An affine multiparameter quantum space \(A\) over an algebraically closed field \(k\) of characteristic zero is generated by elements \(x_1,\dots,x_n\) with defining relations \(x_ix_j=q_{ij}x_jx_i\) where \(q_{ij}\in k^*\). It is assumed that the multiplicative subgroup \(G\) in \(k^*\) generated by all \(q_{ij}\) is torsion-free. Let \(R=k[z_1,\dots,z_n]\) be the ordinary polynomial algebra. There is defined a Poisson bracket \(\{z_i,z_j\}=q_{ij}'z_iz_j\) in \(R\) for some \(q_{ij}'\in G\).
The main result of the paper establishes a homeomorphism between the Poisson-prime spectrum of \(R\) and the prime spectrum of \(A\). This homeomorphism maps Poisson-primitive ideals of \(R\) onto primitive ideals of \(A\).