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

### Tomasz Brzeziński

Swansea University, UK and University of Białystok, Poland### Wojciech Szymański

University of Southern Denmark, Odense, Denmark

## Abstract

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.

## Cite this article

Tomasz Brzeziński, Wojciech Szymański, The $C^*$-algebras of quantum lens and weighted projective spaces. J. Noncommut. Geom. 12 (2018), no. 1, pp. 195–215

DOI 10.4171/JNCG/274