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 $L^{p}$?



Regarding Haagerup $L^{P}$ spaces
For general background on Haagerup $L^p$ spaces, I rather like Terp's lic.scient. thesis. This is old (e.g. is typeset, not LaTeX), but very self-contained and easy to read. It is fortunately available [here](https://www.fuw.edu.pl/~kostecki/scans/terp1981.pdf).


I'm not aware of a more modern "introduction" in this style. It would be nice to know if such a thing exists.


