Measures on the quantum logic of subspaces of aJ-space

Vladova

2005

Int J Theor Phys

The aim of the paper is to measure the logic of J-projections from inductive limit of W J-algebras studied. The main result is Theorem. Let A be a W * J -factor of countable type (type of A is different from I 2 ) and let A be the inductive limit of W * J -factors A α different from I 2 . If (1) A be a W * P -factor or (2) A and all A α are W * K-factors, then any indefinite measure ν : ∪ α A h α → R can be unique by the strong operator topology extended to an indefinite measure on A h .

Section: Proposition 5 Let Conditions Of Proposition 4 Are Fulfilledmentioning

confidence: 90%

“…In Matvejchuk (2000) Theorem 4, we examined indefinite measures on W * J −algebra (the case B(H) see also Matvejchuk (1991). We have proved:…”

Section: Proposition 5 Let Conditions Of Proposition 4 Are Fulfilledmentioning

confidence: 95%

On a Measure on the Inductive Limit of Projection Logics

Vladova

2005

Int J Theor Phys

“…Clearly P e is a quantum sublogic of P . In [3], [4] the logic P e is called a hyperbolic logic. Denote by P + e (P − e ) the set of all p ∈ P e for which the subspace pH e of H e is positive (i.e., ∀z ∈ pH e , z = 0, [z, z] > 0) ( respectively, negative, i.e., ∀z ∈ H e , z = 0, [z, z] < 0).…”

Section: Hyperbolic Sublogics Of the Logic Pmentioning

confidence: 99%

“…In this case, the set P of all J-orthogonal projections serves to be an analog to the logic Π. There is an indefinite analog to the Gleason theorem [3].…”

Section: Introductionmentioning

confidence: 99%

Probability measures in 𝑊*𝐽-algebras in Hilbert spaces with conjugation

1998

Proc. Amer. Math. Soc.

Abstract. Let M be a real W * -algebra of J-real bounded operators containing no central summand of type I 2 in a complex Hilbert space H with conjugation J. Denote by P the quantum logic of all J-orthogonal projections in the von Neumann algebra N = M + iM. Let µ : P → [0, 1] be a probability measure. It is shown that N contains a finite central summand and there exists a normal finite trace τ on N such that µ(p) = τ(p), ∀p ∈ P .

Gleason's Theorem in a Space with Indefinite Metric

Mathematische Nachrichten

1997

An analog to the Gleason theorem for measures on logics of projections in indefinite metric spaces is proved. IntroductionThe problem of describing measures on logics is well-known [B]. A celebrated theorem of Gleason [GI serves to be a base for the Quantum Measure Theory. The theorem asserts every nonnegative measure p on the orthogonal projections of a Hilbert space K with dim K 2 3 to be of the form p ( p ) = t r (Tp), where T 2 0 is a uniquely determined trace class operator. Here we give an analogy to the Gleason's theorem for a real measure on an indefinite metric space.We present the necessary definitions and notation. Let H be a space with an indefinite metric [ a , a ] , a canonical decomposition H = H+ [+I H-, and with a canonical symmetry J. In the terminology of [AI] H is a K r e h space (= J-space). H is a Hilbert space with respect to the inner product (see [AI]) (2, z ) = [Jz, 4. There exist orthogonal projections Q+ and Qsuch that Q++Q-= I , J = Q+-Q-, Q + H = H+, Q-H = Hwith [z, z] = (Jz, z ) , for any z, z E H. Let S = {z E H : (2, z) = 1). Put I?+ = {z E H : [z,z] = 1) and I?-= {x E H : [z,z] = -1). The set r = r+ uris an indefinite analogy to the unit sphere S. Let B(H)P be the set of all orthogonal projections in B ( H ) and P be the set of all J-selfadjoint projections in B ( H ) , i. e., P = { p E B ( H ) : p2 = p , [p., z] = [x,pz], for any z, z E H} It is easy to see that any one-dimensional projection p E P can be represented in the form p f = [ f, f ] [ . , f l f , f E r . With respect to the ordering, p 5 q if and 1991 Mathematics Subject Classification. Primary 81 P 10, 46 L 50, 46 B 09, 46 C 20; Secondary Keywords and phrases. Quantum logics, measure, indefinite metric, W* -algebra. 28 A 80. 230Math. Nachr. 184 (1997) only if pq = qp = p and the orthocomplementation, p -+ pL 5 Ip , the set P is a quantum logic. Now let P+(P-) be the set of all projections p E P , for which the subspace pH is positive (for all x E pH, x # 0, [x, x] > 0) (respectively, negative, i. e., for all x E pH, x # 0, [x,x] < 0). Any projection e E P is representable in the form e = e+ + e-, where e+ E P+, e-E P-. Definition 1.1. A mapping p : P -+ IR is called a measure if p ( e ) = C p ( e L ) for any representation e = C e , (where eLea = 0, L # (Y and the sum is understood in the strong sense). A measure p is said to be indefinite if p / p + 1 0 and p / p -5 0;bounded if cfi = sup { lp(p)l IIpII-' : p E P } < m. The main resultsOur main result is the following: Theorem 2.1. Let Y : P + R be a measure on an infinite-dimensional Kreiiz space H , dim H+ 5 dim H-. Then there exist a unique J -selfadjoint trace -class operator T and a unique number c such that (2.1) v(e) = tr(Te) + cdim(etH), for all e E P . Moreover, if dim H+ = 00, then c = 0 (0m 0 ) .Remark 2.2. An indefinite measure is an analog to a probability measure for the logic P of the J-selfadjoint projections. In [MI, we proved that for any indefinite measure p in a Kre'in space, dim H 2 3, the formula (2.1) is true.At first we prove the followin...