First, make a simple syrup by combining the sugar and water in a small saucepan.
Bring the mixture to a boil, stirring constantly until the sugar melts. Allow it to cool.
Place diced cherimoya in a blender or food processor, along with the simple syrup and lime juice, and purée until smooth.
If you have an ice cream maker, spoon the mixture into it and follow the manufacturer’s instructions.
If you do not have an ice cream maker, place the mixture in a tall sealing container and freeze it for about 90 minutes.
Remove it and stir the sorbet with a whisk.
Return it to the freezer and stir once every hour for about four hours.
This will incorporate air into the mixture, making it light and creamy.