Group algebra

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.

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 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

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

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)

where the dot stands for the product in 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. With the norm:

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

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

\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

$\pi_U (f) = \int_G f(g) U(g)\, d \mu(g)$

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

$U \mapsto \pi_U$
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 π of C_c(G) on a Hilbert space H_π means that

\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)

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:

\|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:

\|\pi (f)\| \leq \| f \|_1,

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:

\mathbf {C} [G]\hookrightarrow 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

\|f\|_{C^*_r} := \sup \left \{ \|f*g\|_2: \|g\|_2 = 1 \right \},

where

\|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 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.

Stereotype group algebras

In stereotype theory there is a series of natural group algebras that includes the following four main examples.

On each locally compact group $G$ one can consider the algebra ${\mathcal {C}}(G)$ of all continuous functions $f:G\to {\mathbb {C} }$ with the topology of uniform convergence on compact sets $T\subseteq G$ . The stereotype dual space ${\mathcal {C}}^{\star }(G)$ , which consists of Radon measures with compact support on a locally compact group $G$ , is a stereotype algebra with respect to the operation of convolution:^[1] $\alpha *\beta (f)=\int _{G}\left(\int _{G}f(s\cdot t)\,\alpha (ds)\right)\beta (dt),\quad \alpha ,\beta \in {\mathcal {C}}^{\star }(G),\ f\in {\mathcal {C}}(G).$

The algebra

{\mathcal {C}}^{\star }(G)

is called the stereotype group algebra of measures on the locally compact group

G

.^[2]

On each real Lie group $G$ one can consider the algebra ${\mathcal {E}}(G)$ of all smooth functions $f:G\to {\mathbb {C} }$ with the topology of uniform convergence with all derivatives on compact sets $T\subseteq G$ . The stereotype dual space ${\mathcal {E}}^{\star }(G)$ , which consists of distributions with compact support on $G$ , is a stereotype algebra with respect to the operation of convolution of distributions. The algebra ${\mathcal {E}}^{\star }(G)$ is called the stereotype group algebra of distributions on the real Lie group $G$ .

On each Stein group^[3] $G$ one can consider the algebra ${\mathcal {O}}(G)$ of all holomorphic functions $f:G\to {\mathbb {C} }$ with the topology of uniform convergence on compact sets $T\subseteq G$ . The stereotype dual space ${\mathcal {O}}^{\star }(G)$ , which consists of holomorphic fuhctionals on $G$ , is a stereotype algebra with respect to the operation of convolution of functionals. The algebra ${\mathcal {O}}^{\star }(G)$ is called the stereotype group algebra of analytic functionals on the Stein group $G$ .

On each affine algebraic group $G$ one can consider the algebra ${\mathcal {P}}(G)$ of all polynomials (or regular functions) $f:G\to {\mathbb {C} }$ with the strongest locally convex topology. The stereotype dual space ${\mathcal {P}}^{\star }(G)$ , which consists of currents on $G$ , is a stereotype algebra with respect to the operation of convolution of currents. The algebra ${\mathcal {P}}^{\star }(G)$ is called the stereotype group algebra of currents on the affine algebraic group $G$ .

A map $\pi :G\to A$ of a group $G$ into an associative unital algebra $A$ is called a representation of $G$ in $A$ , if it is a group homomorphism into the group of invertible elements of $A$ , i.e. if it satisfies the following identities:

$\pi (1_{G})=1_{A},\qquad \pi (x\cdot y)=\pi (x)\cdot \pi (y),\qquad x,y\in G.$

The representation $\delta :G\to {\mathcal {C}}^{\star }(G)$ , $\delta (x)(u)=u(x)$ , $x\in G$ , $u\in {\mathcal {C}}^{\star }(G)$ is called the representation as delta-functionals.

The representations $\delta :G\to {\mathcal {E}}^{\star }(G)$ , $\delta :G\to {\mathcal {O}}^{\star }(G)$ , $\delta :G\to {\mathcal {P}}^{\star }(G)$ , are defined similarly.

The following two results distinguish the stereotype group algebras among the other models of group algebras in analysis.

Theorem (universal property).^[4] For any stereotype algebra $A$ the formula

\pi =\varphi \circ \delta

establishes a one-to-one correspondence between

Main property of group algebras.

the continuous representations $\pi :G\to A$ of a locally compact group $G$ in the stereotype algebra $A$ and the morphisms of stereotype algebras $\varphi :{\mathcal {C}}^{\star }(G)\to A$ ,

the smooth^[5] representations $\pi :G\to A$ of a real Lie group $G$ in the stereotype algebra $A$ and the morphisms of stereotype algebras $\varphi :{\mathcal {E}}^{\star }(G)\to A$ ,

the holomorphic^[6] representations $\pi :G\to A$ of a Stein group $G$ in the stereotype algebra $A$ and the morphisms of stereotype algebras $\varphi :{\mathcal {O}}^{\star }(G)\to A$ ,

the polynomial (regular)^[7] representations $\pi :G\to A$ of an affine algebraic group $G$ in the stereotype algebra $A$ and the morphisms of stereotype algebras $\varphi :{\mathcal {P}}^{\star }(G)\to A$ .

Theorem.^[8] The group algebras ${\mathcal {C}}^{\star }(G)$ , ${\mathcal {E}}^{\star }(G)$ , ${\mathcal {O}}^{\star }(G)$ , ${\mathcal {P}}^{\star }(G)$ are Hopf algebras in the monoidal category (Ste, $\circledast$ , ${{\mathbb C}}$ ) of stereotype spaces.

Notes

↑ Akbarov 2003, p. 272.
↑ If $G$ is an infinite locally compact group then the algebra ${\mathcal {C}}^{\star }(G)$ of measures on $G$ is not a Fréchet algebra. In the case when $G$ is compact, ${\mathcal {C}}^{\star }(G)$ is a Smith space. If $G$ is $\sigma$ -compact, then ${\mathcal {C}}^{\star }(G)$ is a Brauner space.
↑ A Stein group is a complex Lie group $G$ which is a Stein manifold.
↑ Akbarov 2003, p. 275.
↑ A map $\xi :M\to X$ of a smooth manifold $M$ into a stereotype space $X$ is said to be smooth if for each functional $f\in X^{\star }$ the composition $f\circ \xi$ is a smooth function on $M$ , and the map $f\in X^{\star }\mapsto f\circ \xi \in {\mathcal {E}}(M)$ is continuous.
↑ A map $\xi :M\to X$ of a Stein manifold $M$ into a stereotype space $X$ is said to be holomorphic if for each functional $f\in X^{\star }$ the composition $f\circ \xi$ is a holomorphic function on $M$ , and the map $f\in X^{\star }\mapsto f\circ \xi \in {\mathcal {O}}(M)$ is continuous.
↑ A map $\xi :M\to X$ of an affine algebraic variety $M$ over ${{\mathbb C}}$ into a stereotype space $X$ is said to be polynomial (or regular) if for each functional $f\in X^{\star }$ the composition $f\circ \xi$ is a polynomial on $M$ , and the map $f\in X^{\star }\mapsto f\circ \xi \in {\mathcal {P}}(M)$ is continuous.
↑ Akbarov 2009, p. 507.

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
A.I. Shtern (2001) [1994], "Group algebra of a locally compact group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 This article incorporates material from Group $C^*$-algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra". Journal of Mathematical Sciences. 113 (2): 179–349. doi:10.1023/A:1020929201133.
Akbarov, S.S. (2009). "Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity". Journal of Mathematical Sciences. 162 (4): 459–586. arXiv:0806.3205. doi:10.1007/s10958-009-9646-1.

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[FOOTNOTEAkbarov2003272-1] Akbarov 2003, p. 272.

[2] If $G$ is an infinite locally compact group then the algebra ${\mathcal {C}}^{\star }(G)$ of measures on $G$ is not a Fréchet algebra. In the case when $G$ is compact, ${\mathcal {C}}^{\star }(G)$ is a Smith space. If $G$ is $\sigma$ -compact, then ${\mathcal {C}}^{\star }(G)$ is a Brauner space.

[Stein-group-3] A Stein group is a complex Lie group $G$ which is a Stein manifold.

[FOOTNOTEAkbarov2003275-4] Akbarov 2003, p. 275.

[smooth-maps-5] A map $\xi :M\to X$ of a smooth manifold $M$ into a stereotype space $X$ is said to be smooth if for each functional $f\in X^{\star }$ the composition $f\circ \xi$ is a smooth function on $M$ , and the map $f\in X^{\star }\mapsto f\circ \xi \in {\mathcal {E}}(M)$ is continuous.

[holomorphic-maps-6] A map $\xi :M\to X$ of a Stein manifold $M$ into a stereotype space $X$ is said to be holomorphic if for each functional $f\in X^{\star }$ the composition $f\circ \xi$ is a holomorphic function on $M$ , and the map $f\in X^{\star }\mapsto f\circ \xi \in {\mathcal {O}}(M)$ is continuous.

[polynomial-maps-7] A map $\xi :M\to X$ of an affine algebraic variety $M$ over ${{\mathbb C}}$ into a stereotype space $X$ is said to be polynomial (or regular) if for each functional $f\in X^{\star }$ the composition $f\circ \xi$ is a polynomial on $M$ , and the map $f\in X^{\star }\mapsto f\circ \xi \in {\mathcal {P}}(M)$ is continuous.

[FOOTNOTEAkbarov2009507-8] Akbarov 2009, p. 507.

Group algebra

Group algebras of topological groups: Cc(G)

The convolution algebra L1(G)

The group C*-algebra C*(G)

The reduced group C*-algebra Cr*(G)