The owner does not solve the problems in our rented accommodation

Can you politely refuse to pay rent until he honors his side of things. Mention you need to get the items fixed and you will get it done and deduct it from rental.

Unfortunately i made a full prepayment.
The owner refuses to return the money to me, claims that he made me a significant discount in view of faults.

Any reason why he is refusing to solve those problems? Is it because he made a discount?
Does not sounds good at all. With such type of landlord, you can be sure of having more problems in the future.

I would ask him for the money and move out to another place. if he refuses, you can obviously hire a lawyer. But this can take time and sometimes can complicate matters.

I paid the money to his wife in London, to an account in Britain, on the basis of an invoice, where it is written for renting an apartment.  She does not own the apartment.  I live in a residential complex where housing is sold only to residents of Mauritius.  We do not have a lease agreement. 
Is it possible to recognize the transaction illegal under the law of Mauritius?  If so, for which article? 
The owner, or rather his mother, who watches the apartment turned against all my local staff against my family.  They now turn off the water for us, the cleaners insult my wife, the guards do not monitor our belongings and do not react to the alarm siren.

Hello

The advice I will give you:
Go on Radio One with Finlay Salesse. See if you can talk with him before broadcasting  your dilemma and to ensure how far and how many details  you can provide on the radio.
http://www.r1.mu

You would like to stress the fact that some (diaspora) Mauritians in the UK are stealing from expatriates via their network of friends and family on the island.
It is those scammers who give  a bad name to Mauritius.

I second External advice. Bringing this to RadioOne will definitely help you.
Meanwhile, I would suggest you keep trying to get your money back and move out.

And as a general advice, next time, make sure you visit to visit a rental first before making any payment.
If you are booking online, make sure to gather all the details first and know who you are dealing with.
I would recommend you deal with registered real estate agencies if you will be living here long term. By doing this, you will be sure the properties being rented are withing all legal frameworks and all the procedures will be met.

Is there information on property owners on the public Internet? 
If not, how can I request it?

You can find all information about rentals and the real estate agents thought which they are marketed here at L'Express Property. It is the largest real estate directory in Mauritius.

https://www.lexpressproperty.com/en/

I am assuming you are renting for long term and you have paid one month deposit and one month rent advance.
How long you have been in the property?

Dave

Thank you all for the answers.
We have already left this apartment. I want to go to court to recover money back. Since the payment was on the account of his wife in London, to the bank HSBC. She is not a homeowner.
Is there a lawyer who will do this?