It seems that my livetable working like a charm now. :-)
 By the way, it's a shame that boolean property didn't work
 with columnsProperties with this :
 #set ($columnsProperties = {
   'MyBooleanPropertyField':
 {"type":"list",'class':'xxxxx'},
 It didn't work with text either.
 Then I suppose I must change the property field type in a
 database list or in static list. 
Right, I found this issue for that: 
http://jira.xwiki.org/browse/XWIKI-6118: "Add
boolean property support on livetables"