Weak containment and Fell s topology in .NET Generating barcode 39 in .NET Weak containment and Fell s topology

Weak containment and Fell s topology using .net todraw code39 in web,windows application Web app Example F.4.2 For the le ANSI/AIM Code 39 for .

NET ft regular representation G of G, the -representation G : L1 (G) L(L2 (G)) is given by convolution: G (f ) = f , for f L1 (G) and L2 (G).. De ne the universal repr esentation univ of G to be the direct sum of all cyclic unitary representations of G. We have a -representation univ : L1 (G) L(Huniv ) and we de ne the maximal norm of f L1 (G) by f Observe that G (f ) f. = univ ( f ) .. De nition F.4.3 The comp letion of L1 (G) with respect to the norm f f max is a C -algebra called the maximal C -algebra of G, and is denoted by C (G).

Let be a unitary representation of G. As is a direct sum of cyclic representations (Proposition C.4.

9), (f ) f max for all f in L1 (G). Hence, f (f ) extends to a -representation of C (G), also denoted by . In this way, we obtain a one-to-one correspondence between unitary representations of G and non-degenerate -representations of the C algebra C (G).

The notion of weak containment introduced in Section F.1 has the following neat interpretation in terms of C (G). For the proof, see [Dixmi 69, Section 18].

Theorem F.4.4 Let and be unitary representations of G.

Denote by C Ker and C Ker the kernels of the corresponding representations of C (G). The following properties are equivalent: (i) ; (ii) C Ker C Ker ; (iii) (f ) (f ) for all f in L1 (G). Remark F.

4.5 Let Prim(C (G)) be the primitive ideal space of C (G), that is, the set of the kernels C Ker of all irreducible representations of G. The Jacobson topology on Prim(C (G)) is a natural topology de ned as follows: the closure of a subset S of Prim(C (G)) is the set of all C Ker in Prim(C (G)) such that C Ker C Ker .

. De ne the Jacobson topol .net framework USS Code 39 ogy on G to be the inverse image of the mapping : G Prim(C (G)), C Ker . F.5 Direct integrals of unitary representations (that is, the closed sub 3 of 9 barcode for .NET sets in G are the sets 1 (S) where S is closed in Prim(C (G)). It can be shown that Fell s topology on G as de ned in Section F.

2 coincides with Jacobson s topology (see [Dixmi 69, Section 18]). There is another C -algebra one can associate to G. De nition F.

4.6 The norm closure of { G ( f ) : f L1 (G)} in L(H) is a C -algebra called the reduced C -algebra of G, and is denoted by Cred (G)..

The C -algebra Cred (G) can also be described as the completion of L1 (G) with respect to the norm f G ( f ) .. Example F.4.7 Let G be a 3 of 9 barcode for .

NET locally compact abelian group. The Fourier transform F : L2 (G) L2 (G) is a unitary equivalence between G and the unitary representation of G on L2 (G) de ned by ( (x) )(x ) = x(x) ( x ), L2 (G), x G, x G. (see Remark D.1.4).

The representation of L1 (G) in L2 (G) associated to is given, for f in L1 (G), by ( f ) = TF f , where TF f is the multiplication operator by the bounded function Ff . By the Stone Weierstra Theorem, {Ff : f L1 (G)} is a dense subalgebra of C0 (G). Hence, G (f ) TF f extends to an isomorphism between Cred (G) and C0 (G).

The regular representation de nes a surjective -homomorphism. G : C (G) Cred (G .net vs 2010 Code 39 Full ASCII ), so that Cred (G) is a quotient of C (G). Observe that, by the previous theorem, G is an isomorphism if and only if f max = G ( f ) for all f in L1 (G), that is, if and only if every unitary representation of G is weakly contained in G .

We will characterise the groups for which this holds in Appendix G..
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