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

Matvejchuk, Marjan

doi:10.1090/s0002-9939-98-04176-8

Cited by 12 publications

(4 citation statements)

References 6 publications

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“…By JpJ, p * ∈ B(H), it will suffice to show that Jpx = JpJx = p * x for all x ∈ H . It is easily to see this fact from (9), and equalities (10), (11). Thus JpJ = p * .…”

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confidence: 76%

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Idempotents as J-projections: II

Matvejchuk

2012

Lobachevskii J Math

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Let B(H) Id be the set of all bounded idempotents on a complex Hilbert space H and let J be a conjugation operator on H. Fix p ∈ B(H) Id . At the paper we describe of J-projections. We prove that for a given p there exists a conjugation operator J 0 such that p is a J 0 -projection. In ([1], Chapter XII) the problem of construction of probability theory for quantum mechanics is posed. An analog of boolean algebra of events is quantum logic. An important interpretation of a quantum logic is the set B(H) or of all orthogonal (=self-adjoint) projections on a Hilbert space H. The problem to construct a quantum field theory sometimes leads to an indefinite metric space. In the indefinite case, the set B(H) Jo of all J-orthogonal projections is an analog of the logics B(H) or . It is important to know the properties and structure of projections in construction of measure theory on projections logic. J-projections were extensively studied at [2-6]. Let H be a complex Hilbert space with the Hilbert product (·, ·). Let B(H) be the set of all bounded operators on H. Let J be a conjugation operator in H (see [7] $50), i.e. 1) J 2 = I, 2) (Jx, Jy) = (y, x), for all x, y ∈ H. Note by 1), and 2), J(λx + βy) = λJx + βJy, for all λ, β ∈ C. A vector x ∈ H is said to be J-real if Jx = x. The vectors x := 1 2 (x + Jx) and x := 1 2i (x − Jx) = − 1 2 (ix + Jix) are J-real,∀x ∈ H and x = x + ix . The set H of all J-real vectors is a real Hilbert space with respect to the inner product (·, ·). Put x, y := (Jx, y). It is clear that an operator A ∈ B(H) is J-self-adjoint, i.e. Ax, y = x, Ay , ∀x, y ∈ H iff A = JA * J. Any bounded J-self-adjoint idempotent (=projection) p is said to be J-projection.Ap is a J 0 -projection (i.e. p = J 0 p * J 0 ) with respect to the symmetry J 0 := 2p − I and a J 1 -projection with respect to any conjugation operator J 1 such that J 1 pH = pH, J 1 (I − p)H = (I − p)H.We can formulate our main result Theorem 2. Let p be a bounded idempotent on a complex Hilbert space H. Then there exists a conjugation operator J on H with (·, ·) such that p is the J-projection.

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“…By JpJ, p * ∈ B(H), it will suffice to show that Jpx = JpJx = p * x for all x ∈ H . It is easily to see this fact from (9), and equalities (10), (11). Thus JpJ = p * .…”

mentioning

confidence: 76%

“…Proposition 3 [11]. The projection p or is the greatest orthogonal projection with p or ≤ p. If p = Jp * J then p or = Jp or J.…”

mentioning

confidence: 99%

Idempotents as J-projections: II

Matvejchuk

2012

Lobachevskii J Math

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“…Пусть r -ограниченный проектор в H. Обозначим через r or ортогональный про-ектор на подпространство rH ∩ r * H. По предложению 4 [3] r or -наибольший орто-гональный проектор со свойством r or r. Проектор r назовем чисто косым, если r or = 0. Отметим, для любого неортогонального r проектор r − r or чисто косой.…”

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“…Доказательство можно провести по схеме доказательства похожей теоремы из [3]. Специфика же J-алгебр Неймана типа (B) позволяет дать прямое доказа-тельство.…”

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$J$-Алгебры Неймана В Пространстве С Двумя Симметриями

Матвейчук¹,

Matveichuk²

2007

Mat. Zametki

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В данной работе с помощью абелевой алгебры Неймана N вводятся J N -неотри-цательные векторы и изучаются некоторые особенности операторов в J-алгебрах типа (B), связанные с положительностью. Мы убеждаемся, что в J-алгебрах ти-па (B) J-самосопряженные операторы (см. предложение 7 и следствие 8) и некото-рые классы плюс-операторов (см. теорема 12) имеют специфическое строение.

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Real-Orthogonal Projections as Quantum Pseudo-Logic

Matvejchuk

Widdows

2016

Quantum Interaction

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

Cited by 12 publications

References 6 publications

Idempotents as J-projections: II

Idempotents as J-projections: II

$J$-Алгебры Неймана В Пространстве С Двумя Симметриями

Real-Orthogonal Projections as Quantum Pseudo-Logic

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