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Metrics on state spaces
 Doc. Math
, 1999
"... This article is dedicated to Richard V. Kadison in anticipation of his completing his seventyfifth circumnavigation of the sun. Abstract. In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly noncommu ..."
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This article is dedicated to Richard V. Kadison in anticipation of his completing his seventyfifth circumnavigation of the sun. Abstract. In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly noncommutative compact spaces are usually not determined by the restriction of the metric they define on the state space, to the extreme points of the state space. We characterize the Lipschitz norms which are determined by their metric on the whole state space as being those which are lower semicontinuous. We show that their domain of Lipschitz elements can be enlarged so as to form a dual Banach space, which generalizes the situation for ordinary Lipschitz seminorms. We give a characterization of the metrics on state spaces which come from Lipschitz seminorms. The natural (broader) setting for these results is provided by the “function spaces” of Kadison. A variety of methods for constructing Lipschitz seminorms is indicated. In noncommutative geometry (based on C ∗algebras), the natural way to specify a metric is by means of a suitable “Lipschitz seminorm”. This idea was first suggested by Connes [C1] and developed further in [C2, C3]. Connes pointed out [C1, C2] that from a Lipschitz seminorm one obtains in a simple way an ordinary metric on the state space of the C ∗algebra. This metric generalizes the Monge–Kantorovich metric on probability measures [KA, Ra, RR]. In this article we make more precise the relationship between metrics on the state space and Lipschitz seminorms. Let ρ be an ordinary metric on a compact space X. The Lipschitz seminorm, Lρ, determined by ρ is defined on functions f on X by (0.1) Lρ(f) = sup{f(x) − f(y)/ρ(x, y) : x ̸ = y}.
Unitary orbits of normal operators in von Neumann algebras
, 2005
"... Abstract. The starting points for this paper are simple descriptions of the norm and strong * closures of the unitary orbit of a normal operator in a von Neumann algebra. The statements are in terms of spectral data and do not depend on the type or cardinality of the algebra. We relate this to sever ..."
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Cited by 9 (4 self)
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Abstract. The starting points for this paper are simple descriptions of the norm and strong * closures of the unitary orbit of a normal operator in a von Neumann algebra. The statements are in terms of spectral data and do not depend on the type or cardinality of the algebra. We relate this to several known results and derive some consequences, of which we list a few here. Exactly when the ambient von Neumann algebra is a direct sum of σfinite algebras, any two normal operators have the same normclosed unitary orbit if and only if they have the same strong*closed unitary orbit if and only if they have the same strongclosed unitary orbit. But these three closures generally differ from each other and from the unclosed unitary orbit, and we characterize when equality holds between any two of these four sets. We also show that in a properly infinite von Neumann algebra, the strongclosed unitary orbit of any operator, not necessarily normal, meets the center in the (nonvoid) left essential central spectrum of Halpern. One corollary is a “type III Weylvon NeumannBerg theorem ” involving containment of essential central spectra. 1.
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 7 (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.
ON THE DIMENSION THEORY OF VON NEUMANN ALGEBRAS
, 2005
"... Abstract. In this paper we study three aspects of (P(M) / ∼), the set of Murrayvon Neumann equivalence classes of projections in a von Neumann algebra M. First we determine the topological structure that (P(M) / ∼) inherits from the operator topologies on M. Then we show that there is a version of ..."
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Abstract. In this paper we study three aspects of (P(M) / ∼), the set of Murrayvon Neumann equivalence classes of projections in a von Neumann algebra M. First we determine the topological structure that (P(M) / ∼) inherits from the operator topologies on M. Then we show that there is a version of the centervalued trace which extends the dimension function, even when M is not σfinite. Finally we prove that (P(M) / ∼) is a complete lattice, a fact which has an interesting reformulation in terms of representations. 1.
DIVISIBLE OPERATORS IN VON NEUMANN ALGEBRAS
, 2008
"... Abstract. Relativizing an idea from multiplicity theory, we say that an element x of a von Neumann algebra M is ndivisible if W ∗ (x) ′ ∩ M unitally contains a factor of type In. We decide the density of the ndivisible operators, for various n, M, and operator topologies. The most sensitive case ..."
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Cited by 5 (1 self)
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Abstract. Relativizing an idea from multiplicity theory, we say that an element x of a von Neumann algebra M is ndivisible if W ∗ (x) ′ ∩ M unitally contains a factor of type In. We decide the density of the ndivisible operators, for various n, M, and operator topologies. The most sensitive case is σstrong density in II1 factors, which is closely related to the McDuff property. We make use of Voiculescu’s noncommutative Weylvon Neumann theorem to obtain several descriptions of the norm closure of the ndivisible operators in B(ℓ 2). Here are two consequences: (1) in contrast to the reducible operators, of which they form a subset, the divisible operators are nowhere dense; (2) if an operator is a norm limit of divisible operators, it is actually a norm limit of unitary conjugates of a single divisible operator. This is related to our ongoing work on unitary orbits by the following theorem, which is new even for B(ℓ 2): if an element of a von Neumann algebra belongs to the norm closure of the ℵ0divisible operators, then the σweak closure of its unitary orbit is convex. 1.
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].
An expanded version of a talk given at the AMS Session
, 2007
"... George Mackey devoted his career to exploring the basic unity of mathematics, and to understanding its relationships to modern physics. He was a major figure in the flowering of group representation theory and related portions of functional analysis that occured in the middle decades of the last ..."
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George Mackey devoted his career to exploring the basic unity of mathematics, and to understanding its relationships to modern physics. He was a major figure in the flowering of group representation theory and related portions of functional analysis that occured in the middle decades of the last
Journal of Functional Analysis 176, 115152 (2000) Noncommutative Riemann Integration and NovikovShubin Invariants for Open Manifolds
, 1998
"... Given a C*algebra A with a semicontinuous semifinite trace { acting on the Hilbert space H, we define the family AR of bounded Riemann measurable elements w.r.t. { as a suitable closure, a la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions, an ..."
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Given a C*algebra A with a semicontinuous semifinite trace { acting on the Hilbert space H, we define the family AR of bounded Riemann measurable elements w.r.t. { as a suitable closure, a la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions, and show that AR is a C*algebra, and { extends to a semicontinuous semifinite trace on AR. 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 AR, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a {a.e. bimodule on AR, denoted by AR, and such a bimodule contains the functional calculi of selfadjoint elements of AR under unbounded Riemann measurable functions. Besides, { extends to a bimodule trace on AR. Type II1 singular traces for C*algebras can be defined on the bimodule of unbounded Riemannmeasurable operators. Noncommutative Riemann integration and singular traces for C*algebras are then used to define NovikovShubin numbers for amenable open manifolds, to show their invariance under quasiisometries, and to prove that they are (noncommutative) asymptotic dimensions. 2000 Academic Press 1.