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HYPERGROUP ALGEBRAS AS TOPOLOGICAL ALGEBRAS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 90 / Issue 3 / December 2014
- Published online by Cambridge University Press:
- 13 June 2014, pp. 486-493
- Print publication:
- December 2014
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- Article
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Let
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$ be a locally compact hypergroup endowed with a left Haar measure and let 
$L^1(K)$ be the usual Lebesgue space of 
$K$ with respect to the left Haar measure. We investigate some properties of 
$L^1(K)$ under a locally convex topology 
$\beta ^1$. Among other things, the semireflexivity of 
$(L^1(K), \beta ^1)$ and of sequentially
$\beta ^1$-continuous functionals is studied. We also show that 
$(L^1(K), \beta ^1)$ with the convolution multiplication is always a complete semitopological algebra, whereas it is a topological algebra if and only if 
$K$ is compact.
