In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both §§ Mixed states and Pure states. Density matrices in turn generalize state vectors, which only represent pure states. For M an operator system in a C*-algebra A with identity, the set of all states of M, sometimes denoted by S(M), is convex, weak-* closed in the Banach dual space M*. Thus the set of all states of M with the weak-* topology forms a compact Hausdorff space, known as the state space of M .
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| - Zustand (Mathematik) (de)
- 상태 (함수해석학) (ko)
- State (functional analysis) (en)
- Estado (análise funcional) (pt)
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| - Ein Zustand ist ein mathematischer Begriff, der in der Funktionalanalysis untersucht wird. Es handelt sich um bestimmte lineare Funktionale auf reellen oder komplexen Vektorräumen, die in gewisser Weise normiert sind. Oft sind die Definitionen so angelegt, dass die Zustände bezüglich einer Ordnungsstruktur positiv sind, das heißt, dass sie die positiven Elemente dieser Ordnung auf nicht-negative reelle Zahlen abbilden. Ferner bildet der Zustandsraum, das ist die Menge der Zustände, einen topologisch oder geometrisch interessanten Raum. (de)
- C* 대수 이론에서, 상태(狀態, 영어: state)는 C* 대수 위에 정의된, 특정한 부등식을 만족시키는, 작용소 노름 1의 복소수 값 유계 작용소이다. 이는 대략 C* 대수를 비가환 공간으로 여겼을 때 일종의 “확률 측도”로 여길 수 있다. 양자역학의 밀도 행렬을 추상화한 개념이다. 겔판트-나이마르크-시걸 구성(Гельфанд-Наймарк-Segal構成, 영어: Gelfand–Naimark–Segal construction, 약자 GNS 구성)에 따라, 상태들은 C* 대수의, 복소수 힐베르트 공간 위의 표현(의 동치류)와 일대일 대응한다. (ko)
- Em análise funcional, um estado numa é uma função linear positiva de norma 1. O conjunto de estados de uma C*-álgebra A, algumas vezes escrita de forma , é sempre um conjunto convexo. Os pontos extremos de são chamados estados puros. Na formulação da C*-álgebra da mecânica quântica, estados neste sentido correspondem a estados físicos, isto é: são mapeamentos dos observáveis físicos dos seus resultados esperados de medição. (pt)
- In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both §§ Mixed states and Pure states. Density matrices in turn generalize state vectors, which only represent pure states. For M an operator system in a C*-algebra A with identity, the set of all states of M, sometimes denoted by S(M), is convex, weak-* closed in the Banach dual space M*. Thus the set of all states of M with the weak-* topology forms a compact Hausdorff space, known as the state space of M . (en)
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| - Ein Zustand ist ein mathematischer Begriff, der in der Funktionalanalysis untersucht wird. Es handelt sich um bestimmte lineare Funktionale auf reellen oder komplexen Vektorräumen, die in gewisser Weise normiert sind. Oft sind die Definitionen so angelegt, dass die Zustände bezüglich einer Ordnungsstruktur positiv sind, das heißt, dass sie die positiven Elemente dieser Ordnung auf nicht-negative reelle Zahlen abbilden. Ferner bildet der Zustandsraum, das ist die Menge der Zustände, einen topologisch oder geometrisch interessanten Raum. (de)
- In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both §§ Mixed states and Pure states. Density matrices in turn generalize state vectors, which only represent pure states. For M an operator system in a C*-algebra A with identity, the set of all states of M, sometimes denoted by S(M), is convex, weak-* closed in the Banach dual space M*. Thus the set of all states of M with the weak-* topology forms a compact Hausdorff space, known as the state space of M . In the C*-algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i.e. mappings from physical observables (self-adjoint elements of the C*-algebra) to their expected measurement outcome (real number). (en)
- C* 대수 이론에서, 상태(狀態, 영어: state)는 C* 대수 위에 정의된, 특정한 부등식을 만족시키는, 작용소 노름 1의 복소수 값 유계 작용소이다. 이는 대략 C* 대수를 비가환 공간으로 여겼을 때 일종의 “확률 측도”로 여길 수 있다. 양자역학의 밀도 행렬을 추상화한 개념이다. 겔판트-나이마르크-시걸 구성(Гельфанд-Наймарк-Segal構成, 영어: Gelfand–Naimark–Segal construction, 약자 GNS 구성)에 따라, 상태들은 C* 대수의, 복소수 힐베르트 공간 위의 표현(의 동치류)와 일대일 대응한다. (ko)
- Em análise funcional, um estado numa é uma função linear positiva de norma 1. O conjunto de estados de uma C*-álgebra A, algumas vezes escrita de forma , é sempre um conjunto convexo. Os pontos extremos de são chamados estados puros. Na formulação da C*-álgebra da mecânica quântica, estados neste sentido correspondem a estados físicos, isto é: são mapeamentos dos observáveis físicos dos seus resultados esperados de medição. (pt)
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