Abstract:
The theory of markovian dynamics of open quantum systems relies on the notion of
one-parameter completely positive semigroup on C$^*$-algebras. Generally, the theory of
operator semigroups on Banach spaces is an important field of study with an abundance
of applications [1].
The development of functional analysis and quantum mechanics lead to the idea that
the completely positive maps are naturally defined on operator systems [2]. Therefore
we may regard the completely positive unital semigroups on operator systems to be the
natural framework for studying quantum markovian dynamics.
The category of operator systems and completely positive maps has a number of good
extension properties, and every operator system has an injective envelope, which is a C$^*$-algebra [3]. Using these properties, we show that every unital completely positive semigroup
on operator system can be uniquely extended to a completely positive semigroup
on its injective envelope.
The work is supported by the Russian Science Foundation under the grant no. 19-11-
00086 and performed in the Steklov Mathematical Institute of the Russian Academy of
Sciences.
Language: English
References
Engel K.-J., Nagel R., Brendle S., One-parameter semigroups for linear evolution equations, Springer \publaddress New York, 2000
Paulsen V., Completely bounded maps and operator algebras, Cambridge University Press, 2002
Hamana M., “Injective envelopes of operator systems”, Publications of the Research Institute for Mathematical Sciences, 15:3 (1979), 773–785
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