I always thought the primary belief of Keynesians was that poor aggregate demand led to recessions. Sticky nominal wages was a factor in high unemployment, but it is weak aggregate demand that was where the horns are. Therefor their solution was to pump money into the demand side so that would drive spending. Demand side spending encourages business investment and this jumpstarts the economy.

As far as I know, most economists would argue that if nominal wages weren't sticky, unemployment would be less of a problem. When you say "Italy should have no unemployment", I'm not sure who you are arguing with, but I don't think it is Keynesians.