Let $\mathcal{H}$ be a unitary universe for some group $G$. As $\mathcal{H}$ is a faithful representation the representation map is an injection $G \to \mathcal{U(H)}$, so there's a free $G$ action on $\mathcal{U(H)}$. By Kuiper's theorem $\mathcal{U(H)}$ is also non-equivariantly contractible. So therefore it's a model for $EG$.

I am unsure as to the choice of topology of $\mathcal{U(H)}$ to make the following work (all the maps and actions are continuous). We can have $\mathcal{U(H)}$ act on the Fredholm operators $\mathcal{F(H)}$ via conjugation. Thus the standard action of $G$ on $\mathcal{F(H)}$ factors though $\mathcal{U(H)}$.

I know that the standard action of $G$ on $\mathcal{F(H)}$ is not continuous, but that in the appendix of "Twisted K-theory" by Atiyah and Segal they produce a replacement set of objects $\mathcal{F} _{G-\text{cont.}} (\mathcal {H})$ and $\mathcal{U} _{G-\text{cont.}} (\mathcal {H})$ where the conjugation action by $G$ is continuous. As my action of $G$ on $\mathcal{U(H)}$ is basically composition (different from theirs) is it a continuous action?

Let $\mathcal{H}$ be a unitary universe for some group $G$. As $\mathcal{H}$ is a faithful representation the representation map is an injection $G \to \mathcal{U(H)}$, so there's a free $G$ action on $\mathcal{U(H)}$. By Kuiper's theorem $\mathcal{U(H)}$ is also non-equivariantly contractible. So therefore it's a model for $EG$.

I am unsure as to the choice of topology of $\mathcal{U(H)}$ to make the following work (all the maps and actions are continuous). We can have $\mathcal{U(H)}$ act on the Fredholm operators $\mathcal{F(H)}$ via conjugation. Thus the standard action of $G$ on $\mathcal{F(H)}$ factors though $\mathcal{U(H)}$.

I know that the standard action of $G$ on $\mathcal{F(H)}$ is not continuous, but that in the appendix of "Twisted K-theory" by Atiyah and Segal they produce a replacement set of objects $\mathcal{F} _{G-\text{cont.}} (\mathcal {H})$ and $\mathcal{U} _{G-\text{cont.}} (\mathcal {H})$ where the conjugation action by $G$ is continuous. As my action of $G$ on $\mathcal{U(H)}$ is basically composition (different from theirs) is it a continuous action?

EDIT: By the paper that @DavidRoberts linked in the comments, we have $\mathcal{U(H)}$ has a free action of $G$ and it is contractable. It however does not have an obvious $G$-CW structure. For it to be a model for $EG$, it needs to be at least homotopic to a $G$-CW complex. I'm not entirely sure about this point, there are notes out there that suggest that this isn't needed but that could be a little bit incorrect.

Let $\mathcal{H}$ be a the unitary universe for some group $G$. As $\mathcal{H}$ is a faithful representation the representation map is an injection $G \to \mathcal{U(H)}$, so there's a free $G$ action on $\mathcal{U(H)}$. By Kuiper's theorem $\mathcal{U(H)}$ is also non-equivariantly contractible. So therefore it's a model for $EG$.

I am unsure as to the choice of topology of $\mathcal{U(H)}$ to make the following work (all the maps and actions are continuous). We can have $\mathcal{U(H)}$ act on the Fredholm operators $\mathcal{F(H)}$ via conjugation. Thus the standard action of $G$ on $\mathcal{F(H)}$ factors though $\mathcal{U(H)}$.

I know that the standard action of $G$ on $\mathcal{F(H)}$ is not continuous, but that in the appendix of "Twisted K-theory" by Atiyah and Segal they produce a replacement set of objects $\mathcal{F} _{G-\text{cont.}} (\mathcal {H})$ and $\mathcal{U} _{G-\text{cont.}} (\mathcal {H})$ where the conjugation action by $G$ is continuous. As my action of $G$ on $\mathcal{U(H)}$ is basically composition (different from theirs) is it a continuous action?

Let $\mathcal{H}$ be a unitary universe for some group $G$. As $\mathcal{H}$ is a faithful representation the representation map is an injection $G \to \mathcal{U(H)}$, so there's a free $G$ action on $\mathcal{U(H)}$. By Kuiper's theorem $\mathcal{U(H)}$ is also non-equivariantly contractible. So therefore it's a model for $EG$.

I am unsure as to the choice of topology of $\mathcal{U(H)}$ to make the following work (all the maps and actions are continuous). We can have $\mathcal{U(H)}$ act on the Fredholm operators $\mathcal{F(H)}$ via conjugation. Thus the standard action of $G$ on $\mathcal{F(H)}$ factors though $\mathcal{U(H)}$.

I know that the standard action of $G$ on $\mathcal{F(H)}$ is not continuous, but that in the appendix of "Twisted K-theory" by Atiyah and Segal they produce a replacement set of objects $\mathcal{F} _{G-\text{cont.}} (\mathcal {H})$ and $\mathcal{U} _{G-\text{cont.}} (\mathcal {H})$ where the conjugation action by $G$ is continuous. As my action of $G$ on $\mathcal{U(H)}$ is basically composition (different from theirs) is it a continuous action?