Is it acceptable to omit 来 in 起来 as a direction complement? In the following sentence:


  在帐篷里，周海提议大家一起唱羌族民歌，渐渐地场面也活跃起来。周海还给他们讲羌族的文化和历史。孩子们渐渐建立起对他的信任。


I wonder why it is 起, not 起来 in the sentence above. In my understanding, the correct complement (direction complement) is 起来, but in this sentence it does not have 来.

So is it correct that this usage of 起 is direction complement as well? In that case, why is it 起, not 起来?
起


  （用在动词后，表示动词的趋向）:
  
  hold high the flag; raise the flag;
  
  举起旗子
  
  take up arms;
  
  拿起武器


As you can see, the verb should be followed by 起 when there is an object at the end, for example 拿起武器. When there's no object, 起来 should be used, e.g. 拿起来. Basically, you can consider 拿起 as a transitive and 拿起来 as an intransitive. 

You can also use 把. It will be like 把...拿起来. e.g.  把武器拿起来.

So, 孩子们渐渐建立起对他的信任 can be paraphrased as 孩子们渐渐把对他的信任建立起来.