mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Kadison–Kastler metric is a

metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...

on the space of C^*-algebras on a fixed

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

. It is the

Hausdorff distance In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty set, non-empty compact space, compact subsets o ...

between the

unit ball Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...

s of the two C^*-algebras, under the norm-induced metric on the space of all

bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...

s on that Hilbert space. It was used by

Richard Kadison Richard Vincent Kadison (July 25, 1925 – August 22, 2018) was an American mathematician known for his contributions to the study of operator algebras. Career Born in New York City in 1925, Kadison was a Gustave C. Kuemmerle Professor in the De ...

and

Daniel Kastler Daniel Kastler (; 4 March 1926 – 4 July 2015) was a French theoretical physicist, working on the foundations of quantum field theory and on non-commutative geometry. Biography Daniel Kastler was born on March 4, 1926, in Colmar, a city of nor ...

to study the perturbation theory of

von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann al ...

Formal definition

Let

\mathcal

be a Hilbert space and

B(\mathcal)

denote the set of all bounded operators on

\mathcal

. If

\mathfrak

and

\mathfrak

are linear subspaces of

B(\mathcal)

and

\mathfrak_1, \mathfrak_1

denote their unit balls, respectively, the ''Kadison–Kastler'' distance between them is defined as, :

\,  \mathfrak - \mathfrak \,  := \sup \.

The above notion of distance defines a metric on the space of C^*-algebras which is called the ''Kadison-Kastler metric''.

References

{{DEFAULTSORT:Kadison-Kastler metric C*-algebras