# A Note on States and Traces from Biorthogonal Sets

## Abstract

**:**

## 1. Introduction and Notations

- If ${a}_{\alpha}\to 0$ with regard to $\tau $ and $\omega \left({a}_{\alpha}\right)\to \ell $, then $\ell =0$.
- $\overline{{G}_{\omega}}$, the closure of ${G}_{\omega}$, does not contain couples $(0,\ell )$ with $\ell \ne 0$.

**Definition**

**1.**

**Proposition**

**1.**

## 2. Stating the Problem and Results

**Definition**

**2.**

**well-behaved**if ${Z}_{\phi \phi}<\infty $ and ${Z}_{\psi \psi}<\infty $.

#### Gibbs States

## 3. A Possible Further Generalization of ${\mathit{\omega}}_{\mathbf{0}}$

- (g1)
- ${G}_{\omega}\subseteq H\subset \overline{{G}_{\omega}}$
- (g2)
- $(0,\ell )\in H$ if, and only if, $\ell =0$.

**Proposition**

**2.**

## 4. Concluding Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Shang, Y. Unveiling robustness and heterogeneity through percolation triggered by random-link breakdown. Phys. Rev. E
**2014**, 90, 032820. [Google Scholar] [CrossRef] [PubMed] - Shang, Y. Effect of link oriented self-healing on resilience of networks. J. Stat. Mech. Theory Exp.
**2016**, 2016, 083403. [Google Scholar] [CrossRef] - Di Salvo, R.; Oliveri, F. An operatorial model for long-term survival of bacterial populations. Ric. Mat.
**2016**, 65, 435–447. [Google Scholar] [CrossRef] - Antoine, J.-P.; Inoue, A.; Trapani, C. Partial *-algebras and Their Operator Realizations; Kluwer: Dordrecht, The Netherlands, 2002. [Google Scholar]
- Bagarello, F.; Trapani, C.; Triolo, S. Gibbs states defined by biorthogonal sequences. J. Phys. A Math. Theor.
**2016**, 49, 405202. [Google Scholar] [CrossRef] [Green Version] - Bratteli, O.; Robinson, D.W. Operator Algebras and Quantum Statistical Mechanics 2; Springer: New York, NY, USA, 1987. [Google Scholar]
- Bratteli, O.; Robinson, D.W. Operator Algebras and Quantum Statistical Mechanics 1; Springer: New York, NY, USA, 1987. [Google Scholar]
- Sewell, G.L. Quantum Mechanics and Its Emergent Macrophysics; Princeton University Press: Princeton, NJ, USA, 2002. [Google Scholar]
- Triolo, S. WQ*-Algebras of measurable operators. Indian J. Pure Appl. Math.
**2012**, 43, 601–617. [Google Scholar] [CrossRef] - Aiena, P.; Triolo, S. Local spectral theory for Drazin invertible operators. J. Math. Anal. Appl.
**2016**, 435, 414–424. [Google Scholar] [CrossRef] - Bagarello, F.; Trapani, C.; Triolo, S. Representable states on quasilocal quasi *-algebras. J. Math. Phys.
**2011**, 52, 013510. [Google Scholar] [CrossRef] - Antoine, J.P.; Trapani, C. Operator (quasi-)similarity, quasi-Hermitian operators and all that Non-Hermitian Hamiltonians in Quantum Physics; Bagarello, F., Passante, R., Trapani, C., Eds.; Springer Proceedings in Physics; Springer: Cham, Switzerland, 2016; Volume 184, pp. 45–65. [Google Scholar]
- Bongiorno, B.; Trapani, C.; Triolo, S. Extensions of positive linear functionals on a Topological *-algebra. Rocky Mt. J. Math.
**2010**, 40, 1745–1777. [Google Scholar] [CrossRef] - Triolo, S. Extensions of the non commutative integration. Complex Anal. Oper. Theory
**2016**, 10, 1551–1564. [Google Scholar] [CrossRef] - Trapani, C.; Triolo, S. Representations of modules over a ∗-algebra and related seminorms. Stud. Math.
**2008**, 184, 133–148. [Google Scholar] [CrossRef] - La Russa, C.; Triolo, S. Radon—Nikodym theorem in topological quasi ∗-algebras. J. Oper. Theory
**2013**, 69. [Google Scholar] [CrossRef] - Trapani, C.; Triolo, S. Representations of certain Banach C*-modules. Mediterranean J. Math.
**2004**, 1, 441–461. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Triolo, S.
A Note on States and Traces from Biorthogonal Sets. *Symmetry* **2019**, *11*, 654.
https://doi.org/10.3390/sym11050654

**AMA Style**

Triolo S.
A Note on States and Traces from Biorthogonal Sets. *Symmetry*. 2019; 11(5):654.
https://doi.org/10.3390/sym11050654

**Chicago/Turabian Style**

Triolo, Salvatore.
2019. "A Note on States and Traces from Biorthogonal Sets" *Symmetry* 11, no. 5: 654.
https://doi.org/10.3390/sym11050654