Non-Commutative Metrics on Matrix State Spaces

Details Non-Commutative Metrics on Matrix State Spaces Non-Commutative Metrics on Matrix State Spaces

2004-11-22

arxiv.org/abs/math/0411475v2

J. Ramanujan Math. Soc. 20 (2005), no. 3, 215--254

Mathematics

Operator Algebras

30 pages, minor changes

Scientific paper

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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