever A © B is defined, and j(Q) = 1. That is, j is a state on A (i.e., an effect

algebra morphism from A to [0,1], considered as effect algebras). We note that here the effect algebra order d coincides with the usual order, i.e., xa d Xb iff Xa(w) < Xb(w) for all w G fi.

In [9] it is suggested to extend the orthosum to all measurements associated with A, which are represented by the set M(A) of all random variables on (fi, A, f). The extension is obtained by defining f ± g if fg = 0 for f,g G M(A). Denote the support of f by supp(f) := {w G fi : f (w) = 0}, and the null space of f by null(f) := {w G fi : f (w) = 0} = f-1 (0). Observe that f ± g iff supp(f) C null(g) iff supp(f) ± supp(g). We define f © g = f + g iff f ± g. Define a partial order d on M(A) by f d g if there is an h G M(A) such that f ± h and f © h = g. Then (M(A), d) is a poset and 0 d f for all f G M(A).

In fact, in [9], it was proved that (M(A), ©, 0) admits a structure of a generalized CT-orthocomplete orthoalgebra. Moreover, the infimum f A g exists for all f,g G M(A), while the supremum of f,g exists iff there is h G M(A) such that f,g d h. In what follows, we give a more precise description of the structure of (M(A), ©, 0). We need the following characterization of the partial order d, proved in [9, Th. 3.1].

THEOREM 3.1 The following statements are equivalent: (i) f d g,

(ii) f (w) = g(w) for all w G supp(f),

(iii) f = g • Xsupp(f),

(iv) fg = f2,

(v) f-1 (A) C g-1 (A) for every A G B(R) with 0 / A. Theorem 3.2. The structure (M(A), ©, 0) is a

a) WGOMP,

b) GOMP.

Proof.

a) Recall that by [5, Th. 1.5.13] and [5, Remark 1.5.16], we have to prove:

(i) If f,g G M(A) and f ± g, then f ©g = f Vg, i.e., f © g is the supremum of f and g.

(ii) If f,g,h G M(A) are such that f ± h, g ± h, and f ± g, then h ± f © g.

(i): Let f,g G M(A), f ± g. Clearly, f,g d f © g. Now let h G M(A): f,g d h. While for every w G supp(f © g) either w G supp(f) or w G supp(g), along with f,g d h we have Vw G supp(f © g): (f © g)(w) = f(w) = h(w) or (f © g)(w) = g(w) = h(w) and therefore by Theorem 3.1, f © g d h.

ii): When f, g, h are mutually orthogonal, their supports are disjoint and so obviously supp(f © g) n supp(h) = [supp(f) U supp(g)] n supp(h) = 0

f © g ± h.

b) A WGOMP is GOMP iff V f,g,hE M(A): f ± h, g ± h and the existence of f V g imply f V g ± h. So let us have f,g,h E M(A) such that f,g ± h and f V g exists. Observe that when the supremum of f, g exists, then the functions are equal on the set supp(f) nsupp(g). So we may define p := f ■ xsupp(f )nsupp(a), fi := f (1 - Xsupp(a)), gi := g(1 - Xsuppf)). Therefore the function fi ©p© gi is defined and it can be easily seen that it is in fact the supremum f Vg = fi ©p©gi. Since supp(fi © p © gi) = supp(f) U supp(g) and f,g ± h, straightforwardly supp(f V g) ± supp(h) and thus f V g ± h. □

Recall that a generalized effect algebra A satisfies the Riesz decomposition property (RDP in short) if for any a,b,c E A, a < b © c implies that there are bi < b, ci < c such that a = bi © ci. More generally, A satisfies the a-Riesz decomposition property if a < (J) bi implies a = (J) a^ where ai < bi for

all i E N. ieN ieN

Theorem 3.3. The structure (M(A), ©, 0) satisfies the a-RDP.

Proof. Assume that f 0 gi. It follows that supp(f) C (J supp(gi), and to

ieN ieN

every w E supp(f) there is i E N with f (w) = gi(w). Define hi := gi ■ xsupp(f). Then supp(hi) = supp(gi) n supp(f), and clearly hi gi for each i E N, and hi ± hj whenever i = j. Therefore 0 hi exists, and since supp(f) = (J supp(hi),

we have f = ® hi. ieN ieN □

Let us define

F := {f E M(A) : supp(f) is finit^.

We recall that a subset I of a generalized effect algebra P is an ideal if for any a,b E P,

(i) a I, b < a implies b I (that is, I is an order ideal),

(ii) a,b E I, a ± b implies a © b E I. An ideal is a Riesz ideal if

(R1) a E I, b,c E P, a < b © c implies a < bi © ci where bi,ci E I and bi < b, ci < c;

(R2) i E I, i < a and a Q i ± b implies 3 j E I, j < b and b Q j ± a.

The importance of Riesz ideals is given by the fact that the quotient P/I of a generalized effect algebra P with respect to a Riesz ideal is again a generalized effect algebra (for more details see [10], [5], [3], [18]).

Proposition 3.4. The set F is an ideal in M(A).

Proof. At first, it is obvious that F is an order ideal, because if f E F and g ^ f then supp(g) C supp(f), so that g E F. As for the second property of an ideal, if f,g E F and f ± g then since supp(f © g) = supp(f) U supp(g), clearly f © g F. □

Theorem 3.5. The set F is a Riesz ideal in M(A). Proof. We have to prove:

(R1) f £ F, g,h £ M(A), f d g © h implies that there are g1 d g, h1 d h such that g1 ,h1 £F and f d g1 © h1,

(R2a) g,h £ M(A), f £ F, h d g, g Q h ± f implies that there is f1 £ F such that f1 d h, (g Q f1) ± f.

We notice that condition (R2a) is equivalent with (R2), which is in the original definition of a Riesz ideal (see [5]), what was proved in [13]. While (R1) follows by Theorem 3.3, we proceed immediately to (R2a):

Let g,h £ M(A), f £F, h d g and g Q h ± f. We put f1 := g ■ Xsupp(f) and show that it has required properties. At first, if w £ supp(f1), then obviously w £ supp(f) and w £ supp(g). But since gQh ± f, this implies also w £ supp(h). Therefore we have f1 d h. In fact, we have supp(f1) = supp(f) nsupp(g), which already implies that g Q f1 ± f. Finally, it is clear from the definition of f1 (supp(f1) C supp(f)), that f1 £ F, which ends the proof. □

4. The quantum case

In this section, S(H) will denote the set of bounded self-adjoint operators on a complex Hilbert space H, and the set of orthogonal projections on H will be denoted by P(H). Usually, P(H) is interpreted as the set of events and S(H) as the set of bounded observables (measurable physical quantities) for a quantum system. If A £ S(H) and Pa(A), A £ S(R), is the spectral measure for A, then Pa(A) is interpreted as the event that A admits a value in A. If p is a density operator on H (i.e., positive with trace 1), then p corresponds to a state of the system and tr(pPA(A)) is interpreted as the probability that A has a value in A in the state p.

For P,Q £ P(H), we have P ± Q if P + Q < I, equivalently, if PQ = 0. In [9], the latter definition is extended to elements of S(H) by defining A ± B if AB = 0, in which case A © B := A + B. In agreement with [9], we denote the closure of the range of A by rañ(A) and the projection on rañ(A) by Pa- By null(A) we denote the kernel of A. The proof of the next lemma ([9, Lemma 1]) is straightforward.

LEMMA 4.1 For A,B £ S(H), the following statements are equivalent.

(i) A ± B.

(ii) rañ(A) C null(5).

(iii) rañ(B) C null(A).

(iv) PaPb = 0.

(v) rañ(A) _L rañ(£>).

By [9, Theorem 4.2], the structure (S(H), ©, 0) is a generalized orthoalgebra. The partial order is then defined by A B if there is a C E S(H) with A ± C and A © C = B.

The following characterizations of the partial order were proved in [9, Lemma 4.3], [9, Theorem 4.6].

PROPOSITION 4.2. For A,B E S(H), the following statements are equivalent.

(i) A ^ B.

(ii) Ax = Bx for all x G ran(A).

(iii) A = BP a.

(iv) AB = A2.

(v) Pa(A) < PB(A) for all A E B(R) with 0 E A.

By [9, Corollary 4.5], every A E S(H) is principal. This implies the following statement, which in turn implies that the unitization of (S(H), ©, 0) is an orthomodular poset.

THEOREM 4.3. The structure (S(H), ©, 0) is a weak generalized orthomodular poset (WGOMP).

Proof. We have to prove

(i) if A ± B, then A © B = A V B,

(ii) A ± B, B ± C, C ± A imply A © B ± C.

To prove (i), observe that A,B A © B, and assume that for a C E S(H), A,B C. Since C is principal, it follows that A © B C, hence A © B is the least upper bound of A and B.

To prove (ii), observe that AC = 0 = BC implies (A + B)C = AC + BC = 0, hence A © B ± C. □

Remark 1.

1. We do not know if (S(H), ©, 0) is a GOMP.

2. In analogy with the classical case, we may consider the set F := {AeS(H): dim(ran(A)) < . It can be easily seen that F is an ideal in S(H). We do not know if it is a Riesz ideal.

In what follows, we need the following lemma. LEMMA 4.4. Let (Pv )v be an arbitrary set of projections on H, and let P = \J Pv

in P(H). If B E S(H) is such that B < I and Pv < B for all v, then P < B.

Proof. Let us denote Mv = ran(Pv). We have ||(I - B)i/2x\\2 < HP^xH2, for every x E H and all v. Therefore, if x E 1JMv, (I — B)i/2x = 0, hence (I—B)x = 0 holds for every x E 1J Mv. It follows that I—B reduces (J Mv, hence also (UM,)1, and if x G ((J MV)L = f| {MV)L, then {(I-B)x,x) < \\x\\2. This entails that {(I — B)x,x) < ||Pp| (Mv )± x\\2 = HA Pj~x\\2 = \PL x\\2. Therefore {Px,x) < {Bx,x) for all x E H. □

The following theorem extends [9, Theorem 4.8].

THEOREM 4.5. Let (Aa)a be a net of elements of S(H) (that is, for every ai, «2 there is [ such that Aai, Aa2 d Ap) such that for every a: Aa d B, B G S(H). Then A = V Aa exists in (S(H), d) and A = lim Aa in the strong operator topology.

Proof. Similarly as in the proof of [9, Theorem 4.8], we obtain that Aa d Ap implies Pa0 < Pap . Therefore (PAa )a is a net of projections. According to Vigier's theorem (see e.g. [20, Lemma 1]), the supremum P = \J Pa0 (in S(H))

exists, and P = lim Pa0 in the strong operator topology. By Lemma 4.4, P coincides with the supremum of (Pa0)a in the complete lattice P(H), hence P is a projection. Since Aa d B for every a, we have Aa = BPAa = Pa0 B, and P = lim Pa0 in strong operator topology, together with the one-sided continuity of product, implies that BP = PB. Proceeding similarly as in the proof of [9, Theorem 4.8], define the operator A G S(H) by A = BP. Since

Aa = BPAa = BPPAa = APAa , we conclude by Proposition 4.2 that Aa d A. Suppose Aa d C for all a, where C G S(H). Then CPAa = Aa = BPAa so that CP = BP = A. Since C(CP) = C2P2 = (CP)2, by Proposition 4.2, A = CP d C. Hence A = V Aa. Since Aa = APa0 , we conclude that lim Aa = A in the strong operator topology. □

In [9], it was proved that (S(H), d) is a near lattice in the sense that for A,B G S(H), A A B and A V B exist in S(H) if there is C G S(H) with A,B d C. By [9, Theorem 4.17], if dimH < to, then A A B exists for every A,B G S(H). In the next theorem, we extend the latter result to every Hilbert space H and to arbitrary subsets of elements.

Corollary 4.6.

(i) For every family (A \)\ C S(H) and B G S(H) such that A\ d B for all X, the supremum \J A\ exists.

(ii) For an arbitrary family (A \)\ C S(H) the infimum f\ A\ exists.

Proof.

(i) For any finite subfamily F of (A^) ^ the supremum exists by [9, Corollary 4.13]. The finite suprema form a net in a natural way (with respect to d) each element of which is above bounded by B. Applying Theorem 4.5 we obtain that the supremum of this net exists, and it is clearly also the supremum of (A \)x.

(ii) Consider the family of all lower bounds of (Ax)x. By (i), supremum of this family exists, and it is the greatest lower bound of (A ) . □

Notice that if A G S(H) is invertible, then Pa = I, therefore A d B implies A = BI = B.

In conclusion we have the following statement.

corollary 4.7. The infimum A A B of any two elements A,B E S(H) exists, while the supremum A V B exists if and only if there is C E S(H) 'with A,B C.

5. Compatibility

We recall that two elements a, b in a (generalized) effect algebra P are called (Mackey) compatible (written a ^ b) if there are elements ai,bi,c E P such that a1 © b1 © c is defined, and a = a1 © c, b = b1 © c. If Q is a subset of P, we say that a, b are compatible in Q if a1, b1, c belong to Q. Notice that if E is the unitization of P, then a,b E P are compatible in E if and only if they are compatible in P. If P is a WGOMP, so that E is an OMP, then c = a A b, and a1 © b1 © c = a1 V b1 V c = a V b. Consequently, any two compatible observables in a WGOMP have a supremum.

proposition 5.1 The unitization of the system (M(A), ©, 0) is not a Boolean algebra.

Proof. Let f,g E M(A) be such that f(w) = 0 for all w E 0, and g ¿ f. Then f V g does not exist in M(A). Indeed, if f h, then f (w) = h(w) for all w E supp(f), hence h = f. Therefore f and g have no upper bound in M(A), consequently, f ^ g does hot hold. Since an OMP is a Boolean algebra iff all pairs of elements are compatible, we conclude that the unitization of (M(A), ©, 0) is not a Boolean algebra. □

We remark that since an OMP with the Riesz decomposition properties is a Boolean algebra, the Riesz decomposition property is not satisfied in the unitization.

In the next theorem, we formulate a necessary and sufficient condition for the compatibility of two elements in S(H).

Theorem 5.2. In (S(H), ©, 0) thee following statements for A, B E S(H) are equivalent.

(i) A ^ B.

(ii) AB = (A A B)2.

Proof.

(i) implies (ii): If A ^ B, then A and B are compatible in the unitization of S(H), which is an OMP. Therefore we may write A = (A © (A A B)) © A A B, B = (B © (A A B)) © A A B, and (A © (A A B)) © (A A B) © (B © (A A B)) is defined. It follows that A ± (B © (A A B)), hence AB - A(A A B) = 0. From A A B ^ A we have A(A A B) = (A A B)2.

(ii) implies (i) can be proved by reversing the order of the reasoning. □

THEOREM 5.3. If H is a finite dimensional Hilbert space, then A,B £ S(H) 'with A A B = 0 are Mackey compatible if and only if their spectral resolutions have the following form:

t r— l

A = XP + S vP + 0 ' pr, (!)

i=1 i=t+1 t s — 1

B = £ Xj Qj + £ f Qj + 0- Qs, (2)

j=i j=t+i

where X's, v 's and f 's are nonzero eigenvalues, and the following conditions are satisfied:

Pi < Qs + Qi, i = 1,2,...,t; Pi < Qs, i = t + - 1;

Qj < Pr + Pj, j = 1 ,2,..., t; Qj < Pr, j = t +1 , ..., s- 1.

Proof. According to [9, Theorem 4.17], operators A and B have a nonzero

lower bound iff they are of the form (1) and (2), and D = A A B = XiPi A Qi.

Now the condition AB = D2 yields PiQj = QjPi for all i, j, and the rest follows by a routine computation. □

Notice that if A A B = 0, then A, B are Mackey compatible iff AB = 0, i.e., A ± B. Moreover, if in the above theorem, 0 is not an eigenvalue of A, then A ^ B iff B d A. Consequently, if none of A, B has zero eigenvalue, then they are compatible iff they are equal.

It is well known that S(H) can be covered by maximal families of mutually commuting elements. If H is separable, then every such family can be represented by the family of all real-valued bounded measurable functions M(A) on a measure space (Q, A,f). It is easily seen that every such M(A) is a subalgebra of S(H) in the sense that 0 £ M(A), and if A,B £ M(A) with A ± B, then A © B £ M(A). Hence S(H) can be covered by subalgebras satisfying RDP, but these subalgebras do not belong to blocks of the unitization-OMP of S(H).

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Received 13. 12. 2005 Mathematical Institute

Slovak Academy of Sciences Stefanikova 49 SK-814 73 Bratislava SLOVAKIA

E-mail : pulmann@mat.savba.sk