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I am recently studying ergodic actions of Lie groups acting on Riemannian symmetric spaces. Since I am also interested in operator algebras, it makes me wonder if there are some very natural noncommutative analogues of Riemannian symmetric spaces and measure preserving ergodic group actions by classical Lie groups?

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You might be interested in the work of Lezter and others on quantum symmetric spaces. See for example this paper

https://arxiv.org/pdf/math/0406193.pdf

These objects seem to be closely connected to the representation theory of quantum groups. People have also considered $C^*$-completions, but I'm not sure about ergodic actions.

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