TOPICS

# Group Algebra

The group algebra , where is a field and a group with the operation , is the set of all linear combinations of finitely many elements of with coefficients in , hence of all elements of the form

 (1)

where and for all . This element can be denoted in general by

 (2)

where it is assumed that for all but finitely many elements of .

is an algebra over with respect to the addition defined by the rule

 (3)

the product by a scalar given by

 (4)

and the multiplication

 (5)

From this definition, it follows that the identity element of is the unit of , and that is commutative iff is an Abelian group.

If the field is replaced by a unit ring , the addition and the multiplication defined above yield the group ring .

If , and is the usual addition of integers, the group ring is isomorphic to the ring formed by all sums

 (6)

where are integers, and for all indices .

Let be a locally compact group and be a left invariant Haar measure on . Then the Banach space under the product given by the convolution for is a commutative Banach algebra that is called the group algebra of .

Algebra, Group, Semigroup Algebra

Portions of this entry contributed by Margherita Barile

Portions of this entry contributed by Mohammad Sal Moslehian

## Explore with Wolfram|Alpha

More things to try:

## References

Bonsall, F. F. and Duncan, J. Complete Normed Algebras. New York: Springer-Verlag, 1973.

Group Algebra

## Cite this as:

Barile, Margherita; Moslehian, Mohammad Sal; and Weisstein, Eric W. "Group Algebra." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GroupAlgebra.html