# Journal of Noncommutative Geometry

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**Volume 12, Issue 1, 2018, pp. 195–215**

**DOI: 10.4171/JNCG/274**

Published online: 2018-03-23

The $C^*$-algebras of quantum lens and weighted projective spaces

Tomasz Brzeziński^{[1]}and Wojciech Szymański

^{[2]}(1) Swansea University, UK and University of Białystok, Poland

(2) University of Southern Denmark, Odense, Denmark

It is shown that the algebra of continuous functions on the quantum $2n+1$-dimensional lens space $C(L^{2n+1}_q(N; m_0,\ldots, m_n))$ is a graph $C^*$-algebra, for arbitrary positive weights $m_0,\ldots, m_n$. The form of the corresponding graph is determined from the skew product of the graph which defines the algebra of continuous functions on the quantum sphere $S_q^{2n+1}$ and the cyclic group $\mathbb Z_N$, with the labelling induced by the weights. Based on this description, the $K$-groups of specific examples are computed. Furthermore, the $K$-groups of the algebras of continuous functions on quantum weighted projective spaces $C(\mathbb W\mathbb P_q^n(m_0,\ldots, m_n))$, interpreted as fixed points under the circle action on $C(S_q^{2n+1})$, are computed under a mild assumption on the weights.

*Keywords: *Quantum lens space, quantum weighted projective space, graph $C^*$-algebra

Brzeziński Tomasz, Szymański Wojciech: The $C^*$-algebras of quantum lens and weighted projective spaces. *J. Noncommut. Geom.* 12 (2018), 195-215. doi: 10.4171/JNCG/274