Quantum real projective space, disc and spheres (Q1811027)

The \(C^*\)-algebra of quantum real projective space is defined, and then the topological \(K\)-groups are calculated for this algebra. Considering \(\mathbb Z_2\)-actions on the two-sphere one obtains a disc \(D^2\) and real projective space \(\mathbb R P^2\): One can obtain a disc \(D^2\) as the quotient of a sphere under the \(\mathbb Z_2\)-action given by the reflection with respect to the equator plane. Real projective space \(\mathbb R P^2\) can also be constructed from the antipodal action of \(\mathbb Z_2\) on the two-sphere. The quantum version for these is considered. First the Podleś quantum sphere by \textit{P. Podleś} [Lett. Math. Phys. 14, 193--202 (1987; Zbl 0634.46054)] is considered. By taking a \(\mathbb Z_2\)-action on the quantum sphere, corresponding to the reflection for classical case, it was shown that the quotient algebra is equal to the quantum disc by \textit{S. Klimek} and \textit{A. Lesniewski} [J. Funct. Anal. 115, 1--23 (1993; Zbl 0780.58020)]. In the same sense, quantum real projective space is defined as the quotient algebra of the quantum sphere by a \(\mathbb Z_2\)-action corresponding to the antipodal action. The paper under review is a kind of continuation of \textit{P. M. Hajac} [Commun. Math. Phys. 182, 579--617 (1996; Zbl 0873.58007)].