Results 1 
7 of
7
GromovHausdorff distance for quantum metric spaces
 Mem. Amer. Math. Soc
"... Abstract. By a quantum metric space we mean a C ∗algebra (or more generally an orderunit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distanc ..."
Abstract

Cited by 57 (7 self)
 Add to MetaCart
(Show Context)
Abstract. By a quantum metric space we mean a C ∗algebra (or more generally an orderunit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distance. We show that the basic theorems of the classical theory have natural quantum analogues. Our main example involves the quantum tori, Aθ. We show, for consistently defined “metrics”, that if a sequence {θn} of parameters converges to a parameter θ, then the sequence {Aθn} of quantum tori converges in quantum Gromov–Hausdorff distance to Aθ. 1.
Matrix algebras converge to the sphere for quantum GromovHausdorff distance
 Mem. Amer. Math. Soc
"... Abstract. On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2sphere (or to other spaces). There is often careful bookkeeping with lengths, which suggests that one is dealing with “quantum metric spaces”. We show how t ..."
Abstract

Cited by 35 (6 self)
 Add to MetaCart
(Show Context)
Abstract. On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2sphere (or to other spaces). There is often careful bookkeeping with lengths, which suggests that one is dealing with “quantum metric spaces”. We show how to make these ideas precise by means of Berezin quantization using coherent states. We work in the general setting of integral coadjoint orbits for compact Lie groups. On perusing the theoretical physics literature which deals with string theory and related parts of quantum field theory, one finds in many scattered places assertions that the complex matrix algebras, Mn, converge to the twosphere, S 2, (or to related spaces) as n goes to infinity. Here S 2 is viewed as synonymous with the algebra C(S 2) of continuous complexvalued functions on S 2 (of which S 2 is the maximalideal space). Approximating the sphere by matrix algebras is attractive for the following reason. In trying to carry out quantum field theory on S 2 it is natural to try to proceed by approximating S 2 by finite spaces. But “lattice ” approximations coming from choosing a finite set of points in S 2 break the very important symmetry of the action of SU(2) on S 2 (via SO(3)). But SU(2) acts naturally on the matrix algebras, in a way coherent with its action on S 2, as we will recall below. So it is natural to use them to approximate C(S 2). In this setting the matrix algebras are often referred to as “fuzzy spheres”. (See [33], [34], [17], [22], [24] and references therein.) When using the approximation of S 2 by matrix algebras, the precise sense of convergence is usually not explicitly specified in the literature. Much of the literature is at a largely algebraic level, with indications that the notion of convergence which is intended involves how structure constants and important formulas change as n grows. See, for
On the Spectral Theory of Manifolds with Cusps
, 2000
"... We are interested in the spectral properties of Dirac operators on Riemannian manifolds with cuspidal ends. We derive estimates for the essential spectrum and get formulas for the index in the Fredholm case. Introduction Let M be a complete oriented Riemannian manifold. If M is closed and D is a fo ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
We are interested in the spectral properties of Dirac operators on Riemannian manifolds with cuspidal ends. We derive estimates for the essential spectrum and get formulas for the index in the Fredholm case. Introduction Let M be a complete oriented Riemannian manifold. If M is closed and D is a formally selfadjoint elliptic operator which acts on the smooth sections of a Hermitian bundle E over M , then the essential spectrum of D is empty. In particular, D is a Fredholm operator. If M is noncompact, then the essential spectrum of D depends heavily on the coefficients of D and the end structure of M . In this work we are interested in the spectral properties of generalized Dirac operators in the sense of Gromov and Lawson, see [LM]. We study the case when the ends of M are cuspidal, see Section 3 for the precise definition. In particular, we assume that each end U of M is of finite volume and diffeomorphic to a product (0; 1) \Theta N , where N = N U is a closed manifold, and tha...
Continuity of Dirac Spectra
, 2013
"... Abstract. It is a wellknown fact that on a bounded spectral interval the Dirac spectrum can be described locally by a nondecreasing sequence of continuous functions of the Riemannian metric. In the present article we extend this result to a global version. We think of the spectrum of a Dirac oper ..."
Abstract
 Add to MetaCart
Abstract. It is a wellknown fact that on a bounded spectral interval the Dirac spectrum can be described locally by a nondecreasing sequence of continuous functions of the Riemannian metric. In the present article we extend this result to a global version. We think of the spectrum of a Dirac operator as a function Z → R and endow the space of all spectra with an arsinhuniform metric. We prove that the spectrum of the Dirac operator depends continuously on the Riemannian metric. As a corollary, we obtain the existence of a nondecreasing family of functions on the space of all Riemannian metrics, which represents the entire Dirac spectrum at any metric. We also show that in general these functions do not descend to the space of