Mathematical Notes

, Volume 84, Issue 3–4, pp 529–537 | Cite as

Invariant Weighted Algebras ℒ p w (G)

Article

Abstract

The paper is devoted to weighted spaces ℒ p w (G) on a locally compact group G. If w is a positive measurable function on G, then the space ℒ p w (G), p ≥ 1, is defined by the relation ℒ p w (G) = {f: fw ∈ ℒ p (G)}. The weights w for which these spaces are algebras with respect to the ordinary convolution are treated. It is shown that, for p > 1, every sigma-compact group admits a weight defining such an algebra. The following criterion is proved (which was known earlier for special cases only): a space ℒ p w (G) is an algebra if and only if the function w is semimultiplicative. It is proved that the invariance of the space ℒ p w (G) with respect to translations is a sufficient condition for the existence of an approximate identity in the algebra ℒ p w (G). It is also shown that, for a nondiscrete group G and for p > 1, no approximate identity of an invariant weighted algebra can be bounded.

Key words

locally compact group weighted space weighted algebra approximate identity bounded approximate identity σ-compact group measurable function 

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Copyright information

© Pleiades Publishing, Ltd. 2008

Authors and Affiliations

  1. 1.Department of MathematicsAll-Russian Institute of Scientific and Technical Information (VINITI)MoscowRussia

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