Nonseparable CCR algebras

Extending a result of the first author and Katsura, we prove that for every UHF algebra [Formula: see text] of infinite type, in every uncountable cardinality [Formula: see text] there are [Formula: see text] nonisomorphic approximately matricial C*-algebras with the same [Formula: see text] group as [Formula: see text]. These algebras are group C*-algebras “twisted” by prescribed canonical commutation relations (CCR), and they can also be considered as nonseparable generalizations of noncommutative tori.

Solitons of general topological charge over noncommutative tori

We study solitons of general topological charge over noncommutative tori from the perspective of time-frequency analysis. These solitons are associated with vector bundles of higher rank, expressed in terms of vector-valued Gabor frames. We apply the duality theory of Gabor analysis to show that Gaussians are such solitons for any value of a topological charge. Also they solve self/anti-self duality equations resulting from an energy functional for projections over noncommutative tori, and have a reformulation in terms of Gabor frames. As a consequence, the projections generated by Gaussians minimize the energy functional. We also comment on the case of the Moyal plane and the associated continuous vector-valued Gabor frames and show that Gaussians are the only class of solitons there.

Some remarks on $\mathrm{K}_0$ of noncommutative tori

Using Rieffel's construction of projective modules over higher dimensional noncommutative tori, we construct projective modules over some continuous field of C*-algebras whose fibres are noncommutative tori. Using a result of Echterhoff et al., our construction gives generators of $\mathrm {K}_0$ for all noncommutative tori.