Results 1  10
of
21
Metrics on states from actions of compact groups
 Doc. Math
, 1998
"... Abstract. Let a compact Lie group act ergodically on a unital C ∗algebra A. We consider several ways of using this structure to define metrics on the state space of A. These ways involve length functions, norms on the Lie algebra, and Dirac operators. The main thrust is to verify that the correspon ..."
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Cited by 44 (5 self)
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Abstract. Let a compact Lie group act ergodically on a unital C ∗algebra A. We consider several ways of using this structure to define metrics on the state space of A. These ways involve length functions, norms on the Lie algebra, and Dirac operators. The main thrust is to verify that the corresponding metric topologies on the state space agree with the weak∗ topology. Connes [Co1, Co2, Co3] has shown us that Riemannian metrics on noncommutative spaces (C ∗algebras) can be specified by generalized Dirac operators. Although in this setting there is no underlying manifold on which one then obtains an ordinary metric, Connes has shown that one does obtain in a simple way an ordinary metric on the state space of the C ∗algebra, generalizing the MongeKantorovich metric on probability measures [Ra] (called the “Hutchinson metric ” in the theory of fractals [Ba]). But an aspect of this matter which has not received much attention so far [P] is the question of when the metric topology (that is, the topology from the metric coming from a Dirac operator) agrees with the underlying weak ∗ topology on the state space. Note that for locally compact spaces their topology agrees with the weak ∗ topology coming from viewing points as linear functionals (by evaluation) on the algebra of continuous functions vanishing at infinity.
Sobolev inequalities in disguise
 Indiana Univ. Math. J
, 1995
"... We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or subelliptic geometry, as well as on graphs and to certain nonlocal Sobolev norms. It only uses elementary cutoff argu ..."
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Cited by 38 (4 self)
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We present a simple and direct proof of the equivalence of various functional inequalities such as Sobolev or Nash inequalities. This proof applies in the context of Riemannian or subelliptic geometry, as well as on graphs and to certain nonlocal Sobolev norms. It only uses elementary cutoff arguments. This method has interesting consequences concerning Trudinger type inequalities. 1. Introduction. On R n, the classical Sobolev inequality [27] indicates that, for every smooth enough function f with compact support,
Nash Inequalities for Finite Markov Chains
, 1996
"... This paper develops bounds on the rate of decay of powers of Markov kernels on finite state spaces. These are combined with eigenvalue estimates to give good bounds on the rate of convergence to stationarity for finite Markov chains whose underlying graph has moderate volume growth. Roughly, for suc ..."
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Cited by 34 (9 self)
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This paper develops bounds on the rate of decay of powers of Markov kernels on finite state spaces. These are combined with eigenvalue estimates to give good bounds on the rate of convergence to stationarity for finite Markov chains whose underlying graph has moderate volume growth. Roughly, for such chains, order (diameter) 2 steps are necessary and suffice to reach stationarity. We consider local Poincar6 inequalities and use them to prove Nash inequalities. These are bounds on (,_norms in terms of Dirichlet forms and l~norms which yield decay rates for iterates of the kernel. This method is adapted from arguments developed by a number of authors in the context of partial differential equations and, later, in the study of random walks on infinite graphs. The main results do not require reversibility.
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 23 (6 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
YangMills theory and the SegalBargmann transform
 Commun. Math. Phys
, 1999
"... Abstract. We use a variant of the SegalBargmann transform to study canonically quantized YangMills theory on a spacetime cylinder with a compact structure group K. The nonexistent Lebesgue measure on the space of connections is “approximated ” by a Gaussian measure with large variance. The Segal ..."
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Cited by 18 (15 self)
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Abstract. We use a variant of the SegalBargmann transform to study canonically quantized YangMills theory on a spacetime cylinder with a compact structure group K. The nonexistent Lebesgue measure on the space of connections is “approximated ” by a Gaussian measure with large variance. The SegalBargmann transform is then a unitary map from the L2 space over the space of connections to a holomorphic L2 space over the space of complexified connections with a certain Gaussian measure. This transform is given roughly by et∆A/2 followed by analytic continuation. Here ∆A is the Laplacian on the space of connections and is the Hamiltonian for the quantized theory. On the gaugetrivial subspace, consisting of functions of the holonomy around the spatial circle, the SegalBargmann transform becomes et∆K/2 followed by analytic continuation, where ∆K is the Laplacian for the structure group K. This result gives a rigorous meaning to the idea that ∆A reduces to ∆K on functions of the holonomy. By letting the variance of the Gaussian measure tend to infinity we recover the standard realization of the quantized
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 13 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group
 Potential Analysis
"... Abstract. The heat kernel measure µt is constructed on an infinite dimensional complex group using a diffusion in a Hilbert space. Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, HL2 (SOHS, µ ..."
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Cited by 11 (7 self)
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Abstract. The heat kernel measure µt is constructed on an infinite dimensional complex group using a diffusion in a Hilbert space. Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, HL2 (SOHS, µt), is one of two spaces of holomorphic functions we consider. The second space, HL2 (SO(∞)), consists of functions which are holomorphic on an analog of the CameronMartin subspace for the group. It is proved that there is an isometry from the first space to the second one. The main theorem is that an infinite dimensional nonlinear analog of the Taylor expansion defines an isometry from HL2 (SO(∞)) into the Hilbert space associated with a Lie algebra of the infinite dimensional group. This is an extension to infinite dimensions of an isometry of B. Driver and L. Gross for complex Lie groups. All the results of this paper are formulated for one concrete group, the HilbertSchmidt complex orthogonal group, though our methods can be applied in more general situations. 1.
ROBIN BOUNDARY VALUE PROBLEMS ON ARBITRARY DOMAINS
"... Abstract. We develop a theory of generalised solutions for elliptic boundary value problems subject to Robin boundary conditions on arbitrary domains, which resembles in many ways that of the Dirichlet problem. In particular, we establish LpLqestimates which turn out to be the best possible in tha ..."
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Cited by 9 (1 self)
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Abstract. We develop a theory of generalised solutions for elliptic boundary value problems subject to Robin boundary conditions on arbitrary domains, which resembles in many ways that of the Dirichlet problem. In particular, we establish LpLqestimates which turn out to be the best possible in that framework. We also discuss consequences to the spectrum of Robin boundary value problems. Finally, we apply the theory to parabolic equations. 1.
Large time behavior of the heat kernel
, 2002
"... In this paper we study the large time behavior of the (minimal) heat kernel kM P (x, y, t) of a general time independent parabolic operator L = ut+P(x, ∂x) which is defined on a noncompact manifold M. More precisely, we prove that lim t→ ∞ eλ0t k M P (x, y, t) always exists. Here λ0 is the generaliz ..."
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Cited by 8 (2 self)
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In this paper we study the large time behavior of the (minimal) heat kernel kM P (x, y, t) of a general time independent parabolic operator L = ut+P(x, ∂x) which is defined on a noncompact manifold M. More precisely, we prove that lim t→ ∞ eλ0t k M P (x, y, t) always exists. Here λ0 is the generalized principal eigenvalue of the operator P in M.
Eigenfunction expansions for generators of Dirichlet forms
, 2002
"... Dedicated to the memory of Klaus Floret Abstract We present an eigenfunction expansion theorem for generators of strongly local, regular Dirichlet forms. Conditions are phrased in terms of the intrinsic metric. The result covers many cases of Hamiltonians which appear in Mathematical Physics and Geo ..."
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Cited by 5 (2 self)
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Dedicated to the memory of Klaus Floret Abstract We present an eigenfunction expansion theorem for generators of strongly local, regular Dirichlet forms. Conditions are phrased in terms of the intrinsic metric. The result covers many cases of Hamiltonians which appear in Mathematical Physics and Geometry.