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Fredholm modules. Probably the most interesting feature of the M¨obius group is the fact that it acts on the space of Fredholm modules. This action, as we shall see later, has interesting geometric consequences. But first, let us quickly review the standard notions. 1. Let A be an involutive algebra over C. An even Fredholm module over A is given by the data (H, π, γ, F ). Here H is a Z2 -graded Hilbert space with the grading given by a self-adjoint involution γ; π is a faithful ∗-representation of A in H commuting with γ, and F is a self-adjoint involution on H which anticommutes with γ.
In other words, g= ab ba = (1 − m2 )− 2 −m(1 − m2 )− 2 1 1 −m(1 − m2 )− 2 (1 − m2 )− 2 1 u0 0u 1 . 29) In this u = a(a∗ a)− 2 and m = −a−1 b. Similarly g can be written as 1 (1 − m2 )− 2 −m(1 − m2 )− 2 1 1 −m(1 − m2 )− 2 (1 − m2 )− 2 1 g= 1 u0 0u , where now u = (aa∗ )− 2 a and m = −ba−1 . 1 Proof. 26. This result was first proved by Connes [5, Prop. 5, p. 335]. Let us denote by U the group of unitary elements in GL1 (A). 30. The unitary group U is a deformation retract of the group µ(A). Proof.
There will be two special cases that roughly determine the scope of interesting choices of an algebra. One is the essential commutant of the representation π, and the other is the commutant of π. The first case is of interest from the point of view of K-homology, whereas the second is better suited to geometric applications. One of the tools that we shall use in further study of the M¨obius group is the Cayley transform, and so we begin by recalling its basic properties. 25. Let H be a (complex) Hilbert space.