There is a definition in Haagerup's paper on $L^{P}$ spaces for weights, my question is after putting the norm is it become semifinite $L^{P}$ space on the crossed product? I am not clear please help. How the norm on L^{P} space related to weight $\varphi$?

There is a definition in Haagerup's paper on $L^{P}$ spaces for weights, my question is after putting the norm is it become semifinite $L^{P}$ space on the crossed product? I am not clear please help. How the norm on L^{P} space related to weight $\varphi$? I am reading this:https://arxiv.org/pdf/0806.3635.pdf, I have not understood section 1.2. With $L^{0}(R, \tau)$ the $tr$ norm is defined, I am not clear whether $tr$ make $L^{p}(M, \varphi)$ make it semifinite $L^{p}$?

There is a definition in Haagerup's paper on $L^{P}$ spaces for weights, my question is after putting the norm is it become semifinite $L^{P}$ space on the crossed product? I am not clear please help.

There is a definition in Haagerup's paper on $L^{P}$ spaces for weights, my question is after putting the norm is it become semifinite $L^{P}$ space on the crossed product? I am not clear please help. How the norm on L^{P} space related to weight $\varphi$?