Mathematical Sciences: K-Theory for Operator Algebras, Index Theory, Riemann-Roch

9401440 Baum the BC(=Baum-Connes)猜想是一个非常一般的等式，它涉及拓扑学，即连续几何和分析，即建立在多元微积分基础上的数学分支。这个猜想的不同寻常之处在于，它跨越了数学的几个不同领域，从而揭示了一个以前完全未知的潜在统一性。这里所进行的研究集中在对某些类别的例子证明这一猜想，然后得出推论。更确切地说，设G是Hausdorff、局部紧和第二可数的拓扑群。主要的例子有李群、p-进群和离散群。C*G表示G的约化C*-代数，KC*G是它的K-理论。BC猜想，如果为真，则回答了计算Kc*G的问题。该猜想的有效性适用于几何拓扑学和表示论中的著名问题。***

9401440 Baum The BC (= Baum-Connes) conjecture is an extremely general equality relating topology, i.e. continuous geometry, and analysis, i.e. the branch of mathematics based on multi-variable calculus. This conjecture is unusual in that it cuts across several different areas of mathematics and thus reveals an underlying unity which previously was completely unknown. The research pursued here centers on proving this conjecture for certain classes of examples and then obtaining corollaries. More precisely, let G be a topological group which is Hausdorff, locally compact and second countable. The main examples are Lie groups, p-adic groups and discrete groups. C*G denotes the reduced C*-algebra of G, and KC*G is its K-theory. The BC conjecture, if true, gives an answer to the problem of calculating KC*G. Validity of the conjecture has applications to well-known problems in geometry-topology and representation theory. ***