---
tag: KDV7W5EDTJ
kind: concept
name: C* identity
msc: [46L05, 46K05]
sources:
  - url: https://en.wikipedia.org/wiki/C*-algebra
---

## Summary

The C* identity is the condition $\norm{a^* a} = \norm{a}^2$ on a Banach algebra with an involution.
A Banach $*$-algebra whose norm satisfies it is a C*-algebra.
This one equation ties the norm to the algebra so tightly that the algebraic structure determines the norm.
Indeed, $\norm{a}$ is the square root of the spectral radius of $a^* a$.

## Explanations

### L1 human

For a complex number, $\abs{\overline{z} z} = \abs{z}^2$.
The C* identity asks the same of a whole algebra of objects, such as matrices or operators, where $a^*$ plays the role of the complex conjugate.
For a matrix with the operator norm and the conjugate transpose as $a^*$, the identity holds.
Algebras with this property behave like algebras of operators on a Hilbert space, and in fact every one of them is such an algebra.