This is probably very simple and just a matter of definition, but I have not enough experience with the category of C*-algebras.
Let $\mathcal{A}$ be a C*-algebra and let $\mathcal{U}$ be an ultrafilter on $\mathbb{N}$. Then :
- Definition 1 : The ultrapower $(\mathcal{A})_{\mathcal{U}}$ is the C*-algebra defined to be the quotient $l^{\infty}(\mathcal{A})/c_{\mathcal{U}}$, where $l^{\infty}(\mathcal{A})$ is the product C*-algebra of $\omega$ copies of $\mathcal{A}$ and $c_{\mathcal{U}}=\lbrace a\in l^{\infty}(\mathcal{A}):\lim_{\mathcal{U}}||a_i||=0\rbrace$, where $\lim_{\mathcal{U}}$ denotes the limit taken along the ultrafilter $\mathcal{U}$
- Definition 2 : Consider the functor $F:(\mathcal{U},\subset)^{op}\rightarrow$ C*-Alg, such that $F(L)=\prod_{L}\mathcal{A}$, where this is the product in C*-Alg category. The maps are simply the projections. We then define $(\mathcal{A})_{\mathcal{U}}$ to be the colimit of $F$.
My question is : Are these definitions equivalent ? According to my intuition, I would say yes. But I am very confused and I would appreciate some clarification. If they are equivalent, how can one check it ? If they are not equivalent, why is that the analogous work with, say, Sets or Rings?
Thanks