In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra, such that the representations of the operator algebra are related to representations of the group. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a locally compact group is a Topological group G which is locally compact as a Topological space. In Functional analysis, an operator algebra is an algebra of continuous Linear operators on a Topological vector space with the multiplication As such, they are similar to the group ring associated to a discrete group. In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively

Group algebras of topological groups: C_c(G)

For the purposes of functional analysis, and in particular of harmonic analysis, one wishes to carry over the group ring construction to topological groups G. For functional analysis as used in psychology see the Functional analysis (psychology article Harmonic analysis is the branch of Mathematics that studies the representation of functions or signals as the superposition of basic Waves It investigates and generalizes In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In case G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called Haar measure. In Mathematics, a locally compact group is a Topological group G which is locally compact as a Topological space. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of Locally compact topological groups and subsequently define Using the Haar measure, one can define a convolution operation on the space C_c(G) of complex-valued functions on G with compact support; C_c(G) can then be given any of various norms and the completion will be a group algebra. In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In the mathematical area of Order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered

To define the convolution operation, let f and g be two functions in C_c(G). For t in G, define

The fact that f * g is continuous is immediate from the dominated convergence theorem. In Measure theory, a branch of Mathematical analysis, Lebesgue 's dominated convergence theorem provides Sufficient conditions under which two Also

C_c(G) also has a natural involution defined by:

where Δ is the modular function on G. In Mathematics, a modular form is a (complex Analytic function on the Upper half-plane satisfying a certain kind of Functional equation and With this involution, it is a *-algebra. -ring In Mathematics, a *-ring is an Associative ring with a map *: A &rarr A which is an Antiautomorphism

Theorem. If C_c(G) is given the norm

it becomes is an involutive normed algebra with an approximate identity. In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the In Functional analysis, a right approximate identity in a Banach algebra, A, is a net (or a Sequence) \{\e_\lambda

The approximate identity can be indexed on a neighborhood basis of the identity consisting of compact sets. Indeed if V is a compact neighborhood of the identity, let f_V be a non-negative continuous function supported in V such that

Then {f_V}_V is an approximate identity. A group algebra can only have an identity, as opposed to just approximate identity, if and only if the topology on the group is the discrete topology. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated "

Note that for discrete groups, C_c(G) is the same thing as the complex group ring CG.

The importance of the group algebra is that it captures the unitary representation theory of G as shown in the following

Theorem. In Mathematics, a unitary representation of a group G is a Linear representation π of G on a complex Hilbert space Let G be a locally compact group. If U is a strongly continuous unitary representation of G on a Hilbert space H, then

is a non-degenerate bounded *-representation of the normed algebra C_c(G). The map

is a bijection between the set of strongly continuous unitary representations of G and non-degenerate bounded *-representations of C_c(G). This bijection respects unitary equivalence and strong containment. In particular, π_U is irreducible if and only if U is irreducible.

Non-degeneracy of a representation π of C_c(G) on a Hilbert space H_π means that

is dense in H_π.

The convolution algebra L¹(G)

It is a standard theorem of measure theory that the completion of C_c(G) in the L¹(G) norm is isomorphic to the space L¹(G) of functions which are integrable with respect to the Haar measure. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding

Theorem. L¹(G) is a B*-algebra with the convolution product and involution defined above and with the L¹ norm. B*-algebras are mathematical structures studied in Functional analysis. L¹(G) also has an approximate identity.

The group C*-algebra C*(G)

For a locally compact group G, the group C*-algebra of G is defined to be the C*-enveloping algebra of L¹(G). C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. It can also be defined as the completion of C_c(G) with respect to the norm

where π ranges over all non-degenerate *-representations of C_c(G) on Hilbert spaces.

Let C[G] be the group ring of a discrete group G. In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively In Mathematics, a discrete group is a group G equipped with the Discrete topology. It has the following two completions to a C*-algebra.

The maximal group C^*-algebra

The maximal group C*-algebra, C*_max(G) or just C*(G), is defined by the following universal property: any *-homomorphism from C[G] to some B() (the C*-algebra of bounded operators on some Hilbert space ) factors through the inclusion map C[G] C*_max(G). In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, if A is a Subset of B, then the inclusion map (also inclusion function, or canonical injection) is the

The reduced group C*-algebra C^*_r(G)

The reduced group C*-algebra focuses on the left regular representation of G rather than on all unitary representations of G. We thus consider the completion of C_c(G) with respect to the norm

where

is the L² norm. Since the completion of C_c(G) with regard to the L² norm is a Hilbert space, the C^*_r norm is the norm of the bounded operator "convolution by f" acting on L²(G) and thus a C* norm.

The reduced group C*-algebra is isomorphic to the non-reduced group C*-algebra defined above if and only if G is amenable. In Mathematics, an amenable group is a Locally compact Topological group G carrying a kind of averaging operation on bounded functions that is

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. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, an orthonormal basis of an Inner product space V (i 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). In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces The weak closure of this subalgebra, NG, is a von Neumann algebra. In Mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the

The center of NG can be described in terms of those elements of G whose conjugacy class is finite. In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class In particular, if the identity element of G is the only group element with that property (that is, G has the infinite conjugacy class property), the center of NG consists only of complex multiples of the identity. In Mathematics, a group is said to have the infinite conjugacy class property, or to be an icc group, if the Conjugacy class of every group element

NG is isomorphic to the hyperfinite type II₁ factor if and only if G is countable, amenable, and has the infinite conjugacy class property. In Mathematics, there are up to isomorphism exactly two hyperfinite type II factors one infinite and one finite In Mathematics, an amenable group is a Locally compact Topological group G carrying a kind of averaging operation on bounded functions that is

References

• J, Dixmier, C* algebras, ISBN 0-7204-0762-1
• A. A. Kirillov, Elements of the theory of representations, ISBN 0-387-07476-7
• A. I. Shtern (2001), “Group algebra of a locally compact group”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 

This article incorporates material from Group $C^*$-algebra on PlanetMath, which is licensed under the GFDL. The Encyclopaedia of Mathematics is a large reference work in Mathematics. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

• L. H. Loomis, "Abstract Harmonic Analysis", ASIN B0007FUU30