Results 1  10
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11
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.
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].
The general form of γfamily of quantum relative entropies
, 2011
"... We use the Falcone–Takesaki noncommutative flow of weights and the resulting theory of noncommutative Lp spaces in order to define the family of relative entropy functionals that naturally generalise the quantum relative entropies of Jenčová–Ojima and the classical relative entropies of Zhu–Rohwer ..."
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Cited by 2 (2 self)
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We use the Falcone–Takesaki noncommutative flow of weights and the resulting theory of noncommutative Lp spaces in order to define the family of relative entropy functionals that naturally generalise the quantum relative entropies of Jenčová–Ojima and the classical relative entropies of Zhu–Rohwer, and belong to an intersection of families of Petz relative entropies with Bregman relative entropies. For the purpose of this task, we generalise the notion of Bregman entropy to the infinitedimensional noncommutative case using the Legendre–Fenchel duality. In addition, we use the Falcone–Takesaki duality to extend the duality between coarse–grainings and Markov maps to the infinitedimensional noncommutative case. Following the recent result of Amari for the Zhu–Rohwer entropies, we conjecture that the proposed family of relative entropies is uniquely characterised by the Markov monotonicity and the Legendre–Fenchel duality. The role of these results in the foundations and applications of quantum information theory is discussed.
Pentagon equation arising from state equations of a C ∗bialgebra
, 906
"... The direct sum O ∗ of all Cuntz algebras has a noncocommutative comultiplication ∆ϕ such that there exists no antipode of any dense subbialgebra of the C ∗bialgebra (O∗, ∆ϕ). From states equations of O ∗ with respect to the tensor product, we construct an operator W for (O∗, ∆ϕ) such that W ∗ is a ..."
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Cited by 1 (1 self)
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The direct sum O ∗ of all Cuntz algebras has a noncocommutative comultiplication ∆ϕ such that there exists no antipode of any dense subbialgebra of the C ∗bialgebra (O∗, ∆ϕ). From states equations of O ∗ with respect to the tensor product, we construct an operator W for (O∗, ∆ϕ) such that W ∗ is an isometry, W(x ⊗ I)W ∗ = ∆ϕ(x) for each x ∈ O ∗ and W satisfies the pentagon equation. Mathematics Subject Classifications (2000). 16W35, 81R50, 46K10. Key words. C ∗bialgebra, pentagon equation
BrascampLieb Inequalities for NonCommutative Integration
, 2008
"... We formulate a noncommutative analog of the BrascampLieb inequality, and prove it in several concrete settings. ..."
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We formulate a noncommutative analog of the BrascampLieb inequality, and prove it in several concrete settings.
Symmetry, Integrability and Geometry: Methods and Applications Measure Theory in Noncommutative Spaces ⋆
"... doi:10.3842/SIGMA.2010.072 Abstract. The integral in noncommutative geometry (NCG) involves a nonstandard trace called a Dixmier trace. The geometric origins of this integral are well known. From a measuretheoretic view, however, the formulation contains several difficulties. We review results con ..."
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doi:10.3842/SIGMA.2010.072 Abstract. The integral in noncommutative geometry (NCG) involves a nonstandard trace called a Dixmier trace. The geometric origins of this integral are well known. From a measuretheoretic view, however, the formulation contains several difficulties. We review results concerning the technical features of the integral in NCG and some outstanding problems in this area. The review is aimed for the general user of NCG. Key words: Dixmier trace; singular trace; noncommutative integration; noncommutative
Algebraic Quantum Theory  A brief guide
, 2011
"... We present the mathematical setting of an algebraic approach to quantum theory, including the Tomita–Takesaki theory, liouvilleans and quantum dynamical systems, ..."
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We present the mathematical setting of an algebraic approach to quantum theory, including the Tomita–Takesaki theory, liouvilleans and quantum dynamical systems,
W*algebras and integration
, 2012
"... We present the mathematical setting of the theory of W ∗algebras, including the Tomita– Takesaki modular theory, the Connes–Araki relative modular theory, the theory of W ∗dynamical systems (with derivations, liouvilleans, and crossed products), and noncommutative integration up to the Falcone–Tak ..."
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We present the mathematical setting of the theory of W ∗algebras, including the Tomita– Takesaki modular theory, the Connes–Araki relative modular theory, the theory of W ∗dynamical systems (with derivations, liouvilleans, and crossed products), and noncommutative integration up to the Falcone–Takesaki theory of noncommutative Lp(N) spaces. We pay special attention to abstract algebraic formulation of all structural properties (avoiding
Unitary Groups in Finite von Neumann Algebras
, 2010
"... We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group U(H) in a Hilbert space H with U(H) equipped with the strong operator topology. More precisely, for any strongly closed subgroup G of the unitary group U(M) in a finite v ..."
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We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group U(H) in a Hilbert space H with U(H) equipped with the strong operator topology. More precisely, for any strongly closed subgroup G of the unitary group U(M) in a finite von Neumann algebra M, we show that the set of all generators of strongly continuous oneparameter subgroups of G forms a complete topological Lie algebra with respect to the strong resolvent topology. We also characterize the algebra M of all densely defined closed