A commutative Banach algebra with an abundance of discountinuous functions 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.. 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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*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?


*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?


*Is there a $C^{*}$-norm on $A$ which is equivalent to the above norms? In particular is  $A$, semi-simple?

 A: 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 $\beta$, $\iota$ 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 $X\times Y$.
