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Franco Fagnola

Publications and source records attributed to Franco Fagnola.

22 records · Page 2Linked to original sources
arXiv2008

Maximal Commutative Subalgebras Invariant for CP-Maps: (Counter-)Examples

We solve, mainly by counterexamples, many natural questions regarding maximal commutative subalgebras invariant under CP-maps or semigroups of CP-maps on a von Neumann algebra. In particular, we discuss the structure of the generators of norm continuous semigroups on B(G) leaving a maximal commutative subalgebra invariant and show that there exists Markov CP-semigroups on M_d without invariant maximal commutative subalgebras for any d>2.

math.OA↗
arXiv2008

Generators of Detailed Balance Quantum Markov Semigroups

For a quantum Markov semigroup $\T$ on the algebra $\B$ with a faithful invariant state $ρ$, we can define an adjoint $\widetilde{\T}$ with respect to the scalar product determined by $ρ$. In this paper, we solve the open problems of characterising adjoints $\widetilde{\T}$ that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators $H,L_k$ in the Gorini Kossakowski Sudarshan Lindblad representation $\Ll(x)=i[H,x] - {1/2}\sum_k(L^*_kL_k x-2L^*_kxL_k + xL^*_kL_k)$ of the generator of $\T$. We study the adjoint semigroup with respect to both scalar products $ = \tr(ρa^* b)$ and $ = \tr(ρ^{1/2} a^* ρ^{1/2}b)$.

math-ph↗
arXiv2007

Restrictions of CP-Semigroups to Maximal Commutative Subalgebras

We give a necessary and sufficient criterion when a normal CP-map on a von Neumann algebra admits a restriction to a maximal commutative subalgebra. We apply this result to give a far reaching generalization of Rebolledo's sufficient criterion for the Lindblad generator of a Markov semigroup on B(G).

math.OA↗