An algebraic construction of quantum flows with unbounded generators

Belton, Alexander C. R.; Wills, Stephen J.

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 349-375.

Des cocycles de Feller stochastiques quantiques $*$-homomorphes sont construits pour certains générateurs non bornés, et ainsi nous obtenons des dilatations pour des semigroupes dynamiques quantiques fortement continus sur des $C^{*}$ algèbres. Ceci généralise la construction d’un processus de Feller classique et de son semigroupe à partir d’un générateur donné. Notre construction est possible à condition que le générateur satisfasse une propriété d’invariance pour une sous-algèbre dense ${𝒜}_0$ de la $C^{*}$ algèbre $𝒜$ et obéisse aux relations de structure nécessaires; les itérations du générateur, lorsqu’elles sont appliquées à une famille génératrice de ${𝒜}_0$, doivent satisfaire à une condition de croissance. De plus, il est supposé que soit la sous-algèbre ${𝒜}_0$ est engendrée par les isométries et $𝒜$ est universelle, ou bien ${𝒜}_0$ contient ses racines carrées. Ces conditions sont vérifiées dans quatre cas: marches aléatoires classiques sur les groupes discrets, le processus d’exclusion quantique symétrique introduit par Rebolledo et des flux sur le tore non commutatif et l’algèbre de rotation universelle.

It is shown how to construct $*$-homomorphic quantum stochastic Feller cocycles for certain unbounded generators, and so obtain dilations of strongly continuous quantum dynamical semigroups on $C^{*}$ algebras; this generalises the construction of a classical Feller process and semigroup from a given generator. Our construction is possible provided the generator satisfies an invariance property for some dense subalgebra ${𝒜}_0$ of the $C^{*}$ algebra $𝒜$ and obeys the necessary structure relations; the iterates of the generator, when applied to a generating set for ${𝒜}_0$, must satisfy a growth condition. Furthermore, it is assumed that either the subalgebra ${𝒜}_0$ is generated by isometries and $𝒜$ is universal, or ${𝒜}_0$ contains its square roots. These conditions are verified in four cases: classical random walks on discrete groups, Rebolledo’s symmetric quantum exclusion process and flows on the non-commutative torus and the universal rotation algebra.

Classification : 81S25, 46L53, 46N50, 47D06, 60J27
Mots clés : quantum dynamical semigroup, quantum Markov semigroup, cpc semigroup, strongly continuous semigroup, semigroup dilation, Feller cocycle, higher-order itô product formula, random walks on discrete groups, quantum exclusion process, non-commutative torus

[1] L. Accardi and S. V. Kozyrev. On the structure of Markov flows. Chaos Solitons Fractals 12 (14–15) (2001) 2639–2655. | MR | Zbl

[2] J. Anderson and W. Paschke. The rotation algebra. Houston J. Math. 15 (1) (1989) 1–26. | MR | Zbl

[3] S. Attal. Classical and quantum stochastic calculus. In Quantum Probability Communications X 1–52. R. L. Hudson and J. M. Lindsay (Eds). World Scientific, Singapore, 1998. | MR

[4] P. Biane. Calcul stochastique non-commutatif. In Lectures on Probability Theory (Saint-Flour, 1993) 1–96. P. Bernard (Ed.). Lecture Notes in Mathematics 1608. Springer, Berlin, 1995. | MR | Zbl

[5] P. Biane. Itô’s stochastic calculus and Heisenberg commutation relations. Stochastic Process. Appl. 120 (5) (2010) 698–720. | MR | Zbl

[6] O. Bratteli and D. W. Robinson. Operator Algebras and Quantum Statistical Mechanics 1 2002. | MR | Zbl

[7] O. Bratteli and D. W. Robinson. Operator Algebras and Quantum Statistical Mechanics 2. Equilibrium States. Models in Quantum Statistical Mechanics, second printing of the second edition. Springer, Berlin, 2002. | MR | Zbl

[8] P. S. Chakraborty, D. Goswami and K. B. Sinha. Probability and geometry on some noncommutative manifolds. J. Operator Theory 49 (1) (2003) 185–201. | MR | Zbl

[9] P. Beazley Cohen, T. M. W. Eyre and R. L. Hudson. Higher order Itô product formula and generators of evolutions and flows. Internat. J. Theoret. Phys. 34 (8) (1995) 1481–1486. | MR | Zbl

[10] K. R. Davidson. $C^{*}$-Algebras by Example. Fields Institute Monographs 6. Amer. Math. Soc., Providence, RI, 1996. | DOI | MR | Zbl

[11] F. Fagnola. Quantum Markov semigroups and quantum flows. Proyecciones 18 (3) (1999). 1–144. | MR | Zbl

[12] F. Fagnola and K. B. Sinha. Quantum flows with unbounded structure maps and finite degrees of freedom. J. London Math. Soc. (2) 48 (3) (1993) 537–551. | MR | Zbl

[13] J. C. García, R. Quezada and L. Pantaleón-Martínez. Sufficient condition for the existence of invariant states for the asymmetric exclusion QMS. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14 (2) (2011) 337–343. | MR | Zbl

[14] L. Pantaleón-Martínez and R. Quezada. The asymmetric exclusion quantum Markov semigroup. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (3) (2009) 367–385. | MR | Zbl

[15] D. Goswami, L. Sahu and K. B. Sinha. Dilation of a class of quantum dynamical semigroups with unbounded generators on UHF algebras. Ann. Inst. H. Poincaré Probab. Statist. 41 (3) (2005) 505–522. | Numdam | MR | Zbl

[16] R. L. Hudson and K. R. Parthasarathy. Quantum Ito’s formula and stochastic evolutions. Comm. Math. Phys. 93 (3) (1984) 301–323. | MR | Zbl

[17] R. L. Hudson and S. Pulmannová. Chaotic expansion of elements of the universal enveloping algebra of a Lie algebra associated with a quantum stochastic calculus. Proc. London Math. Soc. (3) 77 (2) (1998) 462–480. | MR | Zbl

[18] R. L. Hudson and P. Robinson. Quantum diffusions and the noncommutative torus. Lett. Math. Phys. 15 (1) (1988) 47–53. | MR | Zbl

[19] T. M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin, 1999. | MR | Zbl

[20] J. M. Lindsay. Quantum stochastic analysis – An introduction. In Quantum Independent Increment Processes I 181–271. M. Schürmann and U. Franz (Eds). Lecture Notes in Mathematics 1865. Springer, Berlin, 2005. | MR | Zbl

[21] J. M. Lindsay and S. J. Wills. Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise. Probab. Theory Related Fields 116 (4) (2000) 505–543. | MR | Zbl

[22] J. M. Lindsay and S. J. Wills. Markovian cocycles on operator algebras adapted to a Fock filtration. J. Funct. Anal. 178 (2) (2000) 269–305. | MR | Zbl

[23] J. M. Lindsay and S. J. Wills. Existence of Feller cocycles on a $C^{*}$-algebra. Bull. London Math. Soc. 33 (5) (2001) 613–621. | MR | Zbl

[24] J. M. Lindsay and S. J. Wills. Homomorphic Feller cocycles on a $C^{*}$-algebra. J. London Math. Soc. (2) 68 (1) (2003) 255–272. | MR | Zbl

[25] J. M. Lindsay and S. J. Wills. Quantum stochastic cocycles and completely bounded semigroups on operator spaces. Int. Math. Res. Not. IMRN. To appear, 2014. DOI:10.1093/imrn/rnt001. | DOI | MR | Zbl

[26] P.-A. Meyer. Quantum Probability for Probabilists, 2nd edition. Lecture Notes in Mathematics 1538. Springer, Berlin, 1995. | MR | Zbl

[27] K. R. Parthasarathy and K. B. Sinha. Markov chains as Evan–Hudson diffusions in Fock space. In Séminaire de Probabilités XXIV 362–369. J. Azéma, P.-A. Meyer and M. Yor (Eds). Lecture Notes in Mathematics 1426. Springer, Berlin, 1990. | Numdam | MR | Zbl

[28] R. Rebolledo. Decoherence of quantum Markov semigroups. Ann. Inst. H. Poincaré Probab. Statist. 41 (3) (2005) 349–373. | Numdam | MR | Zbl

[29] P. Robinson. Quantum diffusions on the rotation algebras and the quantum Hall effect326–333. L. Accardi and W. von Waldenfels (Eds). Lecture Notes in Mathematics 1442. Springer, Berlin, 1990. | MR | Zbl

[30] K. B. Sinha and D. Goswami. Quantum Stochastic Processes and Noncommutative Geometry. Cambridge Univ. Press, Cambridge, 2007. | MR | Zbl