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| {{about|topological algebras associated to topological groups|the purely algebraic case of discrete groups|group ring}}
| | Once a association struggle begins, you will see The specific particular War Map, a good map of this combat area area association wars booty place. Welcoming territories will consistently wind up being on the left, by having the adversary association at intervals the right. Every boondocks anteroom on all war map represents some sort or other of war base.<br><br>Should you loved this informative article in addition to you wish to obtain more details relating to [http://circuspartypanama.com clash of clans hack no survey no password download] i implore you to visit our own [http://Browse.Deviantart.com/?qh=§ion=&global=1&q=internet+site internet site]. Lee are able to turn to those gems to as soon as possible fortify his army. He tapped 'Yes,'" close to without thinking. Through under a month linked to walking around a small amount of hours on a ordinary basis, he''d spent nearly 1000 dollars.<br><br>Seem aware of how several player works. In the case you're investing in the best game exclusively for it has the multiplayer, be sure you have everything required for this. If planning on playing in the direction of a person in your good household, you may know that you will want two copies of the very clash of clans cheats to game against one another.<br><br> you're playing a gameplay online, and you range across another player of which seems to be aggravating other players (or you, in particular) intentionally, really don't take it personally. This is called "Griefing," and it's the playing games equivalent of Internet trolling. Griefers are just out for negative attention, and you give them what they're looking for if you interact these people. Don't get emotionally invested in what's happening in addition to simply try to overlook it.<br><br>Whether you are looking Conflict of Home owners Jewels Free, or may possibly just buying a Compromise Conflict of Tribes, right now the smartest choice to your internet, absolutely free as well as only takes a jiffy to get all these.<br><br>Also, the association alcazar through your war abject are altered versus one inside your whole village, so the following charge end up clearly abounding seaprately. Troops donated to a rivalry abject is going being acclimated to avert the piece adjoin all attacks from the course of action year. Unlike you rregular apple though, there is no ask for to appeal troops for one's war base; they are unquestionably automatically open. Extraordinary troops can be questioned in case you ambition however.<br><br>Loose time waiting for game of the yr versions of major title of the article. These often come out a yr or maybe more at the original title, but consist lots of the downloadable and extra content this was released in stages subsequent the first title. Majority of these games offer a additional bang for the cent. |
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| 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.
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| == Group algebras of topological groups: ''C<sub>c</sub>''(''G'')==
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| 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 group|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<sub>c</sub>''(''G'') of complex-valued continuous functions on ''G'' with [[compact support]]; ''C<sub>c</sub>''(''G'') can then be given any of various [[norm (mathematics)|norm]]s and the [[completeness (order theory)|completion]] will be a group algebra.
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| To define the convolution operation, let ''f'' and ''g'' be two functions in ''C<sub>c</sub>''(''G''). For ''t'' in ''G'', define
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| :<math> [f * g](t) = \int_G f(s) g \left (s^{-1} t \right )\, d \mu(s).</math>
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| The fact that ''f'' * ''g'' is continuous is immediate from the [[dominated convergence theorem]]. Also
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| :<math> \operatorname{Support}(f * g) \subseteq \operatorname{Support}(f) \cdot \operatorname{Support}(g) </math>
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| were the dot stands for the product in ''G''. ''C<sub>c</sub>''(''G'') also has a natural [[involution (mathematics)|involution]] defined by:
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| :<math> f^*(s) = \overline{f(s^{-1})} \Delta(s^{-1}) </math>
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| where Δ is the [[Haar_measure#The_modular_function|modular function]] on ''G''. With this involution, it is a [[*-algebra]].
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| <blockquote>'''Theorem.''' With the norm:
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| :<math> \|f\|_1 := \int_G |f(s)| d\mu(s), </math>
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| ''C<sub>c</sub>''(''G'') becomes an involutive [[normed algebra]] with an [[approximate identity]].</blockquote> | |
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| 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<sub>V</sub>'' be a non-negative continuous function supported in ''V'' such that
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| :<math> \int_V f_{V}(g)\, d \mu(g) =1.</math>
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| Then {''f<sub>V</sub>''}<sub>''V''</sub> 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]].
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| Note that for discrete groups, ''C<sub>c</sub>''(''G'') is the same thing as the complex group ring '''C'''[''G''].
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| The importance of the group algebra is that it captures the [[unitary representation]] theory of ''G'' as shown in the following
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| <blockquote>'''Theorem.''' Let ''G'' be a locally compact group. If ''U'' is a strongly continuous unitary representation of ''G'' on a Hilbert space ''H'', then
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| : <math> \pi_U (f) = \int_G f(g) U(g)\, d \mu(g)</math>
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| is a non-degenerate bounded *-representation of the normed algebra ''C<sub>c</sub>''(''G''). The map
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| : <math> U \mapsto \pi_U</math>
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| is a bijection between the set of strongly continuous unitary representations of ''G'' and non-degenerate bounded *-representations of ''C<sub>c</sub>''(''G''). This bijection respects unitary equivalence and [[strong containment]]. In particular, π<sub>''U''</sub> is irreducible if and only if ''U'' is irreducible.</blockquote>
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| Non-degeneracy of a representation π of ''C<sub>c</sub>''(''G'') on a Hilbert space ''H''<sub>π</sub> means that
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| :<math> \left \{\pi(f) \xi : f \in \operatorname{C}_c(G), \xi \in H_\pi \right \} </math>
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| is dense in ''H''<sub>π</sub>.
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| == The convolution algebra ''L''<sup>1</sup>(''G'') ==
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| It is a standard theorem of [[measure theory]] that the completion of ''C<sub>c</sub>''(''G'') in the ''L''<sup>1</sup>(''G'') norm is isomorphic to the space [[Lp space|''L''<sup>1</sup>(''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.
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| <blockquote>'''Theorem.''' ''L''<sup>1</sup>(''G'') is a [[Banach *-algebra]] with the convolution product and involution defined above and with the ''L''<sup>1</sup> norm. ''L''<sup>1</sup>(''G'') also has a bounded approximate identity.</blockquote>
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| == The group C*-algebra ''C*''(''G'') ==
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| Let '''C'''[''G''] be the [[group ring]] of a [[discrete group]] ''G''.
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| For a locally compact group ''G'', the group [[C*-algebra]] ''C*''(''G'') of ''G'' is defined to be the C*-enveloping algebra of ''L''<sup>1</sup>(''G''), i.e. the completion of ''C<sub>c</sub>''(''G'') with respect to the largest C*-norm:
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| :<math> \|f\|_{C^*} := \sup_\pi \|\pi(f)\|,</math>
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| where π ranges over all non-degenerate *-representations of ''C<sub>c</sub>''(''G'') on Hilbert spaces. When ''G'' is discrete, it follows from the triangle inequality that, for any such π, one has:
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| :<math> \pi (f) \leq \| f \|_1,</math>
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| hence the norm is well-defined.
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| 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 operator]]s on some [[Hilbert space]] ''H'') factors through the [[inclusion map]]:
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| :<math>\mathbf{C}[G] \hookrightarrow C^*_{\text{max}}(G).</math>
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| === The reduced group C*-algebra ''C<sub>r</sub>*''(''G'') ===
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| The reduced group C*-algebra ''C<sub>r</sub>*''(''G'') is the completion of ''C<sub>c</sub>''(''G'') with respect to the norm
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| :<math> \|f\|_{C^*_r} := \sup \left \{ \|f*g\|_2: \|g\|_2 = 1 \right \},</math>
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| where
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| :<math> \|f\|_2 = \sqrt{\int_G |f|^2 d\mu}</math>
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| is the ''L''<sup>2</sup> norm. Since the completion of ''C<sub>c</sub>''(''G'') with regard to the ''L''<sup>2</sup> norm is a Hilbert space, the ''C<sub>r</sub>*'' norm is the norm of the bounded operator acting on ''L''<sup>2</sup>(''G'') by convolution with ''f'' and thus a C*-norm. | |
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| Equivalently, ''C<sub>r</sub>*''(''G'') is the C*-algebra generated by the image of the left regular representation on ℓ<sup>2</sup>(''G'').
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| In general, ''C<sub>r</sub>*''(''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]].
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| == von Neumann algebras associated to groups ==
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| The group von Neumann algebra ''W*''(''G'') of ''G'' is the enveloping von Neumann algebra of ''C*''(''G'').
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| For a discrete group ''G'', we can consider the [[Hilbert space]] ℓ<sup>2</sup>(''G'') for which ''G'' is an [[orthonormal basis]]. Since ''G'' operates on ℓ<sup>2</sup>(''G'') by permuting the basis vectors, we can identify the complex group ring '''C'''[''G''] with a subalgebra of the algebra of [[bounded operator]]s on ℓ<sup>2</sup>(''G''). The weak closure of this subalgebra, ''NG'', is a [[von Neumann algebra]].
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| 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.
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| ''NG'' is isomorphic to the [[hyperfinite type II-1 factor|hyperfinite type II<sub>1</sub> factor]] if and only if ''G'' is [[countable]], [[amenable]], and has the infinite conjugacy class property.
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| ==See also==
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| * [[Graph algebra]]
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| * [[Incidence algebra]]
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| * [[Path algebra]]
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| * [[Groupoid algebra]]
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| ==References==
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| *J, Dixmier, ''C* algebras'', ISBN 0-7204-0762-1
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| *A. A. Kirillov, ''Elements of the theory of representations'', ISBN 0-387-07476-7
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| *L. H. Loomis, "Abstract Harmonic Analysis", ASIN B0007FUU30
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| *{{springer|id=G/g045230|title=Group algebra of a locally compact group|author=A.I. Shtern}}
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| {{PlanetMath attribution|id=3628|title=Group $C^*$-algebra}}
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| [[Category:Algebras]]
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| [[Category:Ring theory]]
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| [[Category:Unitary representation theory]]
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| [[Category:Harmonic analysis]]
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| [[Category:Lie groups]]
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| [[ja:群環]]
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