By Candel A., Conlon L.

ISBN-10: 0821808095

ISBN-13: 9780821808092

ISBN-10: 0821832220

ISBN-13: 9780821832226

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**Extra info for Foliations. II**

**Example text**

Let D1 G be the bundle of densities of order one on G. This bundle splits as D1 G = s∗ D1 M ⊗ r∗ D1 M, where D1 M is the bundle of densities of order one on M . Thus the ﬁber at γ ∈ G is the tensor product D1 Ms(γ) ⊗ D1 Mr(γ) . Compactly supported densities are elements of Γc (G, D1 ). 1. A current on the foliated space (M, F) is a positive linear functional m on the space Γc (M, D1 ) of compactly supported densities on M. A current induces a (positive) Radon measure on M , and conversely. Indeed, let σ be an everywhere positive density on M and deﬁne a positive linear functional I on Cc∞ (M ) by I(f ) = f (x)σ(x) · m(x).

5. The Basic Examples 27 pseudonorm f = sup π(f ) , π where π runs through all the involutive representations of Γc (G, D1/2 ) on a separable Hilbert space whose restrictions to the graph G(U ) of each foliated chart U for (M, F) are weakly continuous for the inductive limit topology on Γc (G(U ), D1/2 ). Because each Rx , x ∈ M , is an involutive representation of the convolution algebra of G, and is continuous as the deﬁnition requires, there is an obvious surjection C ∗ (M, F) → Cr∗ (M, F); in general, as with groups, it is not an isomorphism.

Fibrations. In this example (M, F) is a foliated space whose leaves are the ﬁbers of a locally trivial ﬁbration p : M → B. Thus B has a covering by open sets {Bi } so that p−1 (Bi ) ∼ = L × Bi . The C ∗ -algebra of M is built by assembling the C ∗ -algebras of the trivial foliated spaces L × Bi . 11. Let X be a manifold and let Diﬀ(X) be the group of diffeomorphisms, with the topology of uniform convergence on compact subsets. 2, is continuous for the strong operator topology. Proof. A net {Ui } of unitary operators on L2 (X) converges to U in the strong operator topology if the operator norm Ui (ξ) − U (ξ) → 0, for every ξ ∈ L2 (X).

### Foliations. II by Candel A., Conlon L.

by Anthony

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