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301
Processes on unimodular random networks
 In preparation
, 2005
"... Abstract. We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasitransitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amen ..."
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Cited by 48 (4 self)
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Abstract. We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasitransitive graphs. We extend various theorems concerning random walks, percolation, spanning forests, and amenability from the known context of unimodular quasitransitive graphs to the more general context of unimodular random networks. We give properties of a trace associated to unimodular random networks with applications
Doob’s inequality for noncommutative martingales
 J. reine angew. Math
"... Abstract. Let 1 ≤ p < ∞ and (xn)n∈N be a sequence of positive elements in a noncommutative Lp space and (En)n∈N be an increasing sequence of conditional expectations, then En(xn) ∥ ≤ cp xn∥ ..."
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Cited by 46 (27 self)
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Abstract. Let 1 ≤ p < ∞ and (xn)n∈N be a sequence of positive elements in a noncommutative Lp space and (En)n∈N be an increasing sequence of conditional expectations, then En(xn) ∥ ≤ cp xn∥
Noncommutative Burkholder/Rosenthal inequalities
 Ann. Probab
, 2000
"... Abstract. We show norm estimates for the sum of independent random variables in noncommutative Lp spaces for 1 < p < ∞ following previous work by the authors. These estimates generalize Rosenthal’s inequalities in the commutative case. Among other applications, we derive a formula for pnorm of the ..."
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Cited by 46 (25 self)
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Abstract. We show norm estimates for the sum of independent random variables in noncommutative Lp spaces for 1 < p < ∞ following previous work by the authors. These estimates generalize Rosenthal’s inequalities in the commutative case. Among other applications, we derive a formula for pnorm of the eigenvalues for matrices with independent entries, and characterize those symmetric subspaces and unitary ideal spaces which can be realized as subspaces of noncommutative Lp for 2 < p < ∞. 0. Introduction and Notation Martingale inequalities have a long tradition in probability. The applications of the work of Burkholder and his collaborators [B73,?, BDG72, B71a, B71b, BGS71, BG70, B66] ranges from classical harmonic analysis to stochastical differential equations and the geometry of Banach spaces. When proving the estimates for the ‘little square function ’ Burkholder
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.
Scaling algebras and renormalization group in algebraic quantum field theory
 Rev. Math. Phys
, 1995
"... Abstract: The concept of scaling algebra provides a novel framework for the general structural analysis and classification of the short distance properties of algebras of local observables in relativistic quantum field theory. In the present article this method is applied to the simple example of ma ..."
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Cited by 42 (7 self)
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Abstract: The concept of scaling algebra provides a novel framework for the general structural analysis and classification of the short distance properties of algebras of local observables in relativistic quantum field theory. In the present article this method is applied to the simple example of massive free field theory in s = 1,2 and 3 spatial dimensions. Not quite unexpectedly, one obtains for s = 2,3 in the scaling (short distance) limit the algebra of local observables in massless free field theory. The case s = 1 offers, however, some surprises. There the algebra of observables acquires in the scaling limit a nontrivial center and describes charged physical states satisfying Gauss ’ law. The latter result is of relevance for the interpretation of the Schwinger model at short distances and illustrates the conceptual and computational virtues of the method. 1
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 ..."
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Cited by 35 (5 self)
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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.
On the structure of subspaces of noncommutative Lpspaces
 C. R. Acad. Sci. Paris
"... Abstract: We study some structural aspects of the subspaces of the noncommutative (Haagerup) Lpspaces associated with a general (non necessarily semifinite) von Neumann algebra a. If a subspace X of Lp(a) contains uniformly the spaces ℓn p, n ≥ 1, it contains an almost isometric, almost 1complem ..."
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Cited by 33 (3 self)
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Abstract: We study some structural aspects of the subspaces of the noncommutative (Haagerup) Lpspaces associated with a general (non necessarily semifinite) von Neumann algebra a. If a subspace X of Lp(a) contains uniformly the spaces ℓn p, n ≥ 1, it contains an almost isometric, almost 1complemented copy of ℓp. If X contains uniformly the finite dimensional Schatten classes Sn p, it contains their ℓpdirect sum too. We obtain a version of the classical KadecPe̷lczyński dichotomy theorem for Lpspaces, p ≥ 2. We also give operator space versions of these results. The proofs are based on previous structural results on the ultrapowers of Lp(a), together with a careful analysis of the elements of an ultrapower Lp(a) U which are disjoint from the subspace Lp(a). These techniques permit to recover a recent result of N. Randrianantoanina concerning a Subsequence Splitting Lemma for the general noncommutative Lp spaces. Various notions of pequiintegrability are studied (one of which is equivalent to Randrianantoanina’s one) and some results obtained by Haagerup, Rosenthal and Sukochev for Lpspaces based on finite von Neumann algebras concerning subspaces of Lp(a) containing ℓp are extended to the general case.
Characterizing quantum theory in terms of informationtheoretic constraints
 Foundations of Physics
, 2003
"... We show that three fundamental informationtheoretic constraints—the impossibility of superluminal information transfer between two physical systems by performing measurements on one of them, the impossibility of broadcasting the information contained in an unknown physical state, and the impossibil ..."
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Cited by 28 (3 self)
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We show that three fundamental informationtheoretic constraints—the impossibility of superluminal information transfer between two physical systems by performing measurements on one of them, the impossibility of broadcasting the information contained in an unknown physical state, and the impossibility of unconditionally secure bit commitment—suffice to entail that the observables and state space of a physical theory are quantummechanical. We demonstrate the converse derivation in part, and consider the implications of alternative answers to a remaining open question about nonlocality and bit commitment. KEY WORDS: quantum theory; informationtheoretic constraints. Of John Wheeler’s ‘‘Really Big Questions,’ ’ the one on which most progress has been made is It from Bit?—does information play a significant role at the foundations of physics? It is perhaps less ambitious than some of the other Questions, such as How Come Existence?, because it does not necessarily require a metaphysical answer. And unlike, say, Why the Quantum?, it does not require the discovery of new laws of nature: there was room for hope that it might be answered through a better understanding of the laws as we currently know them, particularly those of quantum physics. And this is what has happened: the better understanding is the quantum theory of information and computation. 1