Metric on the C*-algebras on a fixed Hilbert space
In mathematics, the Kadison–Kastler metric is a metric on the space of C*-algebras on a fixed Hilbert space. It is the Hausdorff distance between the unit balls of the two C*-algebras, under the norm-induced metric on the space of all bounded operators on that Hilbert space.
It was used by Richard Kadison and Daniel Kastler to study the perturbation theory of von Neumann algebras.
Formal definition.
Let formula_1 be a Hilbert space and formula_2 denote the set of all bounded operators on formula_1. If formula_4 and formula_5 are linear subspaces of formula_2 and formula_7 denote their unit balls, respectively, the "Kadison–Kastler" distance between them is defined as,
formula_8
The above notion of distance defines a metric on the space of C*-algebras which is called the "Kadison-Kastler metric".
References.
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