## A C∗-algebra of geometric operators on self-similar CW-complexes. Novikov–Shubin and L²-Betti numbers (2006)

Citations: | 2 - 2 self |

### BibTeX

@MISC{Cipriani06ac∗-algebra,

author = {Fabio Cipriani and Daniele Guido and Tommaso Isola},

title = {A C∗-algebra of geometric operators on self-similar CW-complexes. Novikov–Shubin and L²-Betti numbers },

year = {2006}

}

### OpenURL

### Abstract

A class of CW-complexes, called self-similar complexes, is introduced, together with C∗-algebras Aj of operators, endowed with a finite trace, acting on square-summable cellular j-chains. Since the Laplacian ∆j belongs to Aj, L²-Betti numbers and Novikov-Shubin numbers are defined for such complexes in terms of the trace. In particular a relation involving the Euler-Poincaré characteristic is proved. L²-Betti and Novikov-Shubin numbers are computed for some self-similar complexes arising from self-similar fractals.

### Citations

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Citation Context ... ) 1 ) = τ0 . 1 + t∆ The result follows. In order to prove the main result of this section, we need a Tauberian theorem. It is a quite simple modification of a theorem of de Haan and Stadtmüller, cf. =-=[5]-=- thm. 2.10.2, and, on the same book, also thm. 1.7.6, by Karamata, showing that the bound α < 1 below is a natural one. Definition 7.2. (i) Let us denote by OR(1) the space of positive, non increasing... |

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Citation Context ...plexes and basic operators. In this paper we shall consider a particular class of infinite CW-complexes, therefore we start by recalling some notions from algebraic topology, general references being =-=[26, 28]-=-. A CW-complex M of dimension p ∈ N is a Hausdorff space consisting of a disjoint union of (open) cells of dimension j ∈ {0, 1, . . ., p} such that: (i) for each j-cell σj α, there is a continuous map... |

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Citation Context ...ocally commuting with the transformations giving the self-similar structure, on which a Roe-type trace is well defined. The theory of L 2 -invariants was started by Atiyah, who, in a celebrated paper =-=[1]-=-, observed that on covering manifolds Γ → M → X, a trace on Γ-periodic operators may be defined, called Γ-trace, with respect to which the Laplace operator has compact resolvent. Replacing the usual t... |

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Citation Context ...plexes and basic operators. In this paper we shall consider a particular class of infinite CW-complexes, therefore we start by recalling some notions from algebraic topology, general references being =-=[26, 28]-=-. A CW-complex M of dimension p ∈ N is a Hausdorff space consisting of a disjoint union of (open) cells of dimension j ∈ {0, 1, . . ., p} such that: (i) for each j-cell σj α, there is a continuous map... |

41 |
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Citation Context ...PRIANI, DANIELE GUIDO, TOMMASO ISOLA L 2 -Betti numbers were proved to be Γ-homotopy invariants by Dodziuk [7], whereas Novikov-Shubin numbers were proved to be Γ-homotopy invariants by Gromov-Shubin =-=[12]-=-. L 2 -Betti numbers (depending on a generalised limit procedure) were subsequently defined for open manifolds by Roe, and were proved to be invariant under quasi-isometries [30]. The invariance of No... |

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De Rham-hodge theory for L 2 -cohomology of infinite coverings
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Citation Context ... and “Quantum Probability and Applications to Physics, Information and Biology”. 12 FABIO CIPRIANI, DANIELE GUIDO, TOMMASO ISOLA L 2 -Betti numbers were proved to be Γ-homotopy invariants by Dodziuk =-=[7]-=-, whereas Novikov-Shubin numbers were proved to be Γ-homotopy invariants by Gromov-Shubin [12]. L 2 -Betti numbers (depending on a generalised limit procedure) were subsequently defined for open manif... |

33 |
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Citation Context ... of some L 2 -invariants, like the L 2 -Betti numbers and Novikov-Shubin numbers, to geometric structures which are not coverings of compact spaces. The first attempt in this sense is due to John Roe =-=[29]-=-, who defined a trace on finite propagation operators on amenable manifolds, allowing the definition of L 2 -Betti numbers on these spaces. However such trace was defined in terms of a suitable genera... |

29 | Telcs A., Harnack inequalities and sub-Gaussian estimates for random walks
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Citation Context ...ch contains the Laplace operator ∆j of the graph Gj, and possesses a finite trace τj. Assume G1 satisfies (7.6), then G2 does as well. As a consequence, α0(G1) = α0(G2). Proof. It is a consequence of =-=[11]-=- Theorem 3.1, [4] Lemma 1.1, and [20] Theorem 5.11. □ 8. Examples In this section, we compute the Novikov-Shubin numbers of some explicit examples. Our first class of examples is that of nested fracta... |

25 | Approximating L 2 -invariants of amenable covering spaces: A combinatorial approach - Dodziuk, Mathai - 1996 |

24 | Which values of the volume growth and escape time exponent are possible for a graph?, Rev
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Citation Context ...place operator ∆j of the graph Gj, and possesses a finite trace τj. Assume G1 satisfies (7.6), then G2 does as well. As a consequence, α0(G1) = α0(G2). Proof. It is a consequence of [11] Theorem 3.1, =-=[4]-=- Lemma 1.1, and [20] Theorem 5.11. □ 8. Examples In this section, we compute the Novikov-Shubin numbers of some explicit examples. Our first class of examples is that of nested fractal graphs, for mor... |

21 |
l 2 -topological invariants of 3-manifolds
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Citation Context ...corresponding fractals, thus strenghtening the interpretation of such numbers as (asymptotic) spectral dimensions given in [13]. Our framework was strongly influenced by the approach of Lott and Lück =-=[25]-=-, in particular we also consider invariants relative to the boundary, however we are not able to prove the Poincaré duality shown in [25]. The paper is organised as follows. In Section 2 we recall som... |

20 |
Dimensions and singular traces for spectral triples, with applications to fractals
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Citation Context ...ly, one proves σj0 ∈ F(Ej0(γ n k Kn)). Therefore, γn k Kn ∩ γn ℓ Kn = F(γn k Kn) ∩ F(γn ℓ Kn). □ Remark 6.9. The construction above can be easily generalised to the case of translation limit fractals =-=[14, 15, 16, 17, 18]-=-. We now study some properties of polyhedral complexes, which are valid in particular for the prefractal complexes. Definition 6.10. We say that ∆j± is a graph-like Laplacian if there exists a suitabl... |

19 | Random walks on graphical Sierpinski carpets
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Citation Context ...ρ) , where ρ ∈ [ 7 3 6 , 2 ], while computer calculations suggest that ρ ∼ = 1.251. The dual graph of M, as in Definition 6.15, is the graph G2 in figure 7, also associated to M by Barlow and Bass in =-=[2]-=-. It is easy to see that the graphs G1 and G2 are roughly-isometric. Therefore, by Theorem 7.12, α2(M, ∂M) = α(∆2−) = 2γ (so that α2(M, ∂M) ∈ [1.67, 1.87]). Figure 6. Carpet Graph (following Barlow)S... |

17 |
Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries
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Citation Context ...L 2 -Betti or Novikov-Shubin numbers, however when 1-dimensional CW-complexes are considered, and in particular prefractal graphs determined by nested fractals, a result by Hambly and Kumagai applies =-=[20]-=-, implying that Novikov-Shubin numbers are invariant under rough isometries. Further results on invariance will be proved elsewhere [6]. We then show that in some cases L 2 -Betti and Novikov-Shubin n... |

15 |
Estimates of transition densities for Brownian motion on nested fractals, Prob
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Citation Context ...5.9. For the Gasket graph in figure 1 we obtain α0 = β1 = 1 2 . For the Vicsek graph in figure 3 we obtain α0 = β1 = 1 4 . For the Lindstrom graph in figure 2 we obtain α0 = computed numerically, see =-=[23]-=-, β0 = 0, and β1 = 1 3 . Our second class of examples is given by the following 2 log 3 log 5 , see [3], β0 = 0, and 2log 5 log 15 , see [21], β0 = 0, and 2log 7 log 12.89027 Proposition 8.3. Let M be... |

13 | Dimensions and spectral triples for fractals
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(Show Context)
Citation Context ...ly, one proves σj0 ∈ F(Ej0(γ n k Kn)). Therefore, γn k Kn ∩ γn ℓ Kn = F(γn k Kn) ∩ F(γn ℓ Kn). □ Remark 6.9. The construction above can be easily generalised to the case of translation limit fractals =-=[14, 15, 16, 17, 18]-=-. We now study some properties of polyhedral complexes, which are valid in particular for the prefractal complexes. Definition 6.10. We say that ∆j± is a graph-like Laplacian if there exists a suitabl... |

9 | Fractals in Non-commutative Geometry
- Guido, Isola
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(Show Context)
Citation Context ...ly, one proves σj0 ∈ F(Ej0(γ n k Kn)). Therefore, γn k Kn ∩ γn ℓ Kn = F(γn k Kn) ∩ F(γn ℓ Kn). □ Remark 6.9. The construction above can be easily generalised to the case of translation limit fractals =-=[14, 15, 16, 17, 18]-=-. We now study some properties of polyhedral complexes, which are valid in particular for the prefractal complexes. Definition 6.10. We say that ∆j± is a graph-like Laplacian if there exists a suitabl... |

9 | A trace on fractal graphs and the ihara zeta function
- Guido, Isola, et al.
(Show Context)
Citation Context ...ber is computed for two examples of 2-dimensional CW-complexes. In closing this introduction, we note that the C ∗ -algebra and the trace for selfsimilar graphs constructed in this paper, are used in =-=[19]-=- to study the Ihara zeta function for fractal graphs. The results contained in this paper were announced in the Conferences “C ∗ - algebras and elliptic theory” Bedlewo 2006, and “21 st International ... |

8 |
Self-similarity of volume measures for Laplacians on p.c.f. self-similar fractals
- Kigami, Lapidus
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Citation Context ...graph in figure 2 we obtain α0 = computed numerically, see [23], β0 = 0, and β1 = 1 3 . Our second class of examples is given by the following 2 log 3 log 5 , see [3], β0 = 0, and 2log 5 log 15 , see =-=[21]-=-, β0 = 0, and 2log 7 log 12.89027 Proposition 8.3. Let M be a p-irreducible prefractal complex in R p , let G be the dual graph of M, as in Definition 6.15, and assume that (7.6) hold on G. Then the N... |

8 |
Morse theory and von Neumann II1 factors
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Citation Context ...rator has compact resolvent. Replacing the usual trace with the Γ-trace, he defined the L 2 - Betti numbers and proved an index theorem for covering manifolds. Based on this paper, Novikov and Shubin =-=[27]-=- observed that, since for noncompact manifolds the spectrum of the Laplacian is not discrete, new global spectral invariants can be defined, which necessarily involve the density near zero of the spec... |

7 |
Effective resistances for harmonic structures on p.c.f. self-similar sets
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(Show Context)
Citation Context ...d [20]. Assume we are given a nested fractal K in R p determined by similarities w1, . . . , wq, with the same similarity parameter, and let S be the Hausdorff dimension of K in the resistance metric =-=[22]-=-. Let M be the nested fractal graph based on K. Theorem 8.1. Let K, S, and M be as above. Then (7.6) hold for M, with γ = S S+1 ∈ (0, 1). Therefore, α0(M) = 2γ. Proof. The thesis follows from Corollar... |

5 | Heat kernels and sets with fractal structure. In: Heat kernels and analysis on manifolds, graphs, and metric spaces
- Barlow
- 2002
(Show Context)
Citation Context ...in α0 = β1 = 1 4 . For the Lindstrom graph in figure 2 we obtain α0 = computed numerically, see [23], β0 = 0, and β1 = 1 3 . Our second class of examples is given by the following 2 log 3 log 5 , see =-=[3]-=-, β0 = 0, and 2log 5 log 15 , see [21], β0 = 0, and 2log 7 log 12.89027 Proposition 8.3. Let M be a p-irreducible prefractal complex in R p , let G be the dual graph of M, as in Definition 6.15, and a... |

5 |
Combinatorial heat kernels and index theorems
- Elek
- 1995
(Show Context)
Citation Context ...rove that such characteristic coincides with the alternating sum of the L 2 -Betti numbers. An analogous result, though obtained with a different proof, for amenable simplicial complexes is contained =-=[9]-=-. Here we do not prove directly invariance results for L 2 -Betti or Novikov-Shubin numbers, however when 1-dimensional CW-complexes are considered, and in particular prefractal graphs determined by n... |

5 |
Aperiodic order, integrated density of states and the continuous algebras of John von Neumann, e-print math-ph/0606061
- Elek
- 2006
(Show Context)
Citation Context .... For the sake of completeness, we mention that notions related to that of geometric operators have been considered in the literature, see e.g. [24] where they are called tight binding operators, and =-=[10]-=- where they are called pattern invariant operators. In the Γ-covering case, L 2 -Betti numbers are defined as Γ-dimensions of the kernels of Laplace operators, namely as Γ-traces of the corresponding ... |

5 | Aperiodic order and quasicrystals: spectral properties
- Lenz, Stollmann
(Show Context)
Citation Context ... 2 -Betti numbers and Novikov-Shubin numbers are defined. For the sake of completeness, we mention that notions related to that of geometric operators have been considered in the literature, see e.g. =-=[24]-=- where they are called tight binding operators, and [10] where they are called pattern invariant operators. In the Γ-covering case, L 2 -Betti numbers are defined as Γ-dimensions of the kernels of Lap... |

4 | Noncommutative Riemann integration and singular traces for C ∗ - algebras
- Guido, Isola
(Show Context)
Citation Context ...a generalised limit procedure) were subsequently defined for open manifolds by Roe, and were proved to be invariant under quasi-isometries [30]. The invariance of Novikov-Shubin numbers was proved in =-=[13]-=-. The basic idea of the present analysis is the notion of self-similar CW-complex, which is defined as a complex endowed with a natural exhaustion {Kn} in such a way that Kn+1 is a union (with small i... |

4 | Tangential dimensions I. Metric spaces
- Guido, Isola
(Show Context)
Citation Context |

4 |
On the quasi-isometry invariance of L 2 Betti numbers
- Roe
- 1989
(Show Context)
Citation Context ...ants by Gromov-Shubin [12]. L 2 -Betti numbers (depending on a generalised limit procedure) were subsequently defined for open manifolds by Roe, and were proved to be invariant under quasi-isometries =-=[30]-=-. The invariance of Novikov-Shubin numbers was proved in [13]. The basic idea of the present analysis is the notion of self-similar CW-complex, which is defined as a complex endowed with a natural exh... |

2 | Tangential dimensions II. Measures
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(Show Context)
Citation Context |