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18 still hold if a and b are replaced by au = ua, bu = ub, with u ∈ Z(A) ∩ GL1 (A), and so they descend to the group GC ⊂ P GL2 (A). In the case when A is the C ∗ -algebra L of bounded operators in a Hilbert space H we can say a bit more. 18 imply that the same identities should hold for an arbitrary self-adjoint projection. 20) where λi are real numbers. Rewriting these formulae in terms of x and y we find x∗ y = λ1 − iλ2 , xy ∗ = λ3 − iλ4 , or, if we use that x and y are invertible, x = (λ3 + iλ4 )(y ∗ )−1 , y = (λ1 − iλ2 )−1 (x∗ )−1 .

36 P. J. M. Bongaarts, J. Brodzki one can reconstruct the original conformal structure [5, p. 332]. It thus transpires that Fredholm modules are very close to being noncommutative analogues of conformal structures on differentiable manifolds. This fact indicates that they play a more basic role than spectral triples that capture the essence of the Riemannian geometry of smooth manifolds. As is well known, conformal structure is fundamental in general relativity, where it carries information about causality.

Of course, we cannot hope that an arbitrary locally convex topological space could be used to gain useful information about geometric objects. There is, however, a category of spaces which are equipped with indefinite inner products, whose properties make them useful in geometric situations. These are Krein spaces. Let us recall that a Hilbertian space H is a locally convex topological vector space which is isomorphic to a Hilbert space. In other words, there exists an inner product on H which induces the same topology.