Non-Commutative Metrics on Matrix State Spaces

Abstract

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 their matrix metrics on the matrix state spaces. A matrix metric comes from a lower semicontinuous matrix Lip-norm if and only if it is convex, midpoint balanced, and midpoint concave. The operator space of Lipschitz functions with a matrix norm coming from a closed matrix Lip-norm is the operator space dual of an operator space. They generalize Rieffel's results to the quantized situation.

30 pages, minor changes

References in corpus (3)

• Gravity coupled with matter and foundation of non-commutative geometry
• Metrics on State Spaces
• Metrics on states from actions of compact groups

Cited by in corpus (5)

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• Ricci curvature of quantum channels on non-commutative transportation metric spaces
• An operator Arzelà-Ascoli theorem