Some results on non-commutative Banach function spaces.
Symplectic models of groups with noncommutative spaces
The closed Friedman world model with the initial and final singularities as a non-commutative space
The most elegant definition of singularities in general relativity as b-boundary points, when applied to the closed Friedman world model, leads to the disastrous situation: both the initial and final singularities form the single point of the b-boundary which is not Hausdorff separated from the rest of space-time. We apply Alain Connes' method of non-commutative geometry, defined in terms of a C*-algebra, to this case. It turns out that both the initial and final singularities can be analysed as...
The cyclic homology of
The de Rham theorem for the noncommutative complex of Cenkl and Porter.
The Frölicher-Nijenhuis bracket in non-commutative differential geometry.
The noncommutative Ward metric.
The Novikov conjecture for linear groups
Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(n,K) is exact, in the sense of C*-algebra theory.
The structure of higher-dimensional noncommutative tori and metric diophantine approximation.
The topological snake lemma and corona algebras.
Traces on the cone algebra with asymptotics
When is a quantum space not a group?
We give a survey of techniques from quantum group theory which can be used to show that some quantum spaces (objects of the category dual to the category of C*-algebras) do not admit any quantum group structure. We also provide a number of examples which include some very well known quantum spaces. Our tools include several purely quantum group theoretical results as well as study of existence of characters and traces on C*-algebras describing the considered quantum spaces as well as properties...
Ковариантная некоммутативная дифференциальная геометрия