The Podleś quantum sphere S 2 q admits a natural commutative C * -subalgebra Iq with spectrum {0} ∪ {q 2k : k ∈ N0}, which may therefore be considered as a quantized version of a classical interval. We study here the compact quantum metric space structure on Iq inherited from the corresponding structure on S 2 q , and provide an explicit formula for the metric induced on the spectrum. Moreover, we show that the resulting metric spaces vary continuously in the deformation parameter q with respect to the Gromov-Hausdorff distance, and that they converge to a classical interval of length π as q tends to 1.

We investigate quantum lens spaces, C(L 2n+1 q (r; m)), introduced by Brzeziński-Szymański as graph C * -algebras. For n ≤ 3, we give a number-theoretic invariant, when all but one weight are coprime to the order of the acting group r. This builds upon the work of Eilers, Restorff, Ruiz and Sørensen.

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We investigate quantum lens spaces, ( 2 +1 ( ; )), introduced by Brzeziński-Szymański as graph * -algebras. We give a new description of ( 2 +1 ( ; )) as graph * -algebras amending an error in the original paper by Brzeziński-Szymański. Furthermore, for ≤ 3, we give a number-theoretic invariant, when all but one weight are coprime to the order of the acting group . This builds upon the work of Eilers, Restorff, Ruiz and Sørensen.

We construct a general cohomological induction isomorphism from a uniform measure equivalence of locally compact, second countable, unimodular groups which, as a special case, implies that the graded cohomology algebras of quasi-isometric, connected, simply connected nilpotent Lie groups are isomorphic. This unifies results of Shalom and Sauer and also provides new insight into the quasi-isometry classification problem for low-dimensional nilpotent Lie groups.