I've recently watched a TV program about Luciano.
What struck me:
That prosecutor(forgot his name) who put him in jail, then became governor, and after all Lucky's collaboration in WWII signed his deportation to Sicily, appears to be his main enemy. But, Luciano went there without any attempt to have his revenge. Then, of course, in America they didn't follow old codes strictly. But as far as Dickie tells us, avenging your enemies is a matter of respect, if you refrain from vendetta, you will not be respected.
However, after coming to Sicily, he settled there very well, and the way he lived selling heroin makes it seem that he had enough respect from Sicilian families.
My question is: why the fact that his wrongs remained unavenged, did not affect the respect he got? Was it because of his connections/money that were useful, or was this question already theoretical even in Sicily?

The reason was that Luciano, though born in Sicily, was a thoroughly American Don. In Sicily, the Mafia was the law, and the law enforcement. But Luciano believed that,in America, murdering honest law enforcement officials would bring more heat down on organized crime, perhaps destroying it. That's the reason why Luciano ordered the assassination of Dutch Schultz--Schultz was planning to murder Thomas E. Dewey, the special prosecutor (the guy you were thinking of) who also was after Dewey.
As you know, Luciano was deported in 1945. He attempted to make a comeback late in 1946 with a Commission meeting in Havana. Someone (some say Vito Genovese) told the Feds about his presence there, and the US leaned on the Cuban government to send him back to Sicily. He lost ground steadily as the years went by.

To add what Turnbull said, I guess sicilian people, or at least most of them, wanted to go to America to live a better life, after all Lucky Luciano was the former Don in America, and his work experience and connections in USA could be helpful for them.I think they didn`t care about what they think about Luciano, they only cared to take advantage of his experience.Despite it`s common in Sicily to kill politicians and it wasn`t in the USA.

The more strange it seems that he got so much respect and influence in Sicily.

He was respected both in Sicily and Naples, where he spent most of his time. But his influence over the Commission in America waned steadily after that abortive Cuban venture in '46. Payments to him from America dwindled to practically nothing.

Well, then he possibly was more consistent in his dealings with local enemies...