About understanding manifold structure on WAP compactification of $\Bbb{C} \rtimes \Bbb{T}$

Question

Let $G$ be a locally compact topological group. A continuous bounded function $f$ on $G$ is called (weakly) almost periodic if the set $L_Gf$ of left translates is relatively compact in the (weak) norm topology. Then $f$ is uniformly continuous with respect to the left and right uniform structures on $G$ and all (weakly) almost periodic functions forms an abelian $C^*$-algebra with the norm of uniform convergence, denoted with ($WAP(G)$ resp.) $AP(G)$. The group multiplication of $G$ extends to the whole maximal ideal space $G^{WAP}$ of $WAP(G)$, making $G^{WAP}$ a semitopological semigroup.

Ruppert on his book "Compact Semitopological semigroups" (Chapter III, section 8, Page no. 160) considers the following Euclidean motion group $G = \Bbb{C} \rtimes \Bbb{T}$, where $\Bbb{T}$ (unit circle) acts on $\Bbb{C}$ by multiplication, which is an example of minimally weakly almost periodic groups, i.e., groups whose WAP compactifications contain only two idempotents. Then the weakly almost periodic compactification of $G$ is the union (of intrinsic subgroups around these idempotents), $S = \phi(G) \cup M(S)$, where $\phi$ is the compactification map, $M(S)$ is the minimal ideal of the compact semitopological semigroup $S$ and is isomorphic with $\Bbb{T}$ because $WAP(G) = AP(G) \oplus C_0(G)$. He claims that the $G^{WAP}$s of this example are compact manifolds. How do we understand the manifold structure on $S$ ?

Answer 1

Let me recall that the WAP compactification of a locally compact group is a semi-topological compact monoid ("semi-topological" means that both left and right multiplications are continuous, but maybe not jointly) which is universal as such.

To be precise, given a locally compact group $G$ there exists a semi-topological compact monoid $S$ and a continuous injection of monoids $\alpha:G\to S$ such that for every semi-topological compact monoid $S'$ and a continuous morphism of monoids $\beta:G\to S'$ there exists a unique continuous morphism of monoids $\gamma:S\to S'$ such that $\beta=\gamma\circ \alpha$. Such a universal compactification, $\alpha:G\to S$, is clearly unique up to a unique isomorphism. It is called the WAP compactification of $G$ and denoted $G^\text{WAP}$.

Another semi-topological monoid compactification of a locally compact group $G$ is given by the one point compactification $G\cup\{\infty\}$ where the multiplication is defined so that $\infty$ is an absorbing point for both left and right multiplications (this example is good for understanding how the multiplication could be left and right but not jointly continuous). In particular, we get the one-point compactification of the additive group of the complex field, $\mathbb{C}\cup \{\infty\}$. Note that topologically, $\mathbb{C}\cup \{\infty\} \simeq S^2$.

Note that the torus $\mathbb{T}$ acts on $\mathbb{C}\cup \{\infty\}$ in an obvious way (letting $t\infty=\infty$) and for every $t\in \mathbb{T}$ the map $x\mapsto tx$ is an automorphism of the semi-topological monoid $\mathbb{C}\cup \{\infty\}$. On the product space $\mathbb{T}\times (\mathbb{C}\cup \{\infty\})$ we can form a monoid structure by setting $(t,x)\cdot (t',x')=(tt',t'x+x')$. This is easily checked to give the structure of a semi-topological compact monoid, to be denoted the semidirect product monoid $\mathbb{T} \ltimes (\mathbb{C}\cup \{\infty\})$.

Letting $G=\mathbb{T} \ltimes \mathbb{C}$, we clearly have a continuous injection $G \to \mathbb{T} \ltimes (\mathbb{C}\cup \{\infty\})$ which is a semi-topological monoid compactification of the locally compact group $G$, as discussed above. Thus there is a continuous morphism of monoids $G^\text{WAP} \to \mathbb{T} \ltimes (\mathbb{C}\cup \{\infty\})$. It follows from Theorem 8.6 in Ruppert's book (see the example on p. 160) that the latter morphism is in fact an isomorphism of semi-topological monoids (take into account that every bijective continuous map between compact spaces is a homeomorphism). That is, $G^\text{WAP} \simeq \mathbb{T} \ltimes (\mathbb{C}\cup \{\infty\})$.

In particular, we get that $G^\text{WAP}$ is homeomorphic to $\mathbb{T}\times (\mathbb{C}\cup\{\infty\}) \simeq S^1\times S^2$. Obviously, $S^1\times S^2$ has the structure of a 3-dimensional compact manifold. Moreover, it is a standard fact that the standard manifold structure on $S^1\times S^2$ is the unique manifold structure on this topological space. Thus, Ruppert's remark on p. 160 that $G^\text{WAP}$ is a compact manifold makes sense. However I would phrase it differently, saying that $G^\text{WAP}$ has a unique manifold structure. In any case, the manifold stucture alluded to in the question is the one given by the homeomorphism $G^\text{WAP}\simeq S^1\times S^2$ and I hope it is clear from my answer how to understand it.