---
tag: 7W52X4N3NJ
kind: concept
name: Ring with involution
msc: [16W10, 46K05]
specializes: [0B1261B5ED]
sources:
  - url: https://en.wikipedia.org/wiki/*-algebra
same_as:
  wikidata: Q2836000
  wikipedia: "*-algebra"
---

## Summary

A ring with involution, or $*$-ring, is a [[ring|0B1261B5ED]] with an additive map $a \mapsto a^*$ that reverses products and is its own inverse.
It abstracts complex conjugation of numbers and the conjugate transpose of matrices.
When the ring is also an algebra over a commutative $*$-ring and the involution is conjugate-linear, it is a $*$-algebra.
These structures underlie C*-algebras, where the involution is tied to a norm, and the study of Hermitian and unitary elements.

## Explanations

### L1 human

An [[involution|GP56NBN9RY]] is an operation that undoes itself when applied twice.
In a ring with involution this operation also reverses the order of products, as taking inverses does in a group: $(ab)^* = b^* a^*$.
Complex conjugation on $\C$ is the simplest example, and the conjugate transpose of complex matrices is the most important one.
Elements with $a^* = a$, such as real numbers or Hermitian matrices, play the role of "real" elements.