Quantum extensions of classical Jacobi semigroups

We construct a quantum extension of the classical Jacobi semigroup to the non-commutative von Neumann algebra \({\mathcal {B}}( L^2((-1,1),(1-x)^\alpha (1+x)^\beta dx)\) $\mathcal{B}(L^2((-1,1),{(1-x)}^{\alpha }{(1+x)}^{\beta }dx)$ for certain values of \( \alpha , \beta \) $\alpha ,\beta$ and study some of its properties.