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Noncommutative geometry and reality. (English) Zbl 0871.58008

The author adds the final touches to the picture he has drawn of a noncommutative geometry (see the front cover of his well-known book ‘Noncommutative Geometry’, Acad. Press (1994; Zbl 0818.46076)). In a recent paper [Commun. Math. Phys. 182, No. 1, 155–176 (1996; Zbl 0881.58009)] he explicitly states seven axioms that characterize his concept of a (noncommutative) geometric space. Here he introduces the real structure that is needed to establish Poincaré duality and that helps to improve his noncommutative model of particle physics by removing some artificial ingredients. At first, he recalls (from the afore-mentioned monograph and from Lett. Math. Phys. 34, No. 3, 203–238 (1995; Zbl 1042.46515)) the basic tools of a quantized calculus, a spectral triple \(({\mathcal A},{\mathcal H},{\mathcal D})\) consisting of an involutive complex algebra represented on a (graded) Hilbert space \(H\) and a self-adjoint operator \(D\) (of degree 1). The latter is supposed to have compact resolvents \((D-\lambda)^{-1}\), i.e. a discrete spectrum of finite multiplicity – for nonunital \(\mathcal A\) one assumes \(a(D-\lambda)^{-1}\) to be compact for any \(a\in{\mathcal A}\). Moreover, for any \(a\in{\mathcal A}\) the commutator \([D,a]\) is a bounded operator.
There are two examples which prominently feature in the Connes-Lott model of particle physics: (1) the Dirac triple \(({\mathcal A},L^2(M,S),D)\) describing a continuous space, and (2) the discrete two-point space \((\mathbb C(\{a,b\}),\mathbb C^2,D)\). In the first example \(M\) is a Riemannian manifold allowing a spinor bundle \(S\) with corresponding spin Dirac operator \(D\), and \(\mathcal A\) is the algebra of Lipschitz functions acting on \(L^2\)-sections of \(S\) by multiplication operators \(M_f\). From the spectral triple one recovers the geodesic distance of \(M: d(p,q)=\sup\{|f(p)-f(q)|\mid |[D,M_f]|\leq 1\}\). The same formula applies in the second example and yields the distance \(d(a,b)=1/\mu\) with \(\mu>0\) the entry of the off diagonal \(2\times 2\)-matrix \(D\). Moreover, the dimension of \(M\) is determined by the rate of growth of the eigenvalues of \(D\).
In the commutative case, Poincaré duality up to coefficients in \(\mathbb Z[\frac12]\) rests on an orientation class in real \(K\)-theory (equivariant in the non-simply connected case) which contains all the invariants of the manifold in the given homotopy type. In the noncommutative setting, this is provided by a KR-homology class for \({\mathcal A}\otimes{\mathcal A}^\circ\) (\({\mathcal A}^\circ\) the opposite algebra with multiplication reversed). To this end one turns \(H\) into an \({\mathcal A}\)-bimodule using an antilinear isometry \(J\) on \(H\) which, depending on the dimension, is a symmetric or a symplectic form, commutes with \(D\) and up to sign with the grading. A further condition is that \(a\in {\mathcal A}\) and \([D,a]\) commute with \(J{\mathcal A}J^*\) as in Tomita’s theorem, which not only serves as a motivation but provides appropriate \(J\)’s in special cases.
A second new element that comes in in this paper is a new interpretation of infinitesimals (on the quantum level). These are not as often suggested introduced by nonstandard analysis but as compact operators. The size of such an infinitesimal \(T\) is governed by the rate of decay of its singular values \(\mu_n(T)\to 0\). In particular, they include the differentials \(da\) of elements \(a\in {\mathcal A}\) which are defined as \(da=[F,a]\) with \(F=D|D|^{-1}\) (neglecting \(\text{Ker }D\)). A special example is the infinitesimal length element \(ds=|D|^{-1}\) which can be used to define integration with respect to a quantized volume element via the Dixmier trace.

MSC:

46L85 Noncommutative topology
46L87 Noncommutative differential geometry
58B34 Noncommutative geometry (à la Connes)
19K35 Kasparov theory (\(KK\)-theory)
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