Do approximate identities remain approximate identities if one adjoins 1 to a C* Algebra?

Question

If we have a C* Algebra $\mathscr{U}$ without an identity we can adjoin an identity $\mathbb{1}$ in the following way:

1. We take $\mathscr{\tilde U}$ to be the set $\{(\alpha,A); \alpha \in \mathbb{C}, A \in \mathscr{U} \}$
2. On $\mathscr{\tilde U}$ we define addition in the obvious way and define multiplication via $(\alpha,A) \cdot (\beta,B) := (\alpha \beta,\alpha B + \beta A +AB)$. Also we can introduce a shorthand notation $(\alpha, A) := \alpha \mathbb{1} + A$
3. The norm of the new states is defined via: $$\|\alpha\mathbb{1}+A\| := \sup\{\|\alpha B + AB\|\ ; B \in \mathscr{U},\, \|B\|=1\}$$

The result will again be a C* algebra and $\mathscr{U}$ can be identified as the subset of elements in the form $(0,A)$ of $\mathscr{\tilde U}$. My question is whether an approximate identity on $\mathscr{U}$ will still have the property of being an approximate identity on the new C* Algebra. Is the proof easy?

Answer 1

Well the answer was contained in the question linked by rschwieb, I'll write it up here.

If a C* Algebra $\mathscr{U}$ does not contain a unit then approximate identities will not be approximate identities in the C* Algebra $\mathscr{\tilde U}$ with a unit adjoined.

Every approximate identity in a C* Algebra with a unit must converge to the unit, since (in the case of a right approximate identity $E_\alpha$):

$$\lim_\alpha \|A -A E_\alpha \|= 0 \quad \forall A \in \mathscr{\tilde U}$$ implies $$\lim_\alpha \|\mathbb{1}-\mathbb{1}E_\alpha\| = \lim_\alpha \|\mathbb{1}-E_\alpha\|=0$$ i.e. $E_\alpha$ converges to the identity. Since C* Algebras are closed and we can view $\mathscr{U}$ as a (closed) sub C* Algebra of $\mathscr{\tilde U}$, if the elements of an approximate identity of $\mathscr{\tilde U}$ all lay in $\mathscr{U}$ then the limit, which exists as shown above, must also lie in $\mathscr{U}$, which we are assuming not to be true.

More generally an approximate identity on a closed ideal will not act as an approximate identity on anything not in the ideal, since if $E_\alpha$ is an approximate identity on a closed ideal $I$ and $A\not\in I$ so that $\lim_{\alpha}\|A-AE_\alpha\|=0$, then $AE_\alpha \to A$. Since $I$ is an ideal you have $AE_\alpha \in I$ and since $I$ is closed so $A$ must be in $I$.

The condition that we are looking at an approximate identity of an ideal is necessary, as for example on a unital $C^*$ algebra $\mathbb 1$ is an approximate identity on the sub-algebra generated by $\mathbb 1$, but also on the entire algebra.