Group algebra

This page discusses topological algebras associated to topological groups; for the purely algebraic case of discrete groups see group ring.
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.
Contents
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. In case G is a locally compact Hausdorff group, G carries an essentially unique leftinvariant countably additive Borel measure μ called Haar measure. Using the Haar measure, one can define a convolution operation on the space C_{c}(G) of complexvalued continuous 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.
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. Also
C_{c}(G) also has a natural involution defined by:
where Δ is the modular function on G. With this involution, it is a *algebra.
Theorem. If C_{c}(G) is given the norm
 it becomes is an involutive normed algebra with an approximate identity.
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 nonnegative 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.
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. 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 nondegenerate 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 nondegenerate 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.
Nondegeneracy of a representation π of C_{c}(G) on a Hilbert space H_{π} means that
is dense in H_{π}.
The convolution algebra L^{1}(G)
It is a standard theorem of measure theory that the completion of C_{c}(G) in the L^{1}(G) norm is isomorphic to the space L^{1}(G) of equivalence classes of functions which are integrable with respect to the Haar measure, where, as usual, two functions are regarded as equivalent if and only if they differ on a set of Haar measure zero.
Theorem. L^{1}(G) is a Banach *algebra with the convolution product and involution defined above and with the L^{1} norm. L^{1}(G) also has a bounded approximate identity.
The group C*algebra C*(G)
Let C[G] be the group ring of a discrete group G.
For a locally compact group G, the group C*algebra C*(G) of G is defined to be the C*enveloping algebra of L^{1}(G), i.e. the completion of C_{c}(G) with respect to the largest C*norm:
where π ranges over all nondegenerate *representations of C_{c}(G) on Hilbert spaces. When G is discrete, it follows from the triangle inequality that, for any such π, π(f) ≤ f_{1}. So the norm is welldefined.
It follows from the definition that C*(G) has 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).
The reduced group C*algebra C^{*}_{r}(G)
The reduced group C*algebra C^{*}_{r}(G) is the completion of C_{c}(G) with respect to the norm
where
is the L^{2} norm. Since the completion of C_{c}(G) with regard to the L^{2} norm is a Hilbert space, the C^{*}_{r} norm is the norm of the bounded operator "convolution by f" acting on L^{2}(G) and thus a C*norm.
Equivalently, C^{*}_{r}(G) is the C*algebra generated by the image of the left regular representation on l^{2}(G).
In general, C^{*}_{r}(G) is a quotient of C^{*}(G). The reduced group C*algebra is isomorphic to the nonreduced group C*algebra defined above if and only if G is amenable.
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 l^{2}(G) for which G is an orthonormal basis. Since G operates on l^{2}(G) by permuting the basis vectors, we can identify the complex group ring CG with a subalgebra of the algebra of bounded operators on l^{2}(G). The weak closure of this subalgebra, NG, is a von Neumann algebra.
The center of NG can be described in terms of those elements of G whose conjugacy class is finite. 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.
NG is isomorphic to the hyperfinite type II_{1} factor if and only if G is countable, amenable, and has the infinite conjugacy class property.
See also
 Graph algebra
 Incidence algebra
 Path algebra
 Group ring
References
 J, Dixmier, C* algebras, ISBN 0720407621
 A. A. Kirillov, Elements of the theory of representations, ISBN 0387074767
 A.I. Shtern (2001), "Group algebra of a locally compact group", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/G/g045230.htm
This article incorporates material from Group $C^*$algebra on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
 L. H. Loomis, "Abstract Harmonic Analysis", ASIN B0007FUU30
Categories: Algebras
 Ring theory
 Unitary representation theory
 Harmonic analysis
 Lie groups
Wikimedia Foundation. 2010.
Look at other dictionaries:
group algebra — grupių algebra statusas T sritis fizika atitikmenys: angl. group algebra vok. Gruppenalgebra, f rus. груповая алгебра, f pranc. algèbre de groupe, f … Fizikos terminų žodynas
Group ring — This page discusses the algebraic group ring of a discrete group; for the case of a topological group see group algebra, and for a general group see Group Hopf algebra. In algebra, a group ring is a free module and at the same time a ring,… … Wikipedia
Group (mathematics) — This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines … Wikipedia
Group cohomology — This article is about homology and cohomology of a group. For homology or cohomology groups of a space or other object, see Homology (mathematics). In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well… … Wikipedia
Algebra tiles — Algebra tiles are known as mathematical manipulatives that allow students to better understand ways of algebraic thinking and the concepts of algebra. These tiles have proven to provide concrete models for elementary school, middle school, high … Wikipedia
Group theory — is a mathematical discipline, the part of abstract algebra that studies the algebraic structures known as groups. The development of group theory sprang from three main sources: number theory, theory of algebraic equations, and geometry. The… … Wikipedia
algebra — /al jeuh breuh/, n. 1. the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc., in the description of such relations. 2. any of… … Universalium
Algebra — This article is about the branch of mathematics. For other uses, see Algebra (disambiguation). Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from… … Wikipedia
Group Hopf algebra — In mathematics, the group Hopf algebra of a given group is a certain construct related to the symmetries of group actions. Deformations of group Hopf algebras are foundational in the theory of quantum groups. DefinitionLet G be an arbitrary group … Wikipedia
algebra, modern — ▪ mathematics Introduction also called abstract algebra branch of mathematics concerned with the general algebraic structure of various sets (such as real numbers (real number), complex numbers (complex number), matrices (matrix), and… … Universalium