F.3 Continuity of operations in .NET Draw USS Code 39 in .NET F.3 Continuity of operations

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F.3 Continuity of operations using none todraw none with web,windows application Developing with Visual Studio .NET Proof Let K and K none for none be the Hilbert spaces of and . Let = Ind G and H = Ind G , and denote by H and H their Hilbert spaces. For f Cc (G) H and v K , recall that f ,v denotes the element from H de ned by f ,v (x) = By Lemma E.

1.3, the set { f ,v : f Cc (G), v K } is total in H . Hence, by Lemma F.

1.3, it suf ces to show that the functions of positive type of the form ( ) f ,v , f ,v can be approximated by functions of positive type associated to . Let f Cc (G) and v K .

Choose a quasi-invariant measure on G/H dg and set c (g, xH ) = (xH ) as in Section A.6. We have d (g) f ,v , f ,v = =.

f (xh) (h)vdh,. x G. c (g 1 , xH )1/2 f ,v (g 1 x), f ,v (x) d (xH ) c (g 1 , xH )1/2 f (g 1 xh). f (xk) (k 1 h) none none v, v dhdkd (xH ), for every g G. Fix a compact subset Q of G and a positive number > 0. Denote by K the support of f .

Set L = (K 1 QK) H , which is a compact subset of H . Since , there exists w1 , . .

. , wn K such that. sup (h)v, v h L i=1 (h)wi , wi < .. For g Q, we have (g) f ,v , f ,v (g) f ,wi , f ,wi f (g 1 xh)f (xk)D(k 1 h)dhdkd (xH ),. H H n c (g 1 , xH )1/2. where D(k h) = (k h)v, v (k 1 h)wi , wi . Weak containment and Fell s topology Observe that f (g 1 xh)f (xk) = 0 unless g 1 xh K and xk K; these inclusions imply xh QK and k 1 x 1 K 1 , and therefore also k 1 h (K 1 QK) H = L. Hence,. (g) f ,v , f ,v (g) f ,wi , f ,wi f (g 1 xh). dh c (g 1 , xH )1/2. f (xk). dkd (xH ). c (g 1 , xH )1/2 (TH f . )(g 1 xH )(TH f . )(xH )d (xH ).. By the Cauchy Schwarz inequality, we have nally n g Q sup (g) f ,v , f ,v G/H i=1 (g) f ,wi , f ,wi 1/2. c (g 1 , xH )((TH f . )(g 1 xH ))2 d (xH ) 1/2 ((TH f . )(xH ))2 d (xH ) . = TH f . 2 , 2 so that is none for none weakly contained in . Example F.3.

6 Let t be the non-spherical principal series representation of G = SL2 (R), as in Example E.1.8.

Since t = Ind G t and limt 0 t = 0 , P where a b t = sgn(a). a. it , 0 a 1 we have limt 0 t = 0 = Ind G 0 . P It is known that all t are irreducible for t = 0 and that 0 = + for two irreducible representations + and , the so-called mock discrete series representations (see [Knapp 86, s II and VII]). It follows that limt 0 t = + and limt 0 t = + .

In particular, G is not a Hausdorff space.. F.4 The C -algebras of a locally compact group F.4 The C* -algebras of a locally compact group The study of the un none none itary representations of a locally compact group can be cast into the general framework of C -algebras. An overall reference for what follows is [Dixmi 69]. A Banach -algebra A is called a C -algebra if the norm on A satis es ( ) x x = x 2 , for all x A.

. Example F.4.1 (i) F or any locally compact space X , the algebra C0 (X ), with the obvious operations and the uniform norm, is a C -algebra.

(ii) If A is a commutative C -algebra, then the Gelfand transform (see D.1) is an isometric *-isomorphism between A and C0 ( (A)). So, any commutative C -algebra occurs as in (i).

(iii) Let H be a Hilbert space. It is easy to verify that T T = T 2 for all T L(H). Hence, every norm closed *-subalgebra of L(H) is a C -algebra.

(iv) By a theorem of Gelfand and Naimark, any C -algebra occurs as in (iii). Let G be a locally compact group, xed throughout this section. To every unitary representation ( , H) of G is associated a -representation of the Banach -algebra L1 (G) in H, that is, a continuous -algebra homomorphism L1 (G) L(H), again denoted by and de ned by ( f ) =.

f (x) (x)dx L(H),. namely by (f ) , =. f (x) (x) , dx C,. , H,. for f L1 (G). Thi none none s -representation of L1 (G) is non-degenerate, which means that, for every H \ {0}, there exists f L1 (G) such that ( f ) = 0. Conversely, any non-degenerate -representation of L1 (G) is of this form.

This is straightforward if the algebra L1 (G) has a unit, namely if the group G is discrete. In the general case, this follows from a standard argument using approximate units in L1 (G); see Proposition 13.4.

2 in [Dixmi 69]. To any unitary representation ( , H) of G, we can associate the sub-C algebra of L(H) generated by (L1 (G)). De nitions F.

4.3 and F.4.

6 below refer to the two most important cases..
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