Ring with involution

Remarks

Clause 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 $1$. An element $a$ is self-adjoint or Hermitian if $a^* = a$. Example: $R = M_n(\C)$ with $A^* = \overline{A}^{\,T}$, the conjugate transpose. Clauses 1, 3 and 4 are clear, and clause 2 holds because $(AB)^T = B^T A^T$ and conjugation is multiplicative. Example: the transpose on $M_n(F)$ for a field $F$, and $g \mapsto g^{-1}$ extended linearly on a group ring $\Z[G]$. Non-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 $A, B$. If $R$ is an algebra over a commutative $*$-ring $K$, a $*$-algebra also asks $(\lambda a)^* = \lambda^* a^*$ for $\lambda \in K$.