JournalsjncgVol. 15, No. 2pp. 531–580

Invariant Markov semigroups on quantum homogeneous spaces

  • Biswarup Das

    Wroclaw University, Poland
  • Uwe Franz

    Université de Bourgogne Franche-Comté, Besançon, France
  • Xumin Wang

    Université de Bourgogne Franche-Comté, Besançon, France
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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.

The classical sphere {SN1S^{N-1}}, the free sphere {S+N1S^{N-1}_+}, and the half-liberated sphere {SN1S^{N-1}_*} are considered as examples and the generators of Markov semigroups on these spheres are classified. We compute spectral dimensions for the three families of spheres based on the asymptotic behaviour of the eigenvalues of their Laplace operator.

Cite this article

Biswarup Das, Uwe Franz, Xumin Wang, Invariant Markov semigroups on quantum homogeneous spaces. J. Noncommut. Geom. 15 (2021), no. 2, pp. 531–580

DOI 10.4171/JNCG/404