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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] with unit $1$.\nAn *involution* on $R$ is a map $R \\to R$, $a \\mapsto a^*$, such that for all $a, b \\in R$:\n\n1. $(a + b)^* = a^* + b^*$.\n2. $(ab)^* = b^* a^*$.\n3. $1^* = 1$.\n4. $(a^*)^* = a$.\n\nA *ring with involution*, or *$*$-ring*, is a ring together with an involution.\n\n## Remarks\n\nClause 3 follows from the others: $1^* a = (a^* \\cdot 1)^* = a$ and likewise $a 1^* = a$ for all $a$, so $1^*$ is a unit element and equals $1$.\nAn element $a$ is *self-adjoint* or *Hermitian* if $a^* = a$.\nExample: $R = M_n(\\C)$ with $A^* = \\overline{A}^{\\,T}$, the [[conjugate transpose|0EX045E2BP]].\nClauses 1, 3 and 4 are clear, and clause 2 holds because $(AB)^T = B^T A^T$ and conjugation is multiplicative.\nExample: the transpose on $M_n(F)$ for a field $F$, and $g \\mapsto g^{-1}$ extended linearly on a group ring $\\Z[G]$.\nNon-example: the identity map on $M_2(\\R)$ satisfies clauses 1, 3 and 4 but fails clause 2, since $AB \\ne BA$ for some matrices $A, B$.\nIf $R$ is an algebra over a commutative $*$-ring $K$, a *$*$-algebra* also asks $(\\lambda a)^* = \\lambda^* a^*$ for $\\lambda \\in K$.\n",
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