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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...
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.
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