�ber Fortsetzung positiv definiter Funktionen

Henrichs, Rolf Wim

doi:10.1007/bf01421401

Cited by 17 publications

(9 citation statements)

References 19 publications

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“…We thank the referee for indicating to us that our proof could be made to work in a more general setting. We also thank him for mentioning the related results in [1] and for devising the following corollary, whose proof uses Theorem 2 quoted at the beginning and a result in [2].…”

Section: Proof Since Zc(h) Is Not Open There Is a Net {Ka) In Its Cmentioning

confidence: 97%

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On extending characters

Milnes¹

1981

Proc. Amer. Math. Soc.

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Abstract. Amplifying modestly on recent work of R. W. Henrichs, we indicate that, while the impossibility of extending a continuous character uV on a subgroup H of a topological group G to a character on G is often due to algebraic problems, the impossibility of extending uV to a continuous positive definite function on G can often be viewed as a problem with uniform continuity.Recently R. W. Henrichs [3] proved the following attractive results. Theorem 1. Let G be a connected locally compact group which b not the direct product of a vector group and a compact group. Then there is a continuous character on a closed subgroup of G which cannot be extended to a continuous positive definite function on G. (ii) Characters of closed subgroups always have continuous positive definite extensions.(iii) G is the semidirect product of a vector group V and a compact group K such that K acts on V effectively as a finite group. These theorems and our treatment of an example on the affine group of the line in [7] prompted us to investigate the possibility that some characters on the subgroups (as in the theorems) might not extend to uniformly continuous functions on the whole group. (We remind the reader that continuous positive definite functions on topological groups are uniformly continuous [5, Theorem 32.4].)We note first, that it will often be impossible to extend a character of a subgroup to a character of the containing group. For example, such nonextendable characters will always exist on a normal abelian subgroup 77 of a group G if 77 is not contained in the center of G; thus, if 77 is a cyclic subgroup of order 3 of the

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Section: Proof Since Zc(h) Is Not Open There Is a Net {Ka) In Its Cmentioning

confidence: 97%

“…(For terminology and further references here, and in the rest of this note, the reader is referred to [2], [3], [5].) Theorem 2.…”

mentioning

confidence: 99%

On extending characters

Milnes¹

1981

Proc. Amer. Math. Soc.

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“…The interest in and importance of the separation properties arose from the fact that it turned out to be useful in studying the ideal theory of the Fourier algebra A(G) (see [20]). The extension and the separation properties have been studied by several authors (see [5], [7], [9], [13], [14], [19], [24] and [28] for the extension property and [20], [21] and [25] for the separation property). If H is a closed subgroup of G such that G has small H-conjugation invariant neighbourhoods of the identity (G ∈ [SIN ] H ), then H is extending as well as separating ( [7], [10] and [13]).…”

Section: Introductionmentioning

confidence: 99%

Extension and separation properties of positive definite functions on locally compact groups

Kaniuth

Lau

2006

Trans. Amer. Math. Soc.

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Abstract. Continuing earlier work, we investigate two related aspects of the set P (G) of continuous positive definite functions on a locally compact group G. The first one is the problem of when, for a closed subgroup H of G, every function in P (H) extends to some function in P (G). The second one is the question whether elements in G \ H can be separated from H by functions in P (G) which are identically one on H.

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“…We say that G has the extension property when each closed subgroup of G is extending. These properties have been studied by several authors [2], [3], [6], [9], [10], [16], [17]. Fundamental to all this has been Douady's observation (see [5, p. 204] and [12, (34.28)]) that if A is an abelian closed normal subgroup of a locally compact group G and χ is a character of A, then χ extends to some continuous positive definite function on G only if the stabiliser G χ = {x ∈ G : χ(x −1 ax) = χ(a) for all a ∈ A} is open in G.…”

Section: Introductionmentioning

confidence: 99%

“…It was independently shown by Henrichs [9] and Cowling and Rodway [3] that a closed subgroup H of G is extending if G possesses small H-invariant neighbourhoods of the identity. In particular, every SIN-group, i.e., group with small conjugation invariant neighbourhoods of the identity, has the extension property.…”

Section: Introductionmentioning

confidence: 99%