Equivariant bivariant cyclic theory and equivariant Chern-Connes character (Q1880811)

Equivariant bivariant cyclic theory and the Chern-Connes character in the presence of a compact Lie group \(G\) are defined. The author notes that in [\textit{S. Klimek}, \textit{W. Kondracki} and \textit{A. Lesniewski}, \(K\)-theory 4, 201--218 (1991; Zbl 0744.46063); \textit{S. Klimek} and \textit{A. Lesniewski}, \(K\)-theory 4, 219--226 (1991; Zbl 0744.46064)] (hereafter refered to as [1]), the equivariant bivariant Chern-Connes character is defined if \(G\) is a finite group. This paper is a generalization of [1] to the case where \(G\) is a compact Lie group. The outline of the paper is as follows: In \S1, the equivariant bivariant cyclic homology is defined as a combination of equivariant (normalized) entire cyclic homology and noncommutative de Rham theory for unital \(G\)-Banach algebras \(U\) and \({\mathfrak B}\) [\textit{M. Karoubi}, Homologie cyclique et \(K\)-théorie, Astérisque 149 (1987, Zbl 0648.18008), hereafter refered to as [2]], where \(G\) is a compact Lie group. In \S2, first the \(G\)-equivariant \(\theta\)-summable Fredholm module \((A,{\mathcal H}, D)\) over a unital \(C^*\) algebra \(A\) is defined according to [1]. Then a flat \(G\)-invariant \({\mathcal B}\) connection \(\nabla\) and the operator \({\mathcal D}= D\otimes I+ A\), where \([D,A]\) is a bounded linear operator, is introduced. Here, \(I\) is the identity on \({\mathfrak B}\). \({\mathbf A}=\nabla+{\mathcal D}\) is a superconnection on \({\mathcal M}={\mathcal H}\otimes{\mathfrak B}\). The equivariant bivariant Chern-Connes character of the module \(({\mathcal M}, {\mathcal D})\) is defined by \[ Ch^n_G({\mathcal M},{\mathcal D})(u_0,\dots, u_n)(g)= \langle\langle u_0,[{\mathbf A}, u_1],\dots, [{\mathbf A}, u_1],\dots, [{\mathbf A}, u_n]\rangle\rangle(g)\in \Omega_*({\mathfrak B}), \] where \(\Omega_*(B)\) is the graded differential algebra appeared in the definition of the noncommutative de Rham homology (cf.[2]). Properties of the equivariant bivariant Chern-Connes character is studied by using the heat operator \(e^{-t{\mathbf A}^2}\), which is defined by Duhamel expansion, and the \(\theta\)-summability of \(D\). Then the equivariant bivariant Chern-Connes character is shown to be closed and define a homology class in \(HE^G_{ev} (U,\Omega({\mathfrak B}))\) (Theorem 2.6). It is remarked that if \({\mathfrak B}=\mathbb{C}\) and \(G\) is a finite group, then this definition reduces to the defintion in [1] and if \(G= \{e\}\), it reduces to the definition in [\textit{F. Wu}, \(K\)-Theory 11, 35--82 (1997; Zbl 0880.58025)].