Model theory of Banach algebras Let us consider the (metric) theory of Banach algebras. I have a sentence encoding the (possible) openness of multiplication in a given Banach algebra:
$$(\forall x) (\forall y) (\forall \varepsilon > 0)( \exists \delta > 0)(\forall z)\; \big(\|z-xy\| < \delta \Rightarrow (\exists u)(\exists v)[\|u-x\|<\varepsilon \& \|v-y\|<\varepsilon 
\;\&\; z = uv]\big)$$
This sentence looks quite first-order to me. 

Suppose that a Banach algebra $A$ satisfies the above formula. Can we directly conclude that some ultrapower of $A$ would also satisfy it?

By an ultrapower, I mean the metric (Banach-space) ultrapower. I know that people in C*-algebras have mastered such methods but I am not sure to what extent a similar machinery is available in the more general setting of Banach algebras.
(I know that this holds for all ultrapowers if we shift $(\forall x)(\forall y)$ after $(\exists \delta > 0)$ but the reason is not model-theoretic and the condition is too strong for me.)
Should it hold, I would be most grateful for pointing out the relevant literature touching this topic.
 A: What makes this tricky is that metric or continuous logic (in the sense of this document) doesn't directly allow for quantification over $\varepsilon$'s and $\delta$'s like that. You can still express it with a schema, but what's difficult in this case is that you have quantification over elements of the structure before quantifying over $\varepsilon$ and $\delta$. There's also some subtlety with how you formalize implication in this context.
Generally speaking a statement is only going to be preserved by ultrapowers/metric elementary equivalence if it's uniformly true in the structure. This fact is baked into the formalism in some places, such as the requirement to give a modulus of uniform continuity for a given function. If a function in a metric structure fails to be uniformly continuous, then in some elementary extension it won't even be a function. More generally purely 'topological' properties are often too fragile to be preserved under ultrapowers unless they are actually true in some uniform way.
EDIT2: This is not as straightforward as I thought. I can tell you roughly when this is not going to work. If there is an $\varepsilon > 0$ such that for every $\gamma > 0 $ there exists vectors $x,y$ with $\| x\| = \| y\|=1$ such that the $\delta$ needed to satisfy your condition is $<\gamma$, then in an ultrapower (over a countable index set) the condition will fail. So I would guess that in order for the condition to be satisfied in ultrapowers of the structure you actually need the stronger form you mentioned where the $(\forall x)(\forall y)$ quantifiers are after the $(\exists \delta > 0)$ quantifier. And I mean 'need' in a strong sense, i.e. if you have a Banach algebra such that your condition is true in all ultrapowers then the stronger form of the condition (although restricted to norm 1 vectors) actually holds in the first place.
EDIT: I think that maybe you can construct a counterexample from the proof of theorem 4.12 here, specifically some kind of product algebra of the algebras $\ell_1(\mathbb{Z}_{n!})$ with the convolution algebras. EDIT3: This reference is still relevant but I was being too optimistic about openness of multiplication being preserved under products.
