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Group algebra of a locally compact group

Infunctional analysis and related areas of mathematics, the group algebra is any of various constructions to assign to a locally compact groupanoperator 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.

The algebra C_c(G) of continuous functions with compact support[edit]

IfG is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called a Haar measure. Using the Haar measure, one can define a convolution operation on the space C_c(G) of complex-valued 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 tinG, define

${\displaystyle [f*g]($t)=\int _{G}f(s)g\left(s^{-1}t\right)\,d\mu (s).}

The fact that ${\displaystyle f*g}$ is continuous is immediate from the dominated convergence theorem. Also

${\displaystyle \operatorname {Support} (f*g)\subseteq \operatorname {Support} ($f)\cdot \operatorname {Support} (g)}

where the dot stands for the product in G. C_c(G) also has a natural involution defined by:

${\displaystyle f^{*}($s)={\overline {f(s^{-1})}}\,\Delta (s^{-1})}

where Δ is the modular functiononG. With this involution, it is a *-algebra.

Theorem. With the norm:

${\displaystyle \|f\|_{1}:=\int _{G}|f($s)|\,d\mu (s),}

C_c(G) becomes 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 non-negative continuous function supported in V such that

${\displaystyle \int _{V}f_{V}($g)\,d\mu (g)=1.}

Then {f_V}_V is an approximate identity. A group algebra has an identity, as opposed to just an 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 C[G].

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

${\displaystyle \pi _{U}($f)=\int _{G}f(g)U(g)\,d\mu (g)}

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

${\displaystyle U\mapsto \pi _{U}}$

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 πofC_c(G) on a Hilbert space H_π means that

${\displaystyle \left\{\pi ($f)\xi :f\in \operatorname {C} _{c}(G),\xi \in H_{\pi }\right\}}

is dense in H_π.

The convolution algebra L¹(G)[edit]

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 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 only on a set of Haar measure zero.

Theorem. L¹(G) is a Banach *-algebra with the convolution product and involution defined above and with the L¹ norm. L¹(G) also has a bounded approximate identity.

The group C*-algebra C*(G)[edit]

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 L¹(G), i.e. the completion of C_c(G) with respect to the largest C*-norm:

${\displaystyle \|f\|_{C^{*}}:=\sup _{\pi }\|\pi ($f)\|,}

where π ranges over all non-degenerate *-representations of C_c(G) on Hilbert spaces. When G is discrete, it follows from the triangle inequality that, for any such π, one has:

${\displaystyle \|\pi ($f)\|\leq \|f\|_{1},}

hence the norm is well-defined.

It follows from the definition that, when G is a discrete group, C*(G) has the following universal property: any *-homomorphism from C[G] to some B(H) (the C*-algebra of bounded operators on some Hilbert space H) factors through the inclusion map:

${\displaystyle \mathbf {C} [$G]\hookrightarrow C_{\max }^{*}(G).}

The reduced group C*-algebra C_r*(G)[edit]

The reduced group C*-algebra C_r*(G) is the completion of C_c(G) with respect to the norm

${\displaystyle \|f\|_{C_{r}^{*}}:=\sup \left\{\|f*g\|_{2}:\|g\|_{2}=1\right\},}$

where

${\displaystyle \|f\|_{2}={\sqrt {\int _{G}|f|^{2}\,d\mu }}}$

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 acting on L²(G) by convolution with f and thus a C*-norm.

Equivalently, C_r*(G) is the C*-algebra generated by the image of the left regular representation on ℓ²(G).

In general, C_r*(G) is a quotient of C*(G). The reduced group C*-algebra is isomorphic to the non-reduced group C*-algebra defined above if and only if Gisamenable.

von Neumann algebras associated to groups[edit]

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 C[G] with a subalgebra of the algebra of bounded operatorsonℓ²(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 II₁ factor if and only if Giscountable, amenable, and has the infinite conjugacy class property.

See also[edit]

• Graph algebra
• Incidence algebra
• Hecke algebra of a locally compact group
• Path algebra
• Groupoid algebra
• Stereotype algebra
• Stereotype group algebra
• Hopf algebra

Notes[edit]

References[edit]

• Lang, S. (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 978-1-4613-0041-0.
• Vinberg, E. (10 April 2003). A Course in Algebra. American Mathematical Society. doi:10.1090/gsm/056. ISBN 978-0-8218-3413-8.
• Dixmier, Jacques (1982). C*-algebras. North-Holland. ISBN 978-0-444-86391-1.
• Kirillov, Aleksandr A. (1976). Elements of the Theory of Representations. Springer-Verlag. ISBN 978-3-642-66245-4.
• Loomis, Lynn H. (19 July 2011). Introduction to Abstract Harmonic Analysis (Dover Books on Mathematics) by Lynn H. Loomis (2011) Paperback. Dover Publications. ISBN 978-0-486-48123-4.
• A.I. Shtern (2001) [1994], "Group algebra of a locally compact group", Encyclopedia of Mathematics, EMS Press This article incorporates material from Group $C^*$-algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.