Hi,
The search field withing the Live Activity is a Basic Search across different fields of different BO (business objects) . It brings in information from Account, Individual Customer and Contacts and the behaviour would not be same as when you do a specific search in the account field on the list.
If you want to implement a specific search, then you would have to implement a BADI.
check the link for BADI :-https://wiki.wdf.sap.corp/wiki/display/ServiceOD/Live+Activity+Customer+Search+BADI+Exit
Regards,
Suresh