Let $A$ be the algebra of all bounded functions from $[0,\;1]$ to $\mathbb{C}$.


For $f\in A,\;$ $\omega\_{f}$ is the standard [oscillation function.](http://en.wikipedia.org/wiki/Mathematics_of_oscillation). Each of the following two (equivalent) norms on $A$, defines a Banach algebra structure on $A$.


$$\parallel f \parallel=\parallel f\parallel\_{\infty}+ \parallel \omega\_{f}\parallel\_{\infty}$$ or $$\parallel f \parallel=\parallel f\parallel\_{\infty}+ \int\_{[0,\;1]} \omega\_{f}(x)dx$$ The later is well defined, since the oscillation function is a bounded measurable function.


**Questions:**



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> 1. Let $X$ be the Gelfand spectrum of $A$. What is the topological structure of this compact nonmetrizable disconnected Haussdorf space, precisely ? Is it homeomorphic to a known space?
> 2. Assume that $B$ and $C$ are two $C^{\*}$-algebras which are embedded isometrically into $A$. Must their $C^{\*}$-tensor product be embedded in $A$, too?
> 3. Is there a $C^{\*}$-norm on $A$ which is equivalent to the above norms? In particular is $A$, semi-simple?
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> 
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A commutative Banach algebra with an abundance of discountinuous functions
*This is an answer to Q2, because as pointed out by Yemon, Q1 and Q3 follow easily from the fact that this algebra is isomorphic to $C(\beta [0,1]\_d)$. It is enough to prove then that if two compact spaces $X$ and $Y$ are continuous images of $\beta \kappa$ for some cardinal $\kappa$, then so is $X\times Y$.*


Let $X$ be a compact Hausdorff space. Let $D$ be a dense subset of $X$ endowed with the discrete topology. The identity map


$$\iota\colon D \to X$$


is continuous. By the universal property of the [Stone–Čech functor $\beta$](http://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactification#Universal_property_and_functoriality), $\iota$ extends (uniquely) to a continuous map


$$\beta \iota \colon \beta D \to X.$$
Since continuous images of compact spaces are compact, $\beta \iota$ is surjective. Hence we arrive at the following conclusion.


*Conclusion*. Let $X$ be a compact Hausdorff space with density character $\kappa$. Then $X$ is a continuous image of $\beta \kappa$. Consequently, since the product of two compact spaces with character $\kappa$ is compact and has density character $\kappa$, if $X$ are $Y$ are continuous, Hausdorff images of $\beta \kappa$, then so is $X\times Y$.


