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 $L^{p}$?
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5$\begingroup$ It seems hard to know how this could be answered on MathOverFlow: it seems to require sketching the entire construction, including dual weights on crossed products etc. To improve the question, I wonder if you could add some references (and links?) to papers / articles you have read which shows what your knowledge of Haagerup $L^p$ spaces is. Is there some particular place you are then stuck? (e.g. "I have read [1] and [2], and am now interested in [3], but I don't follow the paragraph at the top of page 123 in [3], because...") $\endgroup$– Matthew DawsCommented Mar 27, 2019 at 8:44
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2 Answers
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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.
I'm not aware of a more modern "introduction" in this style. It would be nice to know if such a thing exists.
answered Mar 29, 2019 at 12:58
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$\begingroup$ FWIW, I think Terp has now published a LaTeXed version in Moslehian's AOT journal... $\endgroup$ Commented Mar 30, 2019 at 3:08
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$\begingroup$ @YemonChoi I think you are thinking of Terp's paper "Lp Fourier transformation on non-unimodular locally compact groups." which is different material. $\endgroup$ Commented Mar 30, 2019 at 9:38
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$\begingroup$ And it is actually her PhD thesis, not MS. $\endgroup$ Commented Apr 3, 2019 at 22:35
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1$\begingroup$ This might be my misunderstanding: according to the document itself, it is part of Terp's lic.scient. thesis. According to wikipedia and Danish wikipedia this is something half-way between an MSc and PhD; probaby in 1981 actually much the same as what I think of as a PhD (from the UK). Anyway, I have updated the answer to make it accurate! $\endgroup$ Commented Apr 4, 2019 at 8:28
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How the norm on L^{P} space related to weight φ?
The L^p-spaces and their norms are independent of the choice of the weight φ.
See, for instance, the exposition by Yamagami in “Algebraic Aspects in Modular Theory”. Section 2 contains an exposition of noncommutative L^p-spaces that makes it clear there is no dependence on any choice of a weight.
The paper https://arxiv.org/abs/1309.7856 contains a more modern exposition in Section 6, in particular, it explains how to turn the L^p-space construction into a functor defined on the category of von Neumann algebras with appropriate morphisms.
answered Mar 27, 2019 at 13:39
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$\begingroup$ What my confusion is crossed product with modular automorphism group is semifinite so it has the semifinite trace, then the norm coming from that trace, or are we getting something different? $\endgroup$ Commented Mar 28, 2019 at 4:49
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1$\begingroup$ @user136400: No, it does not. The relation between the L^1-norm and the semifinite trace τ is given by Haagerup's formula τ(1−e(λ))=φ(1)/λ, where e denotes the spectral projection and φ∈L^1. This is explained in Yamagami's paper. $\endgroup$ Commented Mar 28, 2019 at 12:35