  Projecting a quantum theory onto the Hilbert subspace of states with energies
below a cutoff $\overline{E}$ may lead to an effective theory with modified
observables, including a noncommutative space(time). Adding a confining
potential well $V$ with a very sharp minimum on a submanifold $N$ of the
original space(time) $M$ may induce a dimensional reduction to a noncommutative
quantum theory on $N$. Here in particular we briefly report on our application
of this procedure to spheres $S^d\subset\mathbb{R}^D$ of radius $r=1$
($D=d\!+\!1>1$): making $\overline{E}$ and the depth of the well depend on (and
diverge with) $\Lambda\in\mathbb{N}$ we obtain new fuzzy spheres
$S^d_{\Lambda}$ covariant under the {\it full} orthogonal groups $O(D)$; the
commutators of the coordinates depend only on the angular momentum, as in
Snyder noncommutative spaces. Focusing on $d=1,2$, we also discuss uncertainty
relations, localization of states, diagonalization of the space coordinates and
construction of coherent states. As $\Lambda\to\infty$ the Hilbert space
dimension diverges, $S^d_{\Lambda}\to S^d$, and we recover ordinary quantum
mechanics on $S^d$. These models might be suggestive for effective models in
quantum field theory, quantum gravity or condensed matter physics.
