Abstract
| One of the most beautiful results in operator theory are the so-called Cwikel
estimates concerning the singular values of the operator $M_f g(-i\nabla)$ on
$L_2(R^d)$,
where $M_f$ denotes the multiplication operator by the function $f$ and
$\nabla$
denotes the gradient. Several analogues of Cwikel estimates have been proved
for different spectral triples in noncommutative geometry, where they represent
an important tool in identification of the dimension of a locally compact
spectral triple.
In our joint work with F. Sukochev (UNSW) and D. Zanin (UNSW) we propose a new
approach in the definition of noncommutative plane (also known as the Moyal
plane). Using this new definition
we significantly improve Cwikel estimates for noncommutative plane previously
proved by Gayral, Iochum, and V\'arilly. |