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**Generalization of the topological algebra \((C_b (X), \beta)\).**
*(English)*
Zbl 1176.46050

Let \(X\) be a completely regular Hausdorff space, \(B(X)\) the set of all complex-valued bounded functions on \(X\), \(B_0(X)\) the subset of functions in \(B(X)\) which vanish at infinity, \(C(X)\) the set of all complex-valued continuous functions on \(X\) and \(C_0(X)\) the subset of functions in \(C(X)\) which vanish at infinity. Moreover, let \(v\) be an upper semicontinuous real-valued function on \(X\) for which \(\inf _{t\in X}v(t)>0\), \(C_b^v(X)=\{f\in C(X):vf\in B(X)\}\) endowed with the norm \(\|f\|_v=\sup _{t\in X}v(t)|f(t)\), and for locally compact \(X\) let \(C_0^v(X)=\{f\in C(X):vf\in B_0(X)\}\) endowed with the seminorms \(p_{v,\phi}(f)=\sup _{t\in X}v(t)|\phi(t)f(t)|\) for each \(\phi\in B_0(X)\). The topology defined on \(C_b^v(X)\) by the norm \(\|\cdot \|_v\) is denoted by \(\sigma _v\) and the topology on \(C_0^v(X)\) defined by the system \(\{p_{v,\phi}: \phi\in B_0(X)\}\) of seminorms is denoted by \(\beta _v\).

Among other interesting results, the connections between the topologies \(\sigma _v\), \(\beta _v\), the uniform topology and the strict topology are considered. It is shown that \((C_b^v(X),\sigma _v)\) is complete, \((C_0^v(X),\beta _v)\) is complete if \(X\) is a \(k_\mathbb{R}\)-space, every closed maximal ideal of \((C_b^v(X),\beta _v)\) has the form \(\{f\in C_b^v(X): f(t)=0 \}\) for some \(t\in X\), and every closed ideal of \((C_0^v(X),\beta _v)\) has the form \(\{f\in C_b^v(X): f(t)=0\;\text{for\;all}\;t\in E\}\) for some closed subset \(E\) of \(X\). Moreover, properties of the quotient algebra \((C_b^v(X)/I, \tilde{\beta}_v)\), where \(I\) is a closed ideal in \((C_0^v(X),\beta _v)\) and \(\tilde{\beta}_v\) the quotient topology defined by \(\beta _v\), are considered separately.

Among other interesting results, the connections between the topologies \(\sigma _v\), \(\beta _v\), the uniform topology and the strict topology are considered. It is shown that \((C_b^v(X),\sigma _v)\) is complete, \((C_0^v(X),\beta _v)\) is complete if \(X\) is a \(k_\mathbb{R}\)-space, every closed maximal ideal of \((C_b^v(X),\beta _v)\) has the form \(\{f\in C_b^v(X): f(t)=0 \}\) for some \(t\in X\), and every closed ideal of \((C_0^v(X),\beta _v)\) has the form \(\{f\in C_b^v(X): f(t)=0\;\text{for\;all}\;t\in E\}\) for some closed subset \(E\) of \(X\). Moreover, properties of the quotient algebra \((C_b^v(X)/I, \tilde{\beta}_v)\), where \(I\) is a closed ideal in \((C_0^v(X),\beta _v)\) and \(\tilde{\beta}_v\) the quotient topology defined by \(\beta _v\), are considered separately.

Reviewer: Mati Abel (Tartu)