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.

Read more about Group Algebra: Group Algebras of Topological Groups: *C*_{c}(*G*), The Convolution Algebra *L*1(*G*), The Group C*-algebra C*(*G*), Von Neumann Algebras Associated To Groups

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

... 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 ...

... 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 ...

**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**...

... 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)