Let B(H ) I d be the set of all bounded idempotents on a Hilbert space H . Fix p ∈ B(H ) I d . The aim of the paper is to show a set of symmetries J on H for which p is a J -projection.Keywords Hilbert space · Indefinite metric space · Idempotent · Projection In [1] the problem of construction of probability theory for quantum mechanics is posed. An analog of Boolean algebra of events is quantum logic. An impotent interpretation of a quantum logic is the set B(H ) pr of all orthogonal projections on a Hilbert space H . The problem to construct a quantum field theory sometimes leads to an indefinite metric space. In indefinite case, the set P of all J -orthogonal projections is an analog of the logics B(H ) pr . In construction of measure theory on logics of projections it is important to know the properties and structure of projections. J -projections were extensively studied by [2][3][4][5]7].Let a bounded idempotent p be a given. We are interested in finding a set of symmetries J on H for which p is a J -projection, positive (negative, maximal positive, maximal negative) J -projection. We study the conditions under which idempotent p becomes a J -projections of type (B).Let H be a Hilbert space with the Hilbert product (. , .). Let us denote by B(H ) I d (by Π ) the set of all bounded (orthogonal, respectively) projections (=idempotents) on B(H ).

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...

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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 .

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