Reverse order law in C super(&; lowast; -algebras)

We study equivalent conditions for the reverse order law [inline image] in C super([lowast]-algebras. As corollaries, we obtain some recent and special results. A commercial graphite is ball-milled and the pristine and ball-milled graphites are characterised for use as negative electrodes in lithium-ion capacitors (LICs). Ball milling graphite results in a decrease in discharge capacity when the charge rate is relatively slow, whereas, it leads to an increase in discharge capacity when the charge rate is high. When charged at 0.1 C, the discharge capacities of pristine, 3 h, 10 h and 30 h-milled materials at 6 C are 75, 69, 67 and 66% of theoretical capacity, respectively; however, when charged at 60 C, the discharge capacities of pristine, 3 h, 10 h and 30 h-milled materials, at 60 C, fall to 0.9, 13, 23 and 24% of theoretical capacity, respectively (theoretical capacity: 372 mAh g) super(-)1, for LiC sub(6 stoichiometry). This difference in the discharge rate capability behaviour of the pristine and ball-milled graphites with charge rate is attributed to the interplay of two different charge storage mechanisms: Li-ion intercalation and Li-ion adsorption that co-exist; but the later becomes more significant for milled samples. In terms of cycle-life performance, pristine and ball-milled graphites follow similar trends observed for their rate capability behaviour.)