Group Algebra

The group algebra , where is a field and a group with the operation , is the set of all formal 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 Banach algebra that is called the group algebra of .

For the algebraic group algebra over a field , the Kaplansky zero-divisor conjecture asks whether has no nonzero zero divisors when is a torsion-free group. Ivanov (2026) reports counterexamples in every positive field characteristic, using an external characteristic-two construction and a new odd-characteristic extension. The inherited construction has not been independently verified here. Ivanov (2026) discloses substantial Claude assistance with arguments and checks.