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GNS-like representations of a quasi *-algebra in Krein spaces

  • Giorgia Bellomonte
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Abstract

In this paper unbounded representations of quasi *-algebras on an indefinite inner product space are introduced and studied. After introducing the partial *-algebra of unbounded closable operators sharing a common dense domain in a Krein space, the appropriate notion of *-representation in this setting is developed, and a GNS-type construction for quasi *-algebras based on α-positive sesquilinear forms is established. Furthermore, the notion of abstract conditional expectation is generalized to quasi *-algebras and how \emph{admissible} abstract conditional expectations P give rise to representable P-forms is shown. The relation between α-positive forms and P-forms is also clarified. Several examples are provided.

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Cite

BibTeXRIS
BibTeX
@misc{bellomonte2026gnslike,
  title = {{GNS-like representations of a quasi *-algebra in Krein spaces}},
  author = {Giorgia Bellomonte},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08329},
  eprint = {2611.08329},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08329},
}
RIS
TY  - UNPB
TI  - GNS-like representations of a quasi *-algebra in Krein spaces
AU  - Giorgia Bellomonte
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08329
UR  - https://arxiv.org/abs/2611.08329
AB  - In this paper unbounded representations of quasi *-algebras on an indefinite inner product space are introduced and studied. After introducing the partial *-algebra of unbounded closable operators sharing a common dense domain in a Krein space, the appropriate notion of *-representation in this setting is developed, and a GNS-type construction for quasi *-algebras based on α-positive sesquilinear forms is established. Furthermore, the notion of abstract conditional expectation is generalized to quasi *-algebras and how \emph{admissible} abstract conditional expectations P give rise to representable P-forms is shown. The relation between α-positive forms and P-forms is also clarified. Several examples are provided.
ER  -