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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.

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    $\begingroup$ Could you elaborate on what you don't understand about each statement? $\endgroup$ – JSchlather Jan 27 '13 at 14:53
  • $\begingroup$ en.wikipedia.org/wiki/Initial_topology $\endgroup$ – Martin Jan 27 '13 at 21:00
  • $\begingroup$ 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? $\endgroup$ – Ali Qurbani Jan 31 '13 at 14:33

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