New PDF release: Foliations. II

By Candel A., Conlon L.

ISBN-10: 0821808095

ISBN-13: 9780821808092

ISBN-10: 0821832220

ISBN-13: 9780821832226

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Show that the saturation of a Borel transversal is a Borel set. 13. Let m be a current on the foliated space M . The following conditions are equivalent: (1) there exists a modular function δ : G → R+ such that m is of modulus δ; (2) the current m is quasi-invariant. Proof. It is evident that (1) implies (2). Let U = {Ui } be a locally finite regular cover of M by foliated charts Ui . 6. Quasi-invariant Currents 43 invariant current mi for the foliated space (Ui , F|Ui ). Let {φi } be a partition of unity subordinated to the cover U and let m be the current m = i φi mi .

Proof. The restriction of π to the subspace D ⊗ Cc (Z) can be expressed as the product of representations π1 of D and π2 of Cc (Z) on H. 2 guarantees. 10. The full C ∗ -algebra of a trivial foliated space N × Z is the tensor product K(L2 (N )) ⊗ C0 (Z). C. 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 .

It will be shown that, for each x ∈ M , the representation Rx ◦ i of Γc (G(U ), D1/2 ) is equivalent to a direct sum of representations Rxα , xα ∈ U , and the trivial representation. 5. The space L2 (Gx ) can be decomposed into a direct sum by means of a partition of Gx as follows. Since G(U ) is an open subgroupoid of G, then the relation “γ1 ∼ γ2 if and only if γ1 · γ2−1 ∈ G(U )” is an equivalence relation on r−1 (U ) ∩ Gx . Each equivalence class is open and connected (because G(U ) is open and each plaque of U is connected).

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Foliations. II by Candel A., Conlon L.


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Categories: Algebraic Geometry