  We extend the $\lambda$-theory of operator spaces given by Defant and Wiesner
(2014), that generalizes the notion of the projective, Haagerup and Schur
tensor norm for operator spaces to matrix ordered spaces and Banach
$*$-algebras. Given matrix regular operator spaces and operator systems, we
introduce cones related to $\lambda$ for the algebraic tensor product that
respect the matricial structure of matrix regular operator spaces and operator
systems, respectively. The ideal structure of $\lambda$-tensor product of
$C^*$-algebras has also been discussed.
