How to auto -populate user look up field on the case object?

trigger CaseTrigger on Case(before insert, before update){
List<String> emails = new List<String>();     
Map<String, Id> userEmailToIDMap  = new Map<String, Id>();
for(Case c: trigger.new){
     emails.add(c. SuppliedEmail);
     }
List<USer> users = [SELECT Id, Email from user where Email IN :emails];
for(User u : users){
       userEmailToIDMap.put(u.Email, u.Id();
}
for(Case c:trigger.new){
if(userEmailToIDMap.get(c.SuppliedEmail) != null){
    c.Owner = userEmailToIDMap.get(c.SuppliedEmail);
{
}
}