Description:
<jats:title>Abstract</jats:title>
<jats:p> We consider the algebras of holomorphic functions on a free polydisc <jats:inline-formula>
<jats:tex-math><?CDATA $\mathscr{F}^T(\mathbb{D}_R^n)$?></jats:tex-math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_210_9_1288ieqn1.gif" xlink:type="simple" />
</jats:inline-formula>, <jats:inline-formula>
<jats:tex-math><?CDATA $\mathscr{F}(\mathbb{D}_R^n)$?></jats:tex-math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_210_9_1288ieqn2.gif" xlink:type="simple" />
</jats:inline-formula> and the algebra of holomorphic functions on a free ball <jats:inline-formula>
<jats:tex-math><?CDATA $\mathscr{F}(\mathbb{B}_r^n)$?></jats:tex-math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_210_9_1288ieqn3.gif" xlink:type="simple" />
</jats:inline-formula>. We show that the algebra <jats:inline-formula>
<jats:tex-math><?CDATA $\mathscr{F}(\mathbb{D}_R^n)$?></jats:tex-math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_210_9_1288ieqn2.gif" xlink:type="simple" />
</jats:inline-formula> is a localization of a free algebra and, moreover, is a free analytic algebra with <jats:inline-formula>
<jats:tex-math><?CDATA $n$?></jats:tex-math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_210_9_1288ieqn4.gif" xlink:type="simple" />
</jats:inline-formula> generators (in the sense of J. Taylor), while the algebra <jats:inline-formula>
<jats:tex-math><?CDATA $\mathscr{F}(\mathbb{B}_r^n)$?></jats:tex-math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_210_9_1288ieqn3.gif" xlink:type="simple" />
</jats:inline-formula> is not a localization of a free algebra. In addition we prove that the class of localizations of free algebras and the class of free analytic algebras are closed under the operation of the Arens-Michael free product. </jats:p>
<jats:p> Bibliography: 21 titles. </jats:p>