NOVELTY - A lithium-containing complex oxide containing a transition metal element is contacted with a composition (A) to obtain particles (1). To the obtained particles (1) compound containing a metal element (M) is adhered. Then, particles (1) are mixed with a compound (B) generating hydrogen fluoride by heating and mixture is heated to obtain particles (2) having a coating layer containing a metal element and a fluorine element on the surface of a lithium-containing composite oxide.    USE - Manufacture of positive electrode active material for lithium-ion secondary batteries (claimed) for mobile telephone, vehicle-mounted battery and notebook computer.    ADVANTAGE - Positive electrode active material having excellent cycling characteristics even when charged at high voltages and high discharging capacity per unit mass is manufactured with high yield by a simple process.    DETAILED DESCRIPTION - A lithium-containing complex oxide containing a transition metal element is contacted with a composition (A) to obtain particles (1). To the obtained particles (1) compound containing a metal element (M) is adhered. Then, particles (1) are mixed with a compound (B) generating hydrogen fluoride by heating and mixture is heated to obtain particles (2) having a coating layer containing a metal element and a fluorine element on the surface of a lithium-containing composite oxide. The composition (A) is obtained by dispersing a compound that does not contain elemental lithium and containing metal element(s) chosen from magnesium, calcium, strontium, barium, yttrium, titanium, zirconium, hafnium, vanadium, niobium, tantalum, chromium, molybdenum, tungsten, manganese, iron, cobalt, nickel, lead, copper, zinc, aluminum, indium, tin, antimony, bismuth, lanthanum, cerium, praseodymium, neodymium, gadolinium, dysprosium, erbium and ytterbium in a solvent. An INDEPENDENT CLAIM is included for lithium ion secondary battery, which contains positive electrode, negative electrode and nonaqueous electrolyte.