I have some questions about some arguments used in the discussion about consequnces of Bott periodity in in A Concise Course in Algebraic Topology by P. May at page 207. Her the excerpt:
Following context: Denote $U:= colim_n U(n)$ the infinite unitary group induced via canonical inclusions $U(n) \subset U(n+1)$.
FIRST QUESTION: Why is the loop space $\Omega BU := Hom(S^1, BU)$ $H$- equivalent to $U$?
We know that $BU = E/U$ where $E$ is the unique simply connected universal principal $U$-space and after applying $\pi_0(-)$ functor we obtain $\pi_0(\Omega BU) = \pi_1(E/U)= U$ by covering theory since $E$ is universal cover.
But does this already imply that $\Omega BU $ homotopy equivalent to $U$? $\pi_0$ only counts path components.
SECOND QUESTION: We know that $SU$ - the infinite special unitary group - is the universal cover of $U$ (via colimit argument). Futhermore $\pi_1(U)=\mathbb{Z}$.
Why does it imply that $$\Omega U \cong (\Omega SU) \times \mathbb{Z}$$ as $H$-spaces (remark: $X$ is a $H$-space if there exist a "multiplication" $X \times X \to X$.
Consider the map of fibrations from the universal principal $U$-bundle over $BU$ to the path-loop fibration on $BU$:
$\require{AMScd}$ \begin{CD} U @>>> EU @>>> BU \\ @VVV @VVV @| \\ \Omega BU @>>> PBU @>>> BU \end{CD}
One can write down a map $EU \to PBU$; thus there is a map $U \to \Omega BU$. Looking at the long exact sequence in homotopy and using the five-lemma, one deduces that the induced map $U \to \Omega BU$ is a weak homotopy equivalence because $EU \simeq PBU \simeq *$. (Since both source and target can be equipped with a CW structure, this can be promoted to a homotopy equivalence.)
In general, this shows $\Omega BG \simeq G$ for a topological group $G$. See this answer for more details.
For your second question, there is a short exact sequence of topological groups $$1 \to SU \to U \xrightarrow{\det} S^1 \to 1$$ which is split, so $U \cong SU \rtimes S^1$ as topological groups. If we ignore the group structure and only care about the topology, then we have a homeomorphism $U \cong SU \times S^1$. Taking loops on both sides, we get $$\Omega U \cong \Omega SU \times \Omega S^1 \simeq \Omega SU \times \mathbb{Z}$$ as loop spaces, hence as $H$-spaces.