nLab
normed algebra

Contents

Contents

Definition

A normed algebra AA over a field kk of real or complex numbers is a normed vector space equipped with an associative algebra structure, such that the algebra multiplication is continuous with respect to the norm, i.e. such that there is a positive real number C>0C\gt 0 such that

|ab|C|a||b| {\vert a b \vert} \;\leq\; C {\vert a \vert} {\vert b \vert}

for all a,bAa,b\in A. One can rescale the norm to another norm to get C=1C = 1 (absolute value). A normed algebra whose underlying normed space is complete is called a Banach algebra.

A normed algebra with C=1C = 1 is equivalently a normed division algebra. See there for more.

algebraic structuregroupringfieldvector spacealgebra
(submultiplicative) normnormed groupnormed ringnormed fieldnormed vector spacenormed algebra
multiplicative norm (absolute value/valuation)valued field
completenesscomplete normed groupBanach ringcomplete fieldBanach vector spaceBanach algebra

References

  • M. A. Naimark, Normed rings
  • R. Kadison, Introduction to operator algebras
  • A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988
  • Marco Thill, Introduction to normed *\ast-algebras and their representations, Kassel Univ. Press, pdf
  • I. M. Gel'fand, M. A. Naimark, Normed rings with involutions and their representations, Izv. Akad. Nauk SSSR Ser. Mat., 12:5 (1948), 445–480, Russian orig. pdf

Last revised on December 2, 2018 at 10:01:54. See the history of this page for a list of all contributions to it.