Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Bất biến đồng luân của các chỉ số phân tích của các nhà khai thác chữ ký trên C *- đại số. | Copyright by INCREST 1985 J. OPERATOR THEORY 14 1985 . 113- 127 HOMOTOPY INVARIANCE OF THE ANALYTIC INDEX OF SIGNATURE OPERATORS OVER C -ALGEBRAS JEROME KAM1NKER and JOHN G. MILLER INTRODUCTION A proof of the Novikov conjecture for finitely generated free abelian groups was given by Lusztig in 6 using the Atiyah-Singer Index Theorem for families of operators. It was observed by Miscenko 11 that a family of elliptic operators parametrized by a compact space X can be viewed as a single operator over the c -algebra C X . In 6 the space X was the M-dimensional torus T and the operators were signature operators. Since C T C Z Miscenko was led to consider signature operators over the algebras c r where r is a countable discrete group. In this paper we generalize some of the results in 6 to the non-commutative case. The methods used are based on 6 and 14 and on the theory of Hilbert modules as developed by sevểral people . 10 8 . This latter theory allows great simplification of previous work on these questions. Our goal was to prove the homotopy invariance of the signature operator considered as an element of K-homology on a closed manifold. This result is expressed in the form of a commuting diagram which in current terminology relates the Novikov Conjecture to the Strong Novikov Conjecture 13 Our main result Theorem corresponds roughly to Theorem 3 p. 24 Part 2 of Kasparov s Conspectus 7 . In Section 6 we indicate the changes necessary to handle algebras without unit. This has applications to homotopy invariance properties of the signature operator along the leaves of a foliation of a compact manifold as suggested to us by Paul Baum and Alain Connes. We briefly sketch how a Theorem of theirs also follows from our results. 1. PRELIMINARIES ON HILBERT MODULES Definition . LetX be a c -algebra with or without unit. A Hilbert X-module is a complex vector space M which is a right X-module provided with an X-valued 8 -1086 114 JEROME KAMINKER and JOHN G. MILLER .
