Novikov-Shubin invariants and asymptotic dimensions for open manifolds (original) (raw)
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Asimptotic dimension and Novikov-Shubin invariants for open manifolds
1996
A trace on the C *-algebra A of quasi-local operators on an open manifold is described, based on the results in [36]. It allows a descriptioǹ a la Novikov-Shubin [31] of the low frequency behavior of the Laplace-Beltrami operator. The 0-th Novikov-Shubin invariant defined in terms of such a trace is proved to coincide with a metric invariant, which we call asymptotic dimension, thus giving a large scale "Weyl asymptotics" relation. Moreover, in analogy with the Connes-Wodzicki result [7, 8, 45], the asymptotic dimension d measures the singular traceability (at 0) of the Laplace-Beltrami operator, namely we may construct a (type II1) singular trace which is finite on the *-bimodule over A generated by ∆ −d/2 .
Dimensions and singular traces for spectral triples, with applications to fractals
Journal of Functional Analysis, 2003
Given a spectral triple (A, H, D), the functionals on A of the form a → τω(a|D| −α ) are studied, where τω is a singular trace, and ω is a generalised limit. When τω is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the (d-dimensional) Hausdorff functional.
1998
Given a C*-algebra A with a semicontinuous semifinite trace { acting on the Hilbert space H, we define the family AR of bounded Riemann measurable elements w.r.t. { as a suitable closure, a la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions, and show that AR is a C*-algebra, and { extends to a semicontinuous semifinite trace on AR. Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A " and can be approximated in measure by operators in AR, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a {-a.e. bimodule on AR, denoted by AR, and such a bimodule contains the functional calculi of selfadjoint elements of AR under unbounded Riemann measurable functions. Besides, { extends to a bimodule trace on AR. Type II1 singular traces for C*-algebras can be defined on the bimodule of unbounded Riemann-measurable operators. Noncommutative...
A Semicontinuous Trace for Almost Local Operators on an Open Manifold
International Journal of Mathematics, 2001
A semicontinuous semifinite trace is constructed on the C*-algebra generated by the finite propagation operators acting on the L2-sections of a Hermitian vector bundle on an amenable open manifold of bounded geometry. This trace is the semicontinuous regularization of a functional already considered by J. Roe. As an application, we show that, by means of this semicontinuous trace, Novikov–Shubin numbers for amenable manifolds can be defined.
Spectral dimension of quantum geometries
Classical and Quantum Gravity, 2014
The spectral dimension is an indicator of geometry and topology of spacetime and a tool to compare the description of quantum geometry in various approaches to quantum gravity. This is possible because it can be defined not only on smooth geometries but also on discrete (e.g., simplicial) ones. In this paper, we consider the spectral dimension of quantum states of spatial geometry defined on combinatorial complexes endowed with additional algebraic data: the kinematical quantum states of loop quantum gravity (LQG). Preliminarily, the effects of topology and discreteness of classical discrete geometries are studied in a systematic manner. We look for states reproducing the spectral dimension of a classical space in the appropriate regime. We also test the hypothesis that in LQG, as in other approaches, there is a scale dependence of the spectral dimension, which runs from the topological dimension at large scales to a smaller one at short distances. While our results do not give any strong support to this hypothesis, we can however pinpoint when the topological dimension is reproduced by LQG quantum states. Overall, by exploring the interplay of combinatorial, topological and geometrical effects, and by considering various kinds of quantum states such as coherent states and their superpositions, we find that the spectral dimension of discrete quantum geometries is more sensitive to the underlying combinatorial structures than to the details of the additional data associated with them.
Asymptotic cones and Assouad-Nagata dimension
Proceedings of the American Mathematical Society, 2008
We prove that the dimension of any asymptotic cone over a metric space ( X , ρ ) (X,\rho ) does not exceed the asymptotic Assouad-Nagata dimension asdim A N ( X ) \operatorname {asdim}_{AN}(X) of X X . This improves a result of Dranishnikov and Smith (2007), who showed dim ( Y ) ≤ asdim A N ( X ) \dim (Y)\leq \operatorname {asdim}_{AN}(X) for all separable subsets Y Y of special asymptotic cones Cone ω ( X ) \operatorname {Cone}_\omega (X) , where ω \omega is an exponential ultrafilter on natural numbers. We also show that the Assouad-Nagata dimension of the discrete Heisenberg group equals its asymptotic dimension.
arXiv (Cornell University), 2011
We give local upper and lower bounds for the eigenvalues of the modular operator associated to an ergodic action of a compact quantum group on a unital C *-algebra. They involve the modular theory of the quantum group and the growth rate of quantum dimensions of its representations and they become sharp if other integral invariants grow subexponentially. For compact groups, this reduces to the finiteness theorem of Høegh-Krohn, Landstad and Størmer. Consequently, compact quantum groups of Kac type admitting an ergodic action with a non-tracial invariant state must have representations whose dimensions grow exponentially. In particular, S−1U (d) acts ergodically only on tracial C *-algebras. For quantum groups with noninvolutive coinverse, we derive a lower bound for the parameters 0 < λ < 1 of factors of type III λ that can possibly arise from the GNS representation of the invariant state of an ergodic action with a factorial centralizer.