T-dualization of type II superstring theory in double space

In this article we offer the new interpretation of T-dualization procedure of type II superstring theory in double space framework. We use the ghost free action of type II superstring in pure spinor formulation in approximation of constant background fields up to the quadratic terms. T-dualization along any subset of the initial coordinates, $x^a$, is equivalent to the permutation of this subset with subset of the corresponding T-dual coordinates, $y_a$, in double space coordinate $Z^M=(x^\mu,y_\mu)$. Demanding that the T-dual transformation law after exchange $x^a\leftrightarrow y_a$ has the same form as initial one, we obtain the T-dual NS-NS and NS-R background fields. The T-dual R-R field strength is determined up to one arbitrary constant under some assumptions. The compatibility between supersymmetry and T-duality produces change of bar spinors and R-R field strength. If we dualize odd number of dimensions $x^a$, such change flips type IIA/B to type II B/A. If we T-dualize time-like direction, one imaginary unit $i$ maps type II superstring theories to type $II^\star$ ones.