## ultrapowers of Banach algebras

Let $A$ ba a Banach algebra and $u$ be an arbitrary free ultrafilter. Let $A^{\bullet}$ be the unitization of $A$. Can we have $((A)_{u})^{\bullet} = (A^{\bullet})_{u}$?

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How can you expect to get an answer when you do not even make your question readable? I vote to close. – Bill Johnson Oct 7 at 19:54
I assume that in your question you want $A$ to be infinite-dimensional, otherwise the question is trivial – Yemon Choi Oct 7 at 20:30
For any Banach space $X$ we have $(X\oplus Y)_u = (X)_u \oplus (Y)_u$, so taking $X=A$ and $Y={\mathbb C}$ I think the answer is yes, for reasons which have nothing to do with algebras – Yemon Choi Oct 7 at 20:46