Cathode active material for a lithium secondary battery, a positive electrode, and a lithium secondary battery for a lithium secondary battery

An object of the present invention is to provide is to provide a cathode active material for a lithium secondary battery exhibiting high output at a high current rate. Lithium cathode active material for a lithium secondary battery of the present invention, the surface of the lithium-containing composite metal oxide of lithium ion doping and dedoping can primary particles comprise secondary particles formed by agglomerating, with a covering layer a positive electrode active material for a secondary battery, satisfying the following requirements (1) to (3). (1) the lithium-containing composite metal oxide has a crystal structure of alpha-NaFeO (2) the coating layer comprises a metal composite oxide of Li and (3) the average secondary particle diameter of the lithium secondary battery positive electrode active material for is at 2μm or more 20μm or less, BET specific surface area of not more than 0.1 m