The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert C-module l2(A)
We characterize A -linear symmetric and contraction module operator semigroup { T t } t ∈ℝ+ ⊂ L ( l 2 ( A )), where A is a finite-dimensional C *-algebra, and L ( l 2 ( A )) is the C *-algebra of all adjointable module maps on l 2 ( A ). Next, we introduce the concept of operator-valued quadratic fo...
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| Published in | Science China. Mathematics Vol. 57; no. 2; pp. 377 - 387 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.02.2014
|
| Subjects | |
| Online Access | Get full text |
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| Summary: | We characterize
A
-linear symmetric and contraction module operator semigroup {
T
t
}
t
∈ℝ+
⊂
L
(
l
2
(
A
)), where
A
is a finite-dimensional
C
*-algebra, and
L
(
l
2
(
A
)) is the
C
*-algebra of all adjointable module maps on
l
2
(
A
). Next, we introduce the concept of operator-valued quadratic forms, and give a one to one correspondence between the set of non-positive definite self-adjoint regular module operators on
l
2
(
A
) and the set of non-negative densely defined
A
-valued quadratic forms. In the end, we obtain that a real and strongly continuous symmetric semigroup {
T
t
}
t
∈ℝ+
⊂
L
(
l
2
(
A
)) being Markovian if and only if the associated closed densely defined
A
-valued quadratic form is a Dirichlet form. |
|---|---|
| ISSN: | 1674-7283 1869-1862 |
| DOI: | 10.1007/s11425-013-4678-x |