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57
Optimal Hypercontractivity for Fermi Fields and Related NonCommutative Integration Inequalities
, 1992
"... : Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of noncommutativ ..."
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Cited by 15 (1 self)
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: Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of noncommutative integration are established. On leave from School of Math., Georgia Institute of Technology, Atlanta, GA 30332 Work supported by U.S. National Science Foundation grant no. PHY9019433A01. 1 I. INTRODUCTION Observables pertaining to the configuration of a quantum system with n degrees of freedom are operators Q 1 ; Q 2 ; : : : ; Q n which, depending on the system, may or may not commute. Our main concern is with the case in which the configuration variables are amplitudes of certain field modes. For boson fields, these configuration observables do commute, and the state space H can be taken as the space of all complex square integrable functions on their joint spectrum. This is t...
Meeting Descartes and Klein somewhere in a noncommutative space
 Highlights of Mathematical Physics
"... Abstract. We combine the coordinate method and Erlangen program in the framework of noncommutative geometry through an investigation of symmetries of noncommutative coordinate algebras. As the model we use the coherent states construction and the wavelet transform in functional spaces. New examples ..."
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Cited by 10 (8 self)
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Abstract. We combine the coordinate method and Erlangen program in the framework of noncommutative geometry through an investigation of symmetries of noncommutative coordinate algebras. As the model we use the coherent states construction and the wavelet transform in functional spaces. New examples are a three dimensional spectrum of a nonnormal matrix and a quantisation procedure from symplectomorphisms.
Quantum Stochastic Dynamics I: Spin Systems on a Lattice
, 1995
"... : In the context of noncommutative IL p spaces we discuss the conditions for existence and ergodicity of translation invariant stochastic spin flip and diffusion dynamics for quantum spin systems with finite range interactions on a lattice. Key words: Noncommutative IL p spaces, stochastic spin ..."
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Cited by 9 (0 self)
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: In the context of noncommutative IL p spaces we discuss the conditions for existence and ergodicity of translation invariant stochastic spin flip and diffusion dynamics for quantum spin systems with finite range interactions on a lattice. Key words: Noncommutative IL p spaces, stochastic spin flip and diffusion dynamics, quantum spins, systems on a lattice, finite range interactions. 1.Introduction The analysis in the interpolating family of IL p spaces associated to a probability measure plays an essential role in the study of the classical Markov semigroups. In general it is important for their construction as well as for the investigation of the ergodicity properties. It is especially useful if the underlying configuration space is infinite dimensional. In this paper we introduce some basic ideas concerning the application of interpolating IL p spaces to study Markov semigroups in the noncommutative context of quantum spin systems on a lattice. In Section 2 we show that usi...
A reduction method for noncommutative Lpspaces and applications
, 806
"... We consider the reduction of problems on general noncommutative Lpspaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncommutative Lpspace by tracial ones. We show that under ..."
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Cited by 7 (2 self)
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We consider the reduction of problems on general noncommutative Lpspaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncommutative Lpspace by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products by a locally compact abelian group or to their noncommutative Lpspaces. We present applications of these results to the theory of noncommutative martingale inequalities by reducing most recent general noncommutative martingale/ergodic inequalities to those in the tracial case. 0
On the best constants in some noncommutative martingale inequalities
 Bull. London Math. Soc
"... Abstract. We determine the optimal orders for the best constants in the noncommutative BurkholderGundy, Doob and Stein inequalities obtained recently in the noncommutative martingale theory. AMS Classification: 46L53, 46L51 Key words: Noncommutative martingale, inequality, optimal order, triangu ..."
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Cited by 7 (1 self)
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Abstract. We determine the optimal orders for the best constants in the noncommutative BurkholderGundy, Doob and Stein inequalities obtained recently in the noncommutative martingale theory. AMS Classification: 46L53, 46L51 Key words: Noncommutative martingale, inequality, optimal order, triangular projection † Marius Junge is partially supported by the NSF 1
Singular traces, dimensions, and NovikovShubin invariants
 Proceedings of the 17th OT Conference, Theta
, 2000
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Noncommutative Riemann integration and NovikovShubin invariants for Open Manifolds
, 2001
"... Given a C ∗algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [2 ..."
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Cited by 6 (3 self)
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Given a C ∗algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [26], and show that A R is a C ∗algebra, and τ extends to a semicontinuous semifinite trace on A R. 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 A R, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a τa.e. bimodule on A R, denoted by A R, and such bimodule contains the functional calculi of selfadjoint elements of A R under unbounded Riemann measurable functions. Besides, τ extends to a bimodule trace on A R.
Derivations on the Algebra of Measurable Operators Affiliated with a Type I von Neumann
, 2008
"... Let M be a type I von Neumann algebra with the center Z, and let LS(M) be the algebra of all locally measurable operators affiliated with M. We prove that every Zlinear derivation on LS(M) is inner. In particular all Zlinear derivations on the algebras of measurable and respectively totally measur ..."
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Cited by 6 (4 self)
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Let M be a type I von Neumann algebra with the center Z, and let LS(M) be the algebra of all locally measurable operators affiliated with M. We prove that every Zlinear derivation on LS(M) is inner. In particular all Zlinear derivations on the algebras of measurable and respectively totally measurable operators are spatial and implemented by elements from LS(M). 1 Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D53115
Noncommutative Riemann integration and singular traces for C ∗  algebras
"... Given a C ∗algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [1 ..."
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Cited by 4 (4 self)
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Given a C ∗algebra A with a semicontinuous semifinite trace τ acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. τ as a suitable closure, à la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions [16], and show that A R is a C ∗algebra, and τ extends to a semicontinuous semifinite trace on A R. 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 A R, in analogy with improper Riemann integration. Unbounded Riemann measurable operators form a τa.e. bimodule on A R, denoted by AR, and such bimodule contains the functional calculi of selfadjoint elements of A R under unbounded Riemann measurable functions. Besides, τ extends to a bimodule trace on AR. As type II1 singular traces for a semifinite von Neumann algebra M with a normal semifinite faithful (nonatomic) trace τ have been defined as traces on M − Mbimodules of unbounded τmeasurable operators [5], type II1 singular traces for a C ∗algebra A with a semicontinuous semifinite (nonatomic) trace τ are defined here as traces on A − Abimodules of unbounded Riemann measurable operators (in AR) for any faithful representation of A. An application of singular traces for C ∗algebras is contained in [6].