Before we state our question, we give a motivational simple example: Put $X$ for disjoint union of two circles. However $X$ is not a connected space but it has an open dense subset $U$ such that $U$ has a connected compactification (Not necessarily one point compactification). To construct such a $U$, we remove one point from each circle, then we have a disjoint union of 2 copies of $\mathbb{R}$. The later space $U$, is topologically an open dense subset of $S^1$. Hence $U$ has a connected compactification.
So we can define the property of "somewhat connectivity" as follows: A compact Hausdorff space is "somewhat connected" if it has an open dense subset $U$ which has a connected Hausdorff compactification. Obviously the class of all somewhat connected spaces is closed under Cartesian product. The motivation mentioned above gives rise the following question:
Main Question: A unital $C^*$ algebra $A$ is called a "SC" algebra if it has an essential ideal $I$ such that $I$ has a unitization which has no nontrivial idempotent.
Is the spatial tensor product of two "SC" algebras again a SC algebra?
Remark One can ask the same question in the context of K-theory: We dente by "reform", the process of replacing an algebra $A$ with an arbitrary unitization of some essential ideal of $A$. So can ask: What are some obstructions to reform an algebra to a new algebra with trivial K-theory, the K-theory of $A=\mathbb{C}$?
