Italy does not offer a path to residency or citizenship via the purchase of a property.
To get an Italian CBI Visa (Investor Visa) you have to actually invest in a company or make a donation of a minimum of at least €250,000.
Conditions
To be eligible for the Italy CBI Visa, you have to make one of the following investments:
Invest in a Government Bond issued by the Italian Republic.
Invest at least €500,000 Company incorporated and operating in Italy.
Invest at least €250,000 in an innovative start up
Make a philanthropic donation in the fields of culture, education, immigration, management, scientific research. Preservation of cultural and natural heritage.
Benefits
Visa-free travel to Schengen Zone for yourself and your family members
Visa can be renewed every two years if you have maintained the investment
After five years of residency. You can apply for permanent residency
After ten years, you can apply for Italian citizenship