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F.1 Weak containment of unitary representations in .NET Connect Code 39 in .NET F.1 Weak containment of unitary representations

F.1 Weak containment of unitary representations using .net framework tocompose bar code 39 on asp.net web,windows application iReport exists a s visual .net Code 39 equence ( n )n of functions of positive type associated to which converges to uniformly on G. For n large enough, we have (x) n (x)dx = 0.

. Decomposin g as a direct sum = i i of irreducible subrepresentations (i) (i) i , we see that each n is a sum n = i n for functions n of positive type associated to i . Hence for some i and n, (x) n (x)dx = 0..

G (i). By Schur s orthogonality relations for compact groups (see, e.g. [Robet 83, (5.

6) Theorem]), this implies that is equivalent to i . Remark F.1.

9 Let be a discrete group and H a subgroup of . The restriction . H of the r 39 barcode for .NET egular representation of to H is a multiple of the regular representation H of H . Indeed, let T be a set of representatives for the right coset space H \ .

Then 2 ( ) has a direct sum decomposition 2( ) = 2 (Ht) into (H )-invariant subspaces. Since the restriction of t T . H to each subspace 2 (Ht) is equivalent to H , this proves the claim. The next proposition gives a partial generalisation of this fact to an arbitrary locally compact group (see also Remark F.1.

11 below). Proposition F.1.

10 Let G be a locally compact group, and let H be a closed subgroup of G. Then G . H H . P visual .net Code-39 roof Let f Cc (G).

Since the left regular representation of H is equivalent to the right regular representation H of H (Proposition A.4.1), it suf ces, by Lemma F.

1.3, to show that x G (x)f , f can be approximated, uniformly on compact subsets of H , by sums of functions of positive type associated to H . Let dx and dh be left Haar measures on G and H .

Let be a quasi-invariant Borel measure on G/H , with corresponding rho-function . For k H , we have G (k)f , f =. f (k 1 x)f (x)dx = G G (x G 1 f (x 1 k)f (x 1 )dx )f (xk)f (x)dx 1 1. G (h H )f (xhk)f (xh) (xh) 1 dhd (xH ),. Weak containment and Fell s topology using Lemma A.3.4 and Theorem B.1.4. For x G, de ne fx Cc (H ) by fx (h) = Then G (h 1 1 Code 39 Full ASCII for .NET G (h 1 x 1 )f (xh) (xh) 1/2 , . h H. )f (xhk)f (xh) (xh) 1. 1 ). G (k fx (hk)fx (h). (xhk)1/2 (xh)1/2. H (k)fx (hk)fx (h). = ( H (k)fx ) (h)fx (h), using Lemma B.1.3. Thus G (k)f , f = H (k)fx , fx d (xH ).. Since the visual .net Code 39 Extended mapping (x, k) H (k)fx , fx is continuous, it follows that G ( )f , f is a uniform limit on compact subsets of H of linear combinations with positive coef cients of functions of the form H ( )fxi , fxi with xi G. This shows that G .

H H . Remark F.1.11 Conversely, H is weakly contained in G H (Exercise F.6.1) so that G H and H a re actually weakly equivalent. Example F.1.

12 As shown in Remark 1.1.2.

vi, the unit representation 1R of R is weakly contained in the regular representation R . Let us see that every character R is weakly contained in R . Indeed, given a compact subset Q of R and > 0, there exists a function f in L2 (R) with f 2 = 1 such that R (x)f f < , for all x Q.

. Let g = f . Then, since R (x)g, g = (x) R (x)f , f , we obtain (x) R (x)g, g = . 1 R (x)f , f = . f , R (x)f f < , for all x Q. For a more general fact, see Theorem G.3.2. F.2 Fell topology on sets of unitary representations Let G be a Code 3 of 9 for .NET topological group. One would like to de ne a topology on the family of equivalence classes of unitary representations of G.

There is a problem.
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