By using the decomposition of the decoherence-free subalgebra N(T) in direct integrals of factors, we obtain a structure theorem for every uniformly continuous quantum Markov semigroup. Moreover, we prove that when there exists a faithful normal invariant state, N(T) has to be atomic and decoherence takes place.

1.
Ph.
Blanchard
and
R.
Olkiewicz
, “
Decoherence induced transition from quantum to classical dynamics
,”
Rev. Math. Phys.
15
(
03
),
217
243
(
2003
).
2.
E.
Knill
,
R.
Laflamme
, and
L.
Viola
, “
Theory of quantum error correction for general noise
,”
Phys. Rev. Lett.
84
,
2525
2528
(
2000
).
3.
D. A.
Lidar
and
K.
Birgitta Whaley
,
Decoherence-Free Subspaces and Subsystems
(
Springer
,
Berlin, Heidelberg
,
2003
), pp.
83
120
.
4.
R.
Olkiewicz
, “
Environment-induced superselection rules in Markovian regime
,”
Commun. Math. Phys.
208
(
1
),
245
265
(
1999
).
5.
R.
Olkiewicz
, “
Structure of the algebra of effective observables in quantum mechanics
,”
Ann. Phys.
286
(
1
),
10
22
(
2000
).
6.
F.
Ticozzi
and
L.
Viola
, “
Quantum Markovian subsystems: Invariance, attractivity, and control
,”
IEEE Trans. Autom. Control
53
(
9
),
2048
2063
(
2008
).
7.
R.
Carbone
,
E.
Sasso
, and
V.
Umanità
, “
Decoherence for quantum Markov semi-groups on matrix algebras
,”
Ann. Henri Poincare
14
(
4
),
681
697
(
2013
).
8.
R.
Carbone
,
E.
Sasso
, and
V.
Umanità
, “
Environment induced decoherence for Markovian evolutions
,”
J. Math. Phys.
56
(
9
),
092704
(
2015
).
9.
M.
Hellmich
, “
Quantum dynamical semigroups and decoherence
,”
Adv. Math. Phys.
2011
,
625978
.
10.
A.
Dhahri
,
F.
Fagnola
, and
R.
Rebolledo
, “
The decoherence-free subalgebra of a quantum Markov semigroup with unbounded generator
,”
Infinite Dimens. Anal., Quantum Probab. Relat. Top.
13
(
03
),
413
433
(
2010
).
11.
F.
Fagnola
and
R.
Rebolledo
, “
Algebraic conditions for convergence of a quantum Markov semigroup to a steady state
,”
Infinite Dimens. Anal., Quantum Probab. Relat. Top.
11
(
03
),
467
474
(
2008
).
12.
A.
Frigerio
and
M.
Verri
, “
Long-time asymptotic properties of dynamical semigroups on W*-algebras
,”
Math. Z.
180
(
3
),
275
286
(
1982
).
13.
J.
Deschamps
,
F.
Fagnola
,
E.
Sasso
, and
V.
Umanità
, “
Structure of uniformly continuous quantum Markov semigroups
,”
Rev. Math. Phys.
28
(
01
),
1650003
(
2016
).
14.
F.
Fagnola
,
E.
Sasso
, and
V.
Umanità
, “
The role of the atomic decoherence-free subalgebra in the study of quantum Markov semigroups
,”
J. Math. Phys.
60
,
072703
(
2019
).
15.
A.
Bátkai
,
U.
Groh
,
D.
Kunszenti-Kovács
, and
M.
Schreiber
, “
Decomposition of operator semigroups on W*-algebras
,”
Semigroup Forum
84
,
8
24
(
2012
).
16.
R.
Carbone
and
A.
Jenčová
, “
On period cycles and fixed points of a quantum channel
,”
Ann. Henri Poincare
21
,
155
188
(
2020
).
17.
K. R.
Parthasarathy
,
An Introduction to Quantum Stochastic Calculus
, Monographs in Mathematics Vol. 85 (
Birkhäuser
,
1992
).
18.
K. B.
Sinha
and
D.
Goswami
,
Quantum Stochastic Processes and Noncommutative Geometry
, Cambridge Tracts in Mathematics (
Cambridge University Press
,
2007
).
19.
D. E.
Evans
, “
Irreducible quantum dynamical semigroups
,”
Commun. Math. Phys.
54
(
3
),
293
297
(
1977
).
20.
F.
Fagnola
,
E.
Sasso
, and
V.
Umanità
, “
Structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra
,”
Open Syst. Inf. Dyn.
24
(
03
),
1740005
(
2017
).
21.
R. V.
Kadison
and
J. R.
Ringrose
,
Fundamentals of the Theory of Operator Algebras
, Graduate Studies in Mathematics Vol. 2 (
American Mathematical Society
,
1997
).
22.
J.
Tomiyama
, “
On the projection of norm one in W*-algebras III
,”
Tohoku Math. J.
11
,
125
129
(
1959
).
23.
B.
Blackadar
,
Operator Algebra
, Encyclopaedia of Mathematical Sciences Vol. 122 (
Springer-Verlag, Berlin
,
2006
).
24.
B.
Kummerer
and
R.
Nagel
, “
Mean ergodic semigroups on W*-algebras
,”
Acta Sci. Math.
41
(
1–2
),
151
159
(
1979
).
25.
S.
Sakai
,
C*-Algebras and W*-Algebras
(
Springer-Verlag
,
Heidelberg
,
1971
).
26.
M.
Takesaki
,
Theory of Operator Algebras I
(
Springer-Verlag
,
New York; Heidelberg
,
1979
).
27.
P.
Ługiewicz
and
R.
Olkiewicz
, “
Classical properties of infinite quantum open systems
,”
Commun. Math. Phys.
239
,
241
259
(
2003
).
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