nLab
real part

Contents

Contents

Definition

Given a star-algebra (A,*)(A ,\ast) in characteristic zero, then for aAa\in A any element

  • its real part is Re(a)12(a+a *)Re(a) \coloneqq \frac{1}{2}(a + a^\ast);

  • its imaginary part is Im(a)12(aa *)Im(a) \coloneqq \frac{1}{2}(a - a^\ast),

where () *(-)^\ast is the given star-anti-involution (e.g. complex conjugation, quaternionic conjugation, …).

Examples

Standard situations where real and imaginary parts are considered include the real normed division algebra

Last revised on January 26, 2021 at 07:19:33. See the history of this page for a list of all contributions to it.