1000/1000
Hot
Most Recent
In functional analysis and related areas of mathematics, group algebras are constructions that generalize the concept of group ring to some classes of topological groups with the aim to reduce the theory of representations of topological groups to the theory of representations of topological algebras. There are several nonequivalent definitions of group algebra, each of which is considered convenient in a particular situation.
If [math]\displaystyle{ G }[/math] is a finite group then for any field [math]\displaystyle{ K }[/math] the set [math]\displaystyle{ K^G }[/math] of all functions [math]\displaystyle{ u:G\to K }[/math] possesses a natural structure of (finite-dimensional) Hopf algebra over [math]\displaystyle{ K }[/math] with (the usual pointwise summation of functions and multiplication them by scalars, and)
The dual space [math]\displaystyle{ K_G=(K^G)' }[/math] (of all linear functionals [math]\displaystyle{ \alpha:K^G\to K }[/math]) is the dual Hopf algebra, where, in particular, the multiplication [math]\displaystyle{ (\alpha,\beta)\in K_G\times K_G\mapsto \alpha*\beta \in K_G }[/math] can be described by the formula
[math]\displaystyle{ \alpha*\beta(u)=\sum_{s,t\in G}\alpha_s\cdot\beta_t\cdot u(s\cdot t),\quad u\in K^G, }[/math]
or by the formula
[math]\displaystyle{ \alpha*\beta=\sum_{s,t\in G} \alpha_s\cdot\beta_t\cdot\delta_{s\cdot t}, }[/math]
where
[math]\displaystyle{ \delta_t\in K_G, \quad \delta_t(u)=u(t),\quad u\in K^G,\quad t\in G }[/math]
are delta-functionals, and
[math]\displaystyle{ \alpha=\sum_{s\in G} \alpha_s\cdot\delta_s,\quad \beta=\sum_{t\in G} \beta_t\cdot\delta_t }[/math]
are the expansions of [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] along the basis [math]\displaystyle{ \{\delta_t;\ t\in G\} }[/math] in the vector space [math]\displaystyle{ K_G }[/math].
Example: the operation of passage to the delta-functional
[math]\displaystyle{ \delta:G\to K_G,\quad t\in G\mapsto \delta_t\in K_G }[/math]
is a representation of [math]\displaystyle{ G }[/math] in [math]\displaystyle{ K_G }[/math]:
[math]\displaystyle{ \delta_{1_G}=1_{K_G},\quad \delta_{s\cdot t}=\delta_s*\delta_t,\quad s,t\in G. }[/math]
There is a natural correspondence between the representations of the group [math]\displaystyle{ G }[/math] in (unital associative) algebras [math]\displaystyle{ A }[/math] and the homomorphisms of the unital algebras [math]\displaystyle{ \varphi:K_G\to A }[/math]:
This observation has a series of important corollaries, which allow to reduce the theory of representations of finite groups to the theory of representations of finite-dimensional algebras^{[1]}^{[2]}.
The construction of the group algebra [math]\displaystyle{ K_G }[/math] can be easily generalized to arbitrary (not necessarily finite) groups [math]\displaystyle{ G }[/math] (in the purely algebraic sense, without topology) with the same purposes: the generalization is called group ring [math]\displaystyle{ K[G] }[/math], and many results are preserved in this way, including the fact that [math]\displaystyle{ K[G] }[/math] possesses the universal property (and is a Hopf algebra when [math]\displaystyle{ K }[/math] is a field). But up to the recent time the generalizations to the topological groups faced numerous difficulties because of the lack of the convenient categories of topological vector spaces with dualities. The generalizations were mostly constructed in the category of Banach spaces, but the absence of a suitable duality in this category led to various distortions of the properties of these constructions, in particular, they were not Hopf algebras, and even the correspondence between the representations of groups and the homomorphisms of their group algebras was usually violated (however, this correspondence sometimes could be understood in some special sense).
In classical functional analysis, there are several constructions generalizing the purely algebraic idea of group algebra, but preserving only some part of its main properties.
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
The fact that f * g is continuous is immediate from the dominated convergence theorem. Also
where the dot stands for the product in G. 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. 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
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
is dense in H_{π}.
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 only 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.
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 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:
hence the norm is well-defined.
It follows from the definition that 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:
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 acting on L^{2}(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 ℓ^{2}(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.
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 II_{1} factor if and only if G is countable, amenable, and has the infinite conjugacy class property.
The full analogy with the purely algebraic situation appears in the stereotype theory where a series of natural group algebras is constructed including the following four examples.
The representation^{[6]} [math]\displaystyle{ \delta: G\to {\mathcal C}^\star(G) }[/math], [math]\displaystyle{ \delta(x)(u)=u(x) }[/math], [math]\displaystyle{ x\in G }[/math], [math]\displaystyle{ u\in {\mathcal C}^\star(G) }[/math] is called the representation as delta-functionals.
The representations [math]\displaystyle{ \delta: G\to {\mathcal E}^\star(G) }[/math], [math]\displaystyle{ \delta: G\to {\mathcal O}^\star(G) }[/math], [math]\displaystyle{ \delta: G\to {\mathcal P}^\star(G) }[/math], are defined similarly.
The following two results distinguish the stereotype group algebras among the other models of group algebras in analysis.