"Involution" can refer to a particular type of
map on an
algebra. If
A is an
algebra (roughly, a
vector space equipped with a "sensible"
multiplication -- see
here for an
axiomatic
definition), then a map *: A -> A is called an involution on A when it satisfies the conditions below. Regarding the notation: these maps are usually denoted by superscripts, so that
a* means "star of
a", and is commonly pronounced "
a star". The defining characteristics of an involution are that, for all elements
a and
b of
A, we have:
1. (a+b)* = a* + b*
2. (ka)* = k*a* (here k is an arbitrary scalar, so k* denotes complex conjugation)
3. (ab)* = b*a*
4. (a*)* = a
Some of the most common examples should be mentioned: the
complex numbers form an
algebra with involution given by the complex conjugate; the algebra of square complex
matrices has an involution given by the conjugate
transpose; and the
adjoint map on the algebra of
bounded operators on
Hilbert space is an involution.