Answer:  

Dr Kenji Nomiya, who is synthesizing a noble metal complex which has pharmacological actions, kindly answered the question above:
 
 
Hydrated silver ion quickly reacts with chloride ion. So, if tap water is added into silver nitrate solution, silver chloride is generated and you will see white precipitation.
 
On the other hand, silver coated on a water filter is solid. And a trace amount of sliver ion is dissolved little by little into water from the solid silver. Even though the concentration is so low, the silver ion has enough antibacterial activity. (Silver ion inhibits reproduction of bacteria, and bacteria dies out gradually.)
  
Because the concentration of silver ion that is generated from solid silver on the water filter is significantly low, precipitation by combining silver ion with chloride ion in tap water rarely occurs, and the surface of the filter is not covered by the precipitation of silver chloride.
 
 
Acknowledgement
We would like to thank Dr. Kenji Nomiya for his kind answer.
 
 

This article is translated by Chemistryquestion.com from the original article in Chemistryquestion.jp.  Please let us know if you find any errors.