# Group Algebra

In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that representations of the algebra are related to representations of the group. As such, they are similar to the group ring associated to a discrete group.

### Other articles related to "groups, group, group algebra, algebra":

Maschke's Theorem - Reformulation and The Meaning
... One of the approaches to representations of finite groups is through module theory ... Representations of a group G are replaced by modules over its group algebra K ... Let G be a finite group and K a field whose characteristic does not divide the order of G ...
Faithful Representation
... In mathematics, especially in the area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear ... In more abstract language, this means that the group homomorphism ρ G → GL(V) is injective ... de facto the same as -modules (with denoting the group algebra of the group G), a faithful representation of G is not necessarily a faithful module for the ...
Pontryagin Duality and The Fourier Transform - The Group Algebra
... The space of integrable functions on a locally compact abelian group G is an algebra, where multiplication is convolution if f, g are integrable functions then the ... This algebra is referred to as the Group Algebra of G ... convolution is submultiplicative with respect to the L1 norm, making L1(G) a Banach algebra ...
Group Hopf Algebra - Definition
... Let G be an arbitrary group and k a field ... The group Hopf algebra of G over k, denoted kG (or k), is as a set (and vector space) the free vector space on G over k ... As an algebra, its product is defined by linear extension of the group composition in G, with multiplicative unit the identity in G this product is also known as convolution ...
Group Algebra - Von Neumann Algebras Associated To Groups
... The group von Neumann algebra W*(G) of G is the enveloping von Neumann algebra of C*(G) ... For a discrete group G, we can consider the Hilbert space l2(G) for which G is an orthonormal basis ... Since G operates on l2(G) by permuting the basis vectors, we can identify the complex group ring CG with a subalgebra of the algebra of bounded operators on l2(G) ...

### Famous quotes containing the words algebra and/or group:

Poetry has become the higher algebra of metaphors.
José Ortega Y Gasset (1883–1955)

Just as a person who is always asserting that he is too good-natured is the very one from whom to expect, on some occasion, the coldest and most unconcerned cruelty, so when any group sees itself as the bearer of civilization this very belief will betray it into behaving barbarously at the first opportunity.
Simone Weil (1910–1943)