Ring with involution

Summary

A ring with involution, or $*$-ring, is a ring 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.