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  "body": "\n## Summary\n\nA ring with involution, or $*$-ring, is a [[ring|0B1261B5ED]] with an additive map $a \\mapsto a^*$ that reverses products and is its own inverse.\nIt abstracts complex conjugation of numbers and the conjugate transpose of matrices.\nWhen the ring is also an algebra over a commutative $*$-ring and the involution is conjugate-linear, it is a $*$-algebra.\nThese structures underlie C*-algebras, where the involution is tied to a norm, and the study of Hermitian and unitary elements.\n\n## Explanations\n\n### L1 human\n\nAn [[involution|GP56NBN9RY]] is an operation that undoes itself when applied twice.\nIn a ring with involution this operation also reverses the order of products, as taking inverses does in a group: $(ab)^* = b^* a^*$.\nComplex conjugation on $\\C$ is the simplest example, and the conjugate transpose of complex matrices is the most important one.\nElements with $a^* = a$, such as real numbers or Hermitian matrices, play the role of \"real\" elements.\n",
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