Self-Adjoint Algebras of Unbounded Operators

Journal Article · Tue Jun 01 00:00:00 EDT 1971 · Communications in Mathematical Physics

DOI:https://doi.org/10.1007/bf01646746· OSTI ID:4101201

Powers, Robert T. ^[1]

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1. Univ. of Pennsylvania, Philadelphia, PA (United States)

Unbounded *-representations of *-algebras are studied. Representations called self-adjoint representations are defined in analogy to the definition of a self-adjoint operator. It is shown that for self-adjoint representations certain pathologies associated with commutant and reducing subspaces are avoided. A class of well behaved self-adjoint representations, called standard representations, are defined for commutative *-algebras. It is shown that a strongly cyclic self-adjoint representation of a commutative *-algebra is standard if and only if the representation is strongly positive, i.e., the representations preserves a certain order relation. Similar results are obtained for *-representations of the canonical commutation relations for a finite number of degrees of freedom.

• Research Organization: Univ. of Pennsylvania, Philadelphia, PA (United States)
• Sponsoring Organization: US Atomic Energy Commission (AEC)
• NSA Number: NSA-25-003968
• OSTI ID: 4101201
• Alternate ID(s): OSTI ID: 4720424
• Report Number(s): NYO--2171-325
• Journal Information: Communications in Mathematical Physics, Journal Name: Communications in Mathematical Physics Journal Issue: 2 Vol. 21; ISSN 0010-3616
• Publisher: Springer
• Country of Publication: United States
• Language: English

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self-adjoint algebras of unbounded operators