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THE MULTIPLIER ALGEBRA AND BSE PROPERTY OF THE DIRECT SUM OF BANACH ALGEBRAS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 2 / October 2013
- Published online by Cambridge University Press:
- 07 February 2013, pp. 250-258
- Print publication:
- October 2013
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- Article
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The notion of BSE algebras was introduced and first studied by Takahasi and Hatori and later studied by Kaniuth and Ülger. This notion depends strongly on the multiplier algebra
$M( \mathcal{A} )$ of a commutative Banach algebra 
$ \mathcal{A} $. In this paper we first present a characterisation of the multiplier algebra of the direct sum of two commutative semisimple Banach algebras. Then as an application we show that 
$ \mathcal{A} \oplus \mathcal{B} $ is a BSE algebra if and only if 
$ \mathcal{A} $ and 
$ \mathcal{B} $ are BSE. We also prove that if the algebra 
$ \mathcal{A} \hspace{0.167em} {\mathop{\times }\nolimits}_{\theta } \hspace{0.167em} \mathcal{B} $ with 
$\theta $-Lau product is a BSE algebra and 
$ \mathcal{B} $ is unital then 
$ \mathcal{B} $ is a BSE algebra. We present some examples which show that the BSE property of 
$ \mathcal{A} \hspace{0.167em} {\mathop{\times }\nolimits}_{\theta } \hspace{0.167em} \mathcal{B} $ does not imply the BSE property of 
$ \mathcal{A} $, even in the case where 
$ \mathcal{B} $ is unital.
