Document Type: Research Paper

Author

Faculty of Mathematical Sciences, Shahrood University of Technology, P. O. Box 3619995161-316, Shahrood, Iran.

Abstract

‎Let $A$ be a non-zero normed vector space and let $K=\overline{B_1^{(0)}}$ be the closed unit ball of $A$. Also, let $\varphi$ be a non-zero element of $ A^*$ such that $\Vert \varphi \Vert\leq 1$. We first define a new norm $\Vert \cdot \Vert_\varphi$ on $C^b(K)$, that is a non-complete, non-algebraic norm and also non-equivalent to the norm $\Vert \cdot \Vert_\infty$. We next show that for $0\neq\psi\in A^*$ with $\Vert \psi \Vert\leq 1$, the two norms  $\Vert \cdot \Vert_\varphi$ and $\Vert \cdot \Vert_\psi$ are equivalent if and only if $\varphi$ and $\psi$ are linearly dependent. Also by applying the norm $\Vert \cdot \Vert_\varphi $ and a new product `` $\cdot$ '' on $C^b(K)$, we present the normed algebra $ \left( C^{b\varphi}(K), \Vert \cdot \Vert_\varphi \right)$. Finally we  investigate some relations between strongly zero-product preserving maps on $C^b(K)$ and $C^{b\varphi}(K)$.

Keywords

[1] H.G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monogr. Ser., 24 The Clarendon Press, Oxford University Press, New York, (2000).

[2] R.A. Kamyabi-Gol and M. Janfada, Banach algebras related to the elements of the unit ball of a Banach algebra, Taiwan. J. Math., 12 (2008), pp. 1769-1779.

[3] A.R. Khoddami, On maps preserving strongly zero-products, Chamchuri J. Math., 7 (2015), pp. 16-23.

[4] A.R. Khoddami, Strongly zero-product preserving maps on normed algebras induced by a bounded linear functional, Khayyam J. Math., 1 (2015), pp. 107-114.

[5] A.R. Khoddami, On strongly Jordan zero-product preserving maps, Sahand Commun. Math. Anal., 3 (2016), pp. 53-61.

[6] A.R. Khoddami, The second dual of strongly zero-product preserving maps, Bull. Iran. Math. Soc., 43 (2017), pp. 1781-1790.

[7] A.R. Khoddami, Bounded and continuous functions on the closed unit ball of a normed vector space equipped with a new product, U.P.B. Sci. Bull., Series A, 81 (2019), pp. 81-86.

[8] A.R. Khoddami, Biflatness, biprojectivity, $varphi-$amenability and $varphi-$contractibility of a certain class of Banach algebras, U.P.B. Sci. Bull., Series A, 80 (2018), pp. 169-178.

[9] T.W. Palmer, Banach Algebras and The General Theory of $*-$Algebras, Cambridge University Press, (1994).