# Metrics on State Spaces

@article{Rieffel1999MetricsOS, title={Metrics on State Spaces}, author={Marc Rieffel}, journal={arXiv: Operator Algebras}, year={1999} }

In contrast to the usual Lipschitz seminorms associated to ordinary metrics on compact spaces, we show by examples that Lipschitz seminorms on possibly non-commutative 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…

## 179 Citations

Non-Commutative Metrics on Matrix State Spaces

- Mathematics
- 2004

We use the theory of quantization to introduce non-commutative versions of metric on state space and Lipschitz seminorm. We show that a lower semicontinuous matrix Lipschitz seminorm is determined by…

Non-commutative Metrics on Matrix State Spaces

- 2008

We use the theory of quantization to introduce non-commutative versions of metric on state space and Lipschitz seminorm. We show that a lower semicontinuous matrix Lipschitz seminorm is determined by…

Non-commutative metric topology on matrix state space

- Mathematics
- 2004

We present an operator space version of Rieffel's theorem on the agreement of the metric topology, on a subset of the Banach space dual of a normed space, from a seminorm with the weak*-topology. As…

Non-commutative Metric Topology on Matrix State Space

- 2008

We present an operator space version of Rieffel’s theorem on the agreement of the metric topology, on a subset of the Banach space dual of a normed space, from a seminorm with the weak*-topology. As…

NON-COMMUTATIVE METRIC TOPOLOGY ON MATRIX STATE SPACE

- 2005

We present an operator space version of Rieffel’s theorem on the agreement of the metric topology, on a subset of the Banach space dual of a normed space, from a seminorm with the weak*-topology. As…

NON-COMMUTATIVE METRIC TOPOLOGY ON MATRIX STATE SPACE

- 2005

We present an operator space version of Rieffel’s theorem on the agreement of the metric topology, on a subset of the Banach space dual of a normed space, from a seminorm with the weak*-topology. As…

Equivalence of Quantum Metrics with a common domain

- Mathematics
- 2016

We characterize Lipschitz morphisms between quantum compact metric spaces as those *-morphisms which preserve the domain of certain noncommutative analogues of Lipschitz seminorms, namely lower…

Lipschitz functions on topometric spaces

- Mathematics
- 2010

We study functions on topometric spaces which are both (metrically) Lipschitz and (topologically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are…

Commutator estimates on sub-Riemannian manifolds and applications

- Mathematics
- 2013

This article studies sharp norm estimates for the commutator of pseudodifferential operators with non-smooth functions on closed sub-Riemannian manifolds. In particular, we obtain a Calderon…

Dynamics of compact quantum metric spaces

- MathematicsErgodic Theory and Dynamical Systems
- 2021

We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact…

## References

SHOWING 1-10 OF 42 REFERENCES

Lipschitz algebras and derivations II: exterior differentiation

- Mathematics
- 1998

Abstract Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Wiener space,…

Kantorovich-Rubinstein norm and its application in the theory of Lipschitz spaces

- Mathematics
- 1992

We obtain necessary and sufficient conditions on a compact metric space (K, p) that provide a natural isometric isomorphism between completion of the space of Borel measures on K with the…

Monotone Riemannian metrics and relative entropy on noncommutative probability spaces

- Mathematics, Physics
- 1999

We use the relative modular operator to define a generalized relative entropy for any convex operator function g on (0,∞) satisfying g(1)=0. We show that these convex operator functions can be…

On the curvature of monotone metrics and a conjecture concerning the Kubo-Mori metric

- Mathematics, Physics
- 1999

Abstract The manifold of trace one positive complex n×n -matrices represents the space of faithful mixed states of a finite dimensional quantum system. Riemannian monotone metrics on these manifolds…

Defining metric spaces via operators from unital C∗-algebras

- Mathematics
- 1998

For a unital C∗-algebra A and an operator T with DomT ⊆ A, RangeT in a normed space, and ker T = Cmathrm1, we consider the metric dT on S(A), the state space of A, given by dT (φ, ψ) = sup{|φ(a) −…

Compact convex sets and boundary integrals

- Mathematics
- 1971

I Representations of Points by Boundary Measures.- 1. Distinguished Classes of Functions on a Compact Convex Set.- Classes of continuous and semicontinuous, affine and convex functions.- Uniform and…

Metrics on states from actions of compact groups

- Mathematics
- 1998

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…

Analysis on graphs and noncommutative geometry

- Mathematics
- 1993

Abstract We study the form of the continuous time heat kernel for a second order discrete Laplacian on a weighted graph. The analysis is shown to be closely related to the theory of symmetric Markov…

Unitary Invariants for Representations of Operator Algebras

- Mathematics
- 1957

This paper will be concerned with certain developments in the spectral multiplicity (unitary invariants) theory of self-adjoint families of operators. The subject has its roots in the classical…

Gravity coupled with matter and the foundation of non-commutative geometry

- Mathematics, Physics
- 1996

We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond…