Algebra with indefinite involution and its representation in Krein space

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dc.creatorKawamura, Katsunori
dc.date2006-10-02
dc.date2007-11-21
dc.date.accessioned2026-07-07T08:44:06Z
dc.date.available2026-07-07T08:44:06Z
dc.descriptionIt is often inevitable to introduce an indefinite-metric space in quantum field theory, for example, which is explained for the sake of the manifestly covariant quantization of the electromagnetic field. We show two more evident mathematical reasons why such indefinite metric appears. The first idea is the replacement of involution on an algebra. For an algebra ${\cal A}$ with an involution $\sdag$ such that a representation of the involutive algebra $({\cal A},\sdag)$ brings an indefinite-metric space, we replace the involution $\sdag$ with a new one * on ${\cal A}$ such that $({\cal A},*)$ is a well-known involutive algebra acting on a representation space with positive definite metric. This explains that non-isomorphic two involutive algebras are transformed each other by the replacement of involution. The second is that a covariant (Hilbert space) representation $({\cal H},π,U)$ of an involutive dynamical system $(({\cal A},*),{\bf Z}_{2},α)$ brings a Krein space representation of the algebra ${\cal A}$ with the replaced involution. For example, we show representations of abnormal CCRs, CARs and pseudo-Cuntz algebras arising from those of standard CCRs, CARs and Cuntz algebras.
dc.description27pages
dc.identifierhttps://arxiv.org/abs/math/0610059
dc.identifierhttp://arxiv.org/abs/math/0610059
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/142542
dc.subjectOperator Algebras
dc.subjectMathematical Physics
dc.subject47B50, 47L55, 81T05
dc.titleAlgebra with indefinite involution and its representation in Krein space
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