# $\widehat{a}: \Omega(A)‎\rightarrow‎ \mathbb{C}~,~\tau‎ \mapsto \tau(A)‎‎$

Suppose that $A$ is abelian Banach algebra for which the space $\Omega(A)$ is non-empty. If $a \in A$, we define the function $\widehat{a}‎‎$ by $$\widehat{a}: \Omega(A)‎\rightarrow‎ \mathbb{C}~,~\tau‎ \mapsto \tau(a)‎‎$$

Explain about the following statement :

1. The topology on $\Omega(A)$ is the smallest on making all of the functions $\widehat{a}$ continuous.

2. The set $\{\tau‎ \in \Omega(A) : ‎‎\vert‎‎‎\tau(A)‎‎\vert\geqslant\epsilon\}$ is weak$^{*}$ closed in the closed unit ball of $A^{*}$ for each $\epsilon>0‎‎$.

Thanks.

• Could you elaborate on what you don't understand about each statement? – JSchlather Jan 27 '13 at 14:53
• en.wikipedia.org/wiki/Initial_topology – Martin Jan 27 '13 at 21:00
• in statement of 2 ,i dont know that why $\{\tau‎ \in \Omega(A) : ‎‎\vert‎‎‎\tau(A)‎‎\vert\geqslant\epsilon\}$ is contained in the closed unit ball of $A^{*}$? Do i write wrong? – Ali Qurbani Jan 31 '13 at 14:33