Algebraic 𝐾-theory of the two-periodic first Morava 𝐾-theory

Using the root adjunction formalism developed in an earlier work and logarithmic THH, we obtain a simplified computation of T ( 2 ) ∗ K ( k u ) T(2)_*K(ku) for p > 3 p>3 . Through this, we also produce a new algebraic K K -theory computation; namely we obtain T ( 2 ) ∗ K ( k u / p ) T(2)_*K(ku/p) , where k u / p ku/p is the 2 2 -periodic Morava K K -theory spectrum of height 1 1 .