nLab normed algebra

Definition

A normed algebra $A$ over a field $k$ 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 $C\gt 0$ such that

$${\vert a b \vert} \;\leq\; C {\vert a \vert} {\vert b \vert}$$

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

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

Related concepts

• normed division algebra

algebraic structure	group	ring	field	vector space	algebra
(submultiplicative) norm	normed group	normed ring	normed field	normed vector space	normed algebra
multiplicative norm (absolute value/valuation)			valued field
completeness	complete normed group	Banach ring	complete field	Banach vector space	Banach 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