Group algebra of a locally compact group

In functional analysis and related areas of 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.

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

If G 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 t in G, define

[math]\displaystyle{ [f * g](t) = \int_G f(s) g \left (s^{-1} t \right )\, d \mu(s). }[/math]

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

[math]\displaystyle{ \operatorname{Support}(f * g) \subseteq \operatorname{Support}(f) \cdot \operatorname{Support}(g) }[/math]

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

[math]\displaystyle{ f^*(s) = \overline{f(s^{-1})} \, \Delta(s^{-1}) }[/math]

where Δ is the modular function on G. With this involution, it is a *-algebra.

Theorem. With the norm:

[math]\displaystyle{ \|f\|_1 := \int_G |f(s)| \, d\mu(s), }[/math]

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

[math]\displaystyle{ \int_V f_{V}(g)\, d \mu(g) =1. }[/math]

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

[math]\displaystyle{ \pi_U (f) = \int_G f(g) U(g)\, d \mu(g) }[/math]

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

[math]\displaystyle{ U \mapsto \pi_U }[/math]

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

[math]\displaystyle{ \left \{\pi(f) \xi : f \in \operatorname{C}_c(G), \xi \in H_\pi \right \} }[/math]

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 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)

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:

[math]\displaystyle{ \|f\|_{C^*} := \sup_\pi \|\pi(f)\|, }[/math]

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:

[math]\displaystyle{ \|\pi (f)\| \leq \| f \|_1, }[/math]

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:

[math]\displaystyle{ \mathbf{C}[G] \hookrightarrow C^*_{\max}(G). }[/math]

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

[math]\displaystyle{ \|f\|_{C^*_r} := \sup \left \{ \|f*g\|_2: \|g\|_2 = 1 \right \}, }[/math]

where

[math]\displaystyle{ \|f\|_2 = \sqrt{\int_G |f|^2 \, d\mu} }[/math]

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 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 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 operators on ℓ²(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 G is countable, amenable, and has the infinite conjugacy class property.

See also

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

References

• Lang, S. (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 978-1-4613-0041-0. https://www.springer.com/gp/book/9780387953854. 
• Vinberg, E. (10 April 2003) (in en). A Course in Algebra. American Mathematical Society. doi:10.1090/gsm/056. ISBN 978-0-8218-3413-8. https://www.ams.org/books/gsm/056/. 
• Dixmier, Jacques (1982) (in en). C*-algebras. North-Holland. ISBN 978-0-444-86391-1. https://www.google.com/books/edition/C_algebras/P34ZAQAAIAAJ?hl=en&gbpv=0&bsq=C*-Algebras%20Jacques%20Dixmier. 
• Kirillov, Aleksandr A. (1976) (in en). Elements of the Theory of Representations. Springer-Verlag. ISBN 978-3-642-66245-4. https://link.springer.com/book/10.1007/978-3-642-66243-0. 
• Loomis, Lynn H. (19 July 2011) (in English). Introduction to Abstract Harmonic Analysis (Dover Books on Mathematics) by Lynn H. Loomis (2011) Paperback. Dover Publications. ISBN 978-0-486-48123-4. https://www.amazon.com/Introduction-Abstract-Harmonic-Analysis-Mathematics/dp/0486481239. 
• Hazewinkel, Michiel, ed. (2001), "Group algebra of a locally compact group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=G/g045230  This article incorporates material from Group $C^*$-algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.