Gleason's Theorem in a Space with Indefinite Metric

Abstract

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...

Orthogonality Preserving Transformations on Indefinite Inner Product Spaces: Generalization of Uhlhorn's Version of Wigner's Theorem

“…In the present paper, we apply a completely different approach and obtain a much more general result, namely, a result concerning indefinite inner product spaces induced by any invertible bounded linear operator on a real or complex Hilbert space of any dimension (not less than 3). Quantum logics on spaces with such a general indefinite metric have been investigated by, for example, Matvejchuk in [5]. Our result will follow from the main theorem of the paper, which describes the form of all bijective transformations of the set of all rank-one idempotents on a Banach space which preserve zero products in both directions.…”

Vector Measures on the Logic of J-projections of a Krein Space

“…Conversely, let ξ be a bounded linear H-measure and let a linear operatorξ : B(H ) → H be such thatξ (p) = ξ (p), for all p ∈ P. By Theorem 2.3 of Matvejchuk (1997), for all h ∈ H there exist a unique M h ∈ L 1 and a unique number c h , such that ξ h (q) := (ξ (q), h) H = tr(qM h ) + c h dim(q + H ), ∀q ∈ P.…”

“…The main results for the real measure are the following: Matvejchuk (1997). Let H be a Krein space, dim H ≥ 3, dim H + ≤ dim H − and let µ : P → R be a real measure.…”

“…In this case, the set P of all J -orthogonal projections serves to be an analog to the logics B(H ) pr . There is an indefinite analog (Matvejchuk, 1997) to the remarkable Gleason's theorem. In this paper a vector measure on the quantum logic P is studied for the first time.…”

“…Then any real measure µ : P → R is bounded.Let ξ be a bounded H-measure. By Theorem 2.3 ofMatvejchuk (1997), for all h ∈ H there exist a unique M h ∈ L 1 and a unique number c h , such that…”