Group Algebra

GROUP ALGEBRA OF A FINITE GROUP

Given a Finite Group ''G'', define the group algebra ''CG'' as the Vector Space over the Complex Number s, with Basis Vector s

\{e_g\}

corresponding to the elements

g\in G

. The Algebra structure on this vector space is defined as
:

e_g \cdot e_h = e_{gh}

.
A Representation of the algebra ''CG'' on a vector space ''V'' is the algebra homomorphism
:

\mathbb{C} G
ightarrow \mbox{End} (V)

.
That is, a representation is a Left ''CG''-module . Any group representation

ho:G
ightarrow \mbox{Aut}(V)

then extends linearly to an algebra representation

\overline{
ho}:\mathbb{C}G
ightarrow \mbox{End}(V)

. Thus, representations of the group correspond exactly to representations of the algebra, and so, in a certain sense, talking about the one is the same as talking about the other.

The ''center'' of the group algebra is the set of vectors which commute with the action of the group ''G'' on the vector space ''V'':
:

Z(\mathbb{C} G) := \left\{ z \in \mathbb{C} G \mid vzr = vrz \mbox{ for all } v \in V, z, r \in \mathbb{C} G
ight\}

GROUP ALGEBRAS OF TOPOLOGICAL GROUPS: ''C''<SUB>''C''</SUB>(''G'')

For the purposes of Functional Analysis , and in particular of Harmonic Analysis , one wishes to carry over the group ring construction to Topological Group s ''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 Norm s 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

g (t) = \int_G f(s) g(s^{-1} t)\, d \mu(s) \quad

''g'' is continuous is immediate from the Dominated Convergence Theorem . Also

g) \subseteq \operatorname{Support}(f) \cdot \operatorname{Support}(g)

''C''_''c''(''G'') also has a natural involution defined by:

(s) = \overline{f(s^{-1})} \Delta(s^{-1})

-algebra" class="copylinks">
algebra .

Theorem. If ''C''_''c''(''G'') is given the norm

:<math> \f\ {C^} :

\sup_\pi \\pi(f)\ \quad </math>

_r(G) \cong C^---_\mathrm{max}(G).

-algebra C^---_r(''G'')

-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

:<math> \f\ 2 \sqrt{\int_G f^2 d\mu} \quad </math>

	CATEGORIES ABOUT GROUP ALGEBRA

	ring theory

	representation theory of topological groups

	harmonic analysis

	lie groups

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Information About

Group Algebra