{"product_id":"introduction-to-the-baum-connes-conjecture-9783764367060","title":"Introduction to the Baum-Connes Conjecture","description":"\u003cp\u003e • Author(s): Alain Valette\u003cbr\u003e • Publisher: Springer\u003cbr\u003e • Publisher Imprint: Birkhauser\u003cbr\u003e • BISAC: Geometry - Differential\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eA quick description of the conjecture The Baum-Connes conjecture is part of Alain Connes'tantalizing \"noncommuta- tive geometry\" programme [18]. It is in some sense the most \"commutative\" part of this programme, since it bridges with classical geometry and topology. Let r be a countable group. The Baum-Connes conjecture identifies two objects associated with r, one analytical and one geometrical\/topological. The right-hand side of the conjecture, or analytical side, involves the K- theory of the reduced C*-algebra c;r, which is the C*-algebra generated by r in 2 its left regular representation on the Hilbert space C(r). The K-theory used here, Ki(C;r) for i = 0, 1, is the usual topological K-theory for Banach algebras, as described e.g. in [85]. The left-hand side of the conjecture, or geometrical\/topological side RKf(Er) (i=O, I), is the r-equivariant K-homology with r-compact supports of the classifying space Er for proper actions of r. If r is torsion-free, this is the same as the K-homology (with compact supports) of the classifying space Br (or K(r, l) Eilenberg-Mac Lane space). This can be defined purely homotopically.\u003c\/p\u003e","brand":"Springer","offers":[{"title":"Paperback","offer_id":45280503201943,"sku":"9783764367060","price":3639.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9783764367060.webp?v=1769296505","url":"https:\/\/atlanticbooks.com\/products\/introduction-to-the-baum-connes-conjecture-9783764367060","provider":"Atlantic Books","version":"1.0","type":"link"}