Ergodic actions and spectral triples (Q2835263)

A general construction of spectral triple (or unbounded Fredholm module, cf.\ Def.~2.1) \((\pi,\mathcal H,D)\) from an ergodic action of a group \(G\) on \(A\), a \(C^\ast\)-algebra, is given (Th.~1.1, Th.~5.5). As examples, noncommutative tori and quantum Heisenberg manifolds are investigated (\S8, Example 8.1, 8.3). \(G\) is assumed to be an \(n\)-dimensional compact Lie group acting on \(A\) via the strongly continuous action \(\alpha\). Its Lie algebra \(\mathfrak g\) is given an Ad-invariant positive-definite scalar product \(\mu\). It gives a Clifford structure on the underlying vector space \(\mathfrak g\) (Assumption 2.3). From this Clifford structure, the Dirac operator \(D\) is introduced by NEWLINE\[NEWLINED=\sum_{j=1}^n\partial_j\otimes F_j.NEWLINE\]NEWLINE Here, the \(F_i\)'s determine the representation space \(\mathcal S\) (Prop.~2.8; expositions on related topics such as \(n^\pm\) are also given in \S2). \(\alpha\) is called ergodic if the only \(G\)-invariant elements are scalars (Def.~5.1). If \(G\) is compact and the action is ergodic, the unique \(G\)-invariant state is a trace \(\tau\) [\textit{R. Høegh-Krohn} et al., Ann. Math. (2) 114, 75--86 (1981; Zbl 0472.46046)]. Using \(\tau\), we set \(\mathcal H_0=\text{GNS}(A,\tau)\) (the GNS construction is explained in \S3). Then, using ergodicity and an estimate in [\textit{R. Høegh-Krohn} et al., loc. cit.], \(D\) is shown to be essentially selfadjoint and its closure \(\bar{D}\) defines an \(n^+\)-summable spectral triple \((A, \mathcal H_0\otimes S, \bar{D})\) on \(A\) (Th.~5.5). Th.~5.5 also contains estimates of \(D\). They provide formulas on the Dixmier trace \(\text{Tr}_\omega\) (cf. Chap.~IV in [\textit{A. Connes}, Noncommutative Geometry. San Diego, CA: Academic Press (1994; Zbl 0818.46076)]. For the definitions of related function spaces such as \(\mathcal L^{1+}\), cf.\ Def.~5.3. One has:NEWLINENEWLINE1. If \((1+D^2)^{-n_0/2}\in\mathcal L^{1+}\), then there is a scalar \(\lambda\) such that, for every \(a\in A\), NEWLINE\[NEWLINE\text{Tr}_\omega(a(1+D^2)^{-n_0/2})=\lambda \tau(a).NEWLINE\]NEWLINENEWLINENEWLINE2. \(\text{Tr}_\omega(T(1+D^2)^{-n_0/2})\) extends to a trace on \(A\otimes B(s)\) and NEWLINE\[NEWLINE\text{Tr}_\omega(T(1+D^2)^{-n_0/2})=\lambda(\tau\otimes \text{Tr}_S)(T),NEWLINE\]NEWLINE where \(\text{Tr}_S\) on \(B(S)\) satisfies \(\text{Tr}_S(1)=1\) (Prop.~6.1, Th.~6.2). After explaining applications of this construction to the 2-sphere with \(SU(2)\)-action in \S7, the last section states several applications of this construction such as noncommutative tori (Example 8.1), quantum Heisenberg manifolds (Example 8.3, cf. [\textit{M. A. Rieffel}, Commun. Math. Phys. 122, No.~4, 531--562 (1989; Zbl 0679.46055)], Kasparov's Dirac element (Example 8.4, cf. [\textit{G. G. Kasparov}, London Math. Soc. Lect. Note 226, 101--146 (1995; Zbl 0957.58020)] and examples obtained from the Cuntz algebra (Example 8.6 and 8.7). The possibility to extend this construction to non-compact group actions is also discussed (Example 8.5).