By Nomizu K., Sasaki T.

ISBN-10: 0521441773

ISBN-13: 9780521441773

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**Sample text**

This is the standard definition of K1 for a C ∗ -algebra A [189]; to define it thus for a pre-C ∗ -algebra A one needs some arguments from homotopy theory that require A to be also Fréchet [15]. (What we have defined here is a ‘topological’ K-theory: there is a somewhat different ‘algebraic’ K-theory, defined for wider classes of algebras. The K0 -groups in both theories coincide, but the K1 -groups generally do not. ) Both groups are homotopy invariant: if {pt : 0 ≤ t ≤ 1} is a homotopy of projectors in M∞ (A) and if {ut : 0 ≤ t ≤ 1} is a homotopy in U∞ (A), then [p0 ] = [p1 ] in K0 (A) and [u0 ] = [u1 ] in K1 (A).

Now, σξ restricts to the volume form on the unit sphere |ξ | = 1 in each Tx∗ M. On integrating ιR α over these spheres, we get a quantity that transforms under coordinate changes x → y = φ(x), ξ → η = φ (x)t ξ , a−n (x, ξ ) → a˜ −n (y, η) as follows [84], [104]: |η|=1 tr a˜ −n (y, η) ση = | det φ (x)| |ξ |=1 tr a−n (x, ξ ) σξ . 10) The absolute value of the Jacobian det φ (x) appears here because if φ (x)t reverses the orientation on the unit sphere in Tx∗ M then the integral over the sphere also changes sign.

3] that τ (a) = T2 a(φ1 , φ2 ) dφ1 dφ2 , so that τ is just the integral of the classical symbol. The GNS representation space H0 = L2 (Aθ , τ ) may be described as the completion of the vector space Aθ in the Hilbert norm a 2 := τ (a ∗ a). 36 4 Geometries on the noncommutative torus Since τ is faithful, the obvious map Aθ → H0 is injective; to keep the bookkeeping straight, we shall denote by a the image in H0 of a ∈ Aθ . The GNS representation of Aθ is just π0 (a) : b → ab. Notice that the vector 1 is obviously cyclic and separating, so the Tomita involution is given by J0 (a) := a ∗ .

### Affine differential geometry. Geometry of affine immersions by Nomizu K., Sasaki T.

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