## Details

Original language | English |
---|---|

Article number | 20 |

Number of pages | 39 |

Journal | Integral Equations and Operator Theory |

Volume | 96 |

Issue number | 3 |

Early online date | 22 Jun 2024 |

Publication status | Published - Sept 2024 |

## Abstract

In this article, we study a commutative Banach algebra structure on the space L^{1}(R^{2n})⊕T^{1}, where the T^{1} denotes the trace class operators on L^{2}(R^{n}). The product of this space is given by the convolutions in quantum harmonic analysis. Towards this goal, we study the closed ideals of this space, and in particular its Gelfand theory. We additionally develop the concept of quantum Segal algebras as an analogue of Segal algebras. We prove that many of the properties of Segal algebras have transfers to quantum Segal algebras. However, it should be noted that in contrast to Segal algebras, quantum Segal algebras are not ideals of the ambient space. We also give examples of different constructions that yield quantum Segal algebras.

## Keywords

- 43A20, 81S99, Feichtinger algebra, Primary 47B93, Quantum harmonic analysis, Secondary 47B48, Segal algebra

## ASJC Scopus subject areas

- Mathematics(all)
**Analysis**- Mathematics(all)
**Algebra and Number Theory**

## Cite this

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- Apa
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- BibTeX
- RIS

**A Quantum Harmonic Analysis Approach to Segal Algebras.**/ Berge, Eirik; Berge, Stine Marie; Fulsche, Robert.

In: Integral Equations and Operator Theory, Vol. 96, No. 3, 20, 09.2024.

Research output: Contribution to journal › Article › Research › peer review

*Integral Equations and Operator Theory*, vol. 96, no. 3, 20. https://doi.org/10.1007/s00020-024-02771-w

*Integral Equations and Operator Theory*,

*96*(3), Article 20. https://doi.org/10.1007/s00020-024-02771-w

}

TY - JOUR

T1 - A Quantum Harmonic Analysis Approach to Segal Algebras

AU - Berge, Eirik

AU - Berge, Stine Marie

AU - Fulsche, Robert

N1 - Publisher Copyright: © The Author(s) 2024.

PY - 2024/9

Y1 - 2024/9

N2 - In this article, we study a commutative Banach algebra structure on the space L1(R2n)⊕T1, where the T1 denotes the trace class operators on L2(Rn). The product of this space is given by the convolutions in quantum harmonic analysis. Towards this goal, we study the closed ideals of this space, and in particular its Gelfand theory. We additionally develop the concept of quantum Segal algebras as an analogue of Segal algebras. We prove that many of the properties of Segal algebras have transfers to quantum Segal algebras. However, it should be noted that in contrast to Segal algebras, quantum Segal algebras are not ideals of the ambient space. We also give examples of different constructions that yield quantum Segal algebras.

AB - In this article, we study a commutative Banach algebra structure on the space L1(R2n)⊕T1, where the T1 denotes the trace class operators on L2(Rn). The product of this space is given by the convolutions in quantum harmonic analysis. Towards this goal, we study the closed ideals of this space, and in particular its Gelfand theory. We additionally develop the concept of quantum Segal algebras as an analogue of Segal algebras. We prove that many of the properties of Segal algebras have transfers to quantum Segal algebras. However, it should be noted that in contrast to Segal algebras, quantum Segal algebras are not ideals of the ambient space. We also give examples of different constructions that yield quantum Segal algebras.

KW - 43A20

KW - 81S99

KW - Feichtinger algebra

KW - Primary 47B93

KW - Quantum harmonic analysis

KW - Secondary 47B48

KW - Segal algebra

UR - http://www.scopus.com/inward/record.url?scp=85196652788&partnerID=8YFLogxK

U2 - 10.1007/s00020-024-02771-w

DO - 10.1007/s00020-024-02771-w

M3 - Article

AN - SCOPUS:85196652788

VL - 96

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

SN - 0378-620X

IS - 3

M1 - 20

ER -