## 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.
Reviewer: Mati Abel (Tartu)

### MSC:

 46J10 Banach algebras of continuous functions, function algebras 46H10 Ideals and subalgebras 46J20 Ideals, maximal ideals, boundaries
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