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Non-Commutative Metrics on Matrix State Spaces

Wei Wu
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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.

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Wei Wu. 2005-03-28. Non-Commutative Metrics on Matrix State Spaces. https://arxiv.org/abs/math/0411475

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