Riesz Representation on Quantum Filtered L^p Space

Abstract

Let

{\mathcal{C}}_T

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathscr{C}}_{T}$$\end{document}

be the von Neumann algebra generated by certain compactly supported fermion fields and

\mathbb{A}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb{A}}$$\end{document}

be the von Neumann algebra generated by the set of simple adapted processes. We derive the Riesz representation

L_A^p([0,T];L^q({\mathcal{C}}_T))\simeq (L_A^{p^{\mathrm{\prime }}}([0,T];L^{q^{\mathrm{\prime }}}({\mathcal{C}}_T)){)}^{\ast },

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_A^p([0,T];L^q({\mathscr{C}}_T))\simeq(L_A^{p^{\prime}}([0,T];L^{q^{\prime}}({\mathscr{C}}_T)))^*,$$\end{document}

where 1 < p ≤ ∞, 1 < q < ∞ and

\frac{1}{p}+\frac{1}{p^{\mathrm{\prime }}}=1,\frac{1}{q}+\frac{1}{q^{\mathrm{\prime }}}=1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\frac{1}{p}+\frac{1}{p^{\prime}}=1,\,\frac{1}{q}+\frac{1}{q^{\prime}}=1$$\end{document}

. To achieve it, the authors develop an equivalence relation in a more general setting between the above isomorphism and the boundedness of a kind of projection operator. Even in the commutative setting as a special case, their method has certain advantages.

Keywords

Riesz representation / Non-commutative L^p space / Fermion fields / Conditional expectation / 46L51 / 47C15 / 60A10

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