GROUP ALGEBRA

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. As such, they are similar to the group ring associated to a discrete group.

Contents

Group algebras of topological groups: ''C''''c''(''G'')

The convolution algebra ''L''1(''G'')

The group C ★ -algebra C ★ (''G'')

The maximal group C ★ -algebra

The reduced group C ★ -algebra C ★ r(''G'')

von Neumann algebras associated to groups

References

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 left-invariant 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 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.
To define the convolution operation, let ''f'' and ''g'' be two functions in ''C''_''c''(''G''). For ''t'' in ''G'', define
:$[f$
★ g](t) = int_G f(s) g(s^{-1} t), d mu(s) quad
The fact that ''f''
★ ''g'' is continuous is immediate from the dominated convergence theorem. Also
:$operatorname\{Support\}(f$
★ g) subseteq operatorname{Support}(f) cdot operatorname{Support}(g)
''C''_''c''(''G'') also has a natural involution defined by:
:$f^$
★ (s) = overline{f(s^{-1})} Delta(s^{-1})
where Δ is the modular function on ''G''. With this involution, it is a
★ -algebra.
'Theorem'. If ''C''_''c''(''G'') is given the norm
:$|f|\_1\; :=\; int\_G\; |f(s)|\; dmu(s),\; quad$ 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 non-negative continuous function supported in ''V'' such that
:$int\_V\; f\_\{V\}(g),\; d\; mu(g)\; =1.\; quad$
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 '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
: $pi\_U\; (f)\; =\; int\_G\; f(g)\; U(g),\; d\; mu(g)\; quad$
is a non-degenerate bounded
★ -representation of the normed algebra ''C''_''c''(''G''). The map
: $U\; mapsto\; pi\_U\; quad$
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
:$\{pi(f)\; xi:\; f\; in\; operatorname\{C\}\_c(G),\; xi\; in\; H\_pi\; \}$
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 functions which are integrable with respect to the Haar measure.
'Theorem'. ''L''¹(''G'') is a B
★ -algebra with the convolution product and involution defined above and with the ''L''¹ norm. ''L''¹(''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 ''L''¹(''G''). It can also be defined as the completion of ''C''_''c''(''G'') with respect to the norm
:$|f|\_\{C^$
★ } := sup_pi |pi(f)| quad
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 $G.$ 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'($mathcal\{H\}$) (the ''C''
★ -algebra of bounded operators on some Hilbert space $mathcal\{H\}$) factors through the inclusion map 'C'[''G''] $hookrightarrow$ ''C''
★ _max(''G'').

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
:$|f|\_\{C^$
★ _r} := sup { |f
★ g|_2: |g|_2 = 1}, quad
where
:$|f|\_2\; =\; sqrt\{int\_G\; |f|^2\; dmu\}\; quad$
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 ''L''²(''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.

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 'C'''G'' 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 II₁ factor if and only if ''G'' is countable, amenable, and has the infinite conjugacy class property.

References

★ J, Dixmier, ''C
★ algebras'', ISBN 0-7204-0762-1

★ A. A. Kirillov, ''Elements of the theory of representations'', ISBN 0-387-07476-7

★

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