---
tag: VB1J9AA4WX
kind: concept
name: Operator space
msc: [47L25, 46L07]
sources:
  - url: https://en.wikipedia.org/wiki/Operator_space
same_as:
  wikidata: Q7097841
  wikipedia: Operator space
---

## Summary

An operator space is a closed linear subspace of the [[algebra of bounded operators on a Hilbert space|BMDG4JQPEH]].
It comes with the norms it inherits on all matrices with entries in it.
These matrix norms carry more information than the Banach space norm alone.
The natural maps between operator spaces are the completely bounded maps, and the theory is often described as a "quantized" Banach space theory.
Ruan's theorem characterizes operator spaces abstractly by two axioms on the matrix norms.

## Explanations

### L1 human

A Banach space only knows the length of each vector.
An operator space also knows the length of every matrix whose entries are its vectors, because such a matrix acts as one big operator.
Two spaces can look the same as Banach spaces but differ as operator spaces.
For example, the row vectors and the column vectors in $M_n(\C)$, for $n \ge 2$, give the same Banach space $\C^n$ with the Euclidean norm, but they are different operator spaces.