# Off _topic question

I would appreciate if you help me to underestand why is this question counted as off_topic.

Is realy all embedding of $F_{2}$ in $SO3$ an obvious question?

Thank you.

Edit: The MO question has now been deleted by OP.

• Your question is about (infinite, nonabelian) compact subgroups $G$ of $SO(3)$, and one thing that came out in comments was the assertion that the only one is $SO(3)$. I believe that is correct. I think it comes down to checking that the normalizer of an $S^1$ in $SO(3)$ is the same subgroup $S^1$. I think this is something you can check with your bare hands; it might be easier to lift the problem up to the universal cover which is the group of unit quaternions, and check it there. See if that helps. Commented Jan 31, 2015 at 14:28
• In case my last comment was cryptic: certainly a connected compact subgroup of $SO(3)$ is either trivial, $S^1$, or all of $SO(3)$ (this is essentially a Lie algebra computation). And the connected component of a compact subgroup $G$ will be normal in $G$ and of finite index. So the only nontrivial case to check is where the connected component is an $S^1$; since $G$ is contained in the normalizer, it suffices to show the normalizer is just $S^1$ itself. Commented Jan 31, 2015 at 15:44
• @ToddTrimble Thank you very much for your interesting comments Commented Jan 31, 2015 at 18:19
• @ToddTrimble what about the pure algebraic version of my question.(Not taking closure): are there two copy of F2 in SO3 which can not be carry to each other via an automorphism of SO3? Commented Jan 31, 2015 at 18:47
• Ali, I suspect that all Lie group automorphisms $\phi$ of $SO(3)$ are inner (e.g., see: mathoverflow.net/questions/40666/…). Thus, given $g, g' \in SO(3)$, the angle between the axes of rotation for $g$ and $g'$ should equal the angle between the axes of rotation for $\phi(g), \phi(g')$. On the other hand, "most" homomorphisms $F_2 \to SO(3)$ should be embeddings, and by a cardinality argument the countable sets of angles you get from two such embeddings will usually differ. So $Aut(SO(3))$ should not act transitively on images of embeddings. Commented Jan 31, 2015 at 19:33

To clarify: the OP actually posed several questions in a barrage within this one, and my reaction in the comments to the original post was to this part:

We fix an embedding of $F_2$, the free group on two generators, in $SO(3)$ and we put $G=\overline{F_2}$. Then we construct the following $C^∗$ algebra: $A=C^∗(G)\cong C^*_r(G)$. What can be said about $A$? Is it simple? Is it idempotentless? Is its $K$-theory computed already?

The OP has asked many questions which involve $C^*$-algebras, clearly coming from a background which is not familiar with them. In itself that is not a problem, but in my personal view, an experienced user of MO should start to learn from what he or she gains from answers. I don't think MO should be used as a substitute for thinking about one's own questions carefully; for me it is somewhere to turn to when one gets stuck, not somewhere one tries as soon as one has thought of a question. Hence my vote to close.

The $C^*$-algebra of a compact group is isomorphic to a $c_0$-direct sum of full matrix algebras. Even if one has not come across this result in one's own background reading about $C^*$-algebras, surely as soon as one defines the group $G$ in the original question, one should be thinking

"I don't know what $G$ looks like but it is compact, so my question is about the $C^*$-algebra of a compact group, let me look online for ideas or consult a textbook or set of notes that deal with $C^*$-algebras of non-discrete groups."

It is the apparent absence of this first level of self-reflection, which I feel makes the question not really MO-quality. We are all meant to be researchers or potential researchers; we must surely aspire to think through some things on our own sometimes.

• +1 thank you very much for you interesting answer. Commented Jan 31, 2015 at 18:18
• May be I should read the MO aims more carefully. But flexibility of participants could help the beginners in an area. Commented Jan 31, 2015 at 18:42