Idempotent Multipliers on Spaces of Continuous Functions with p-Summable Fourier Transforms (original) (raw)
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Multipliers on spaces of functions on compact groups with p-summable Fourier transforms
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We will start with the set M(X,Y)M(X,Y)M(X,Y), multiplier space, defined by: \[ M(X,Y)=\{a=(a_k)\in \omega \mid ax\in Y \mbox{, for all }x\in X\} \] where omega\omegaomega denotes the space of all complex-valued sequences and XXX and YYY are sequence spaces. Specially, putting Y=csY=csY=cs, where cscscs is the set of convergent series, the multiplier space becomes the beta\betabeta-dual of XXX. We will present some generalized results related to XbetaX^{\beta}Xbeta and extend some of existing. Finally, we will illustrate these generalizations with some examples and applications.
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Throughout the paper, the symbols G1 and G2 will denote two locally compact abelian groups with character groups X1 and X2, respectively. Haar measures on Gj are denoted by μj; the ones on Xj are denoted by θj (j=1,2). The measures μj and θj are normalized so that the Plancherel Theorem holds (see [7, p. 226, Theorem 31.18]).
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Multiplicative functionals on function Algebras
Revista Matemática Complutense, 1988
Leí X be a completely regular Hausdorff space and C(X) the algebra of alí continuous <-valued functions on X (1K = IR o C). If A E C(X) is a subalgebra, in [4] can be found conditions on A under whicls cadi character of A, i.e., each non-zero 1K-linear multiplicative funclional 4xA-* 1K, is given by a point evaluation al sorne point of X. In ibis paper we presení a «Michael» type theorem for the particular case in which X is a real Banach space. As consequence it is showed thai if E is a separable Banach space or E is the topological dual space of a separable Banach space aud A ís the algebra of ah real analytic or the algebra of alí real Cm.functions, ni = 0, 1 oD, on E, then every character 4 of A is a point evaluation at sorne poiní of E. Let E be a real Hanach space with topological dual E' and let C(E) be the algebra of alí continuous IR-valued bunctions on E. Let ll(~~J)={a=(a»)eIRNõ , a t,~09}. n1 Theorem 1. Assume Éhat diere exists («t,)%'1 cE', 114,,II=lforevery nel\J, such that (&) separates points of E. LeÉ A EC(E) be a subalgebra with leA. Assume: (i) IffeA,f(x)#Ofor aH xeE, then 1//EA. (ji) E' a A aná for every í = (í,,) e ¡'(IN), tbe funetion Z~a~belongs to A. n1 Pien every character 4rA-dR, such that /.'(4t,)=#t,(a)for every neN aná sorne acE, is the point evafuation at a.