We introduce a curvature-dimension condition CD ($K$,$N$) for metric measure spaces. It is more restrictive than the curvature bound $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} \right)} > K$ (introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD($K$, ∞). The additional real parameter$N$plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD($K$,$N$) is equivalent to $${\text{Ric}}_{M} {\left( {\xi ,\xi } \right)} > K{\left| \xi \right|}^{2} $$ and dim($M$) ⩽$N$.

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We give complete detail of the description of the GNS representation of the quantum plane A and its dual  as a von Neumann algebra. In particular, we obtain a rather surprising result that the multiplicative unitary W is manageable in this quantum semigroup context. We study the quantum double group construction introduced by Woronowicz, and using Baaj and Vaes' construction of the multiplicative unitary Wm, we give the GNS description of the quantum double D(A) which is equivalent to GLq+(2, ℝ). Furthermore, we study the fundamental corepresentation Tλ, t and its matrix coefficients, and show that it can be expressed by the b-hypergeometric function. We also study the regular corepresentation and representation induced by Wm and prove that the space of L2 functions on the quantum double decomposes into the continuous series representation of Uqq̃(gl(2, ℝ)) with the quantum dilogarithm {pipe}Sb(Q+2iα){pipe}2 as the Plancherel measure. Finally, we describe certain representation theoretic meaning of integral transforms involving the quantum dilogarithm function.

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