By A. Jaffe (Chief Editor)
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Extra resources for Communications in Mathematical Physics - Volume 201
Define x and y in A as x = a + b, y = a − b, with the inverse relations a = 21 (x + y), b = 21 (x − y). The invertibility of a + b and a − b is equivalent to the invertibility of x and y. Matrix multiplication expressed in these coordinates becomes (x1 , y1 )(x2 , y2 ) = (x1 x2 , y1 y2 ), where xi , yi ∈ GL1 (A), which means that the group G is isomorphic to the direct product group GL1 (A) × GL1 (A). Although not necessary here, we mention that using this parametrisation it is also easy to find the inverse of a matrix in G.
Lifting polarized modules. What happens if we lift a given polarized module P = (H, E, π) in two different ways, by choosing two different compatible involutions γ1 and γ2 in P ? The two choices give rise to two Fredholm modules αj = (Hj , π, γj , Fj ), j = 1, 2, where Hj is the Hilbert space equipped with the inner product (·, ·)j = σ(·, γj ·), and Fj is the involution that is +1 on E and −1 on γ(E). We are going to construct a map between these Fredholm modules in a few stages. First we note that the two Hilbert space inner products are related in the following way.
To check the last statement we simply note that (x, Qx)0 = (x, γ0 γ1 x)0 = σ(x, γ1 x) = (x, x)1 ≥ 0. We shall now use the Cayley transform, which establishes a 1-1 correspondence between bounded, positive, invertible operators Q and bounded self adjoint operators m of norm strictly smaller than 1. More precisely, for each operator Q there exists a unique operator m 1−m Q= 1+m with the inverse relation given by the formula m= Thus for a unique m, 1−Q . 1+Q Q = γ0 γ1 = and so 1−m , 1+m γ1 = (1 + m)γ0 (1 + m)−1 .
Communications in Mathematical Physics - Volume 201 by A. Jaffe (Chief Editor)