Abstract
Invariance properties of linear functionals and linear maps on algebras of functions on quantum homogeneous spaces are studied, in particular for the special case of expected co-ideal *-subalgebras. Several one-to-one correspondences between such invariant functionals are established. Adding a positivity condition, this yields one-to-one correspondences of invariant quantum Markov semigroups acting on expected co-ideal *-subalgebras and certain convolution semigroups of states on the underlying compact quantum group. This gives an approach to classifying invariant quantum Markov semigroups on these quantum homogeneous spaces. The generators of these semigroups are viewed as Laplace operators on these spaces.
Highlights
In this paper we introduce an approach for classifying invariant Markov semigroups on noncommutative spaces equipped with an action of a compact quantum group
We may note that expected right coidalgebras of G are examples of quantum homogeneous spaces, i.e. quantum spaces on which the corresponding right action of G is ergodic [P95]
Quantum Markov semigroups coming in this way from convolution semigroups of states are characterized by the invariance property ∆ ◦ the generator L of (Tt) = ◦ ∆), cf. [CFK14, Theorem 3.4]
Summary
Symmetry plays an essential role in many places in mathematics and in the natural sciences. In this paper we introduce an approach for classifying invariant Markov semigroups on noncommutative spaces equipped with an action of a compact quantum group The generators of these semigroups can be considered as natural candidates for Laplace operators. This is slightly more general than in the classical case, where all homogeneous spaces are of quotient type, but follows similar ideas. Conventions: We use ⊗ both for the tensor product of vector spaces and *algebras, and for the minimal tensor product of C∗-algebras, the meaning will be clear from the context
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