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Non-commutative \(L^p\)-spaces. (English) Zbl 1046.46048
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1459-1517 (2003).
The authors present a valuable review of non-commutative \(L^p\) spaces. They start by considering a von Neumann algebra \(N\), equipped with a normal faithful semifinite trace \(\tau\). In this setting, they define the family of (not necessarily bounded) operators affiliated with \(N\), which serves as an analogue of the set of measurable functions. Then they introduce generalized singular values and use complex interpolation to define the space \(L^p(N)\). The authors also present a (more complicated) construction of \(L^p(N)\) for a general von Neumann algebra \(N\). The authors present some properties of non-commutative \(L^p\) spaces. Here is a partial list of the results they mention.
(1) The local theory of non-commutative \(L^p\) spaces is, in some aspects, remarkably similar to that of their commutative analogues. For instance, the moduli of convexity and smoothness behave in the same way. The same is true about type and cotype.
(2) The non-commutative Khintchine’s inequality is different from its classical version. For instance, suppose that \(a_1, \dots, a_n \in L_p(N)\), where \(N\) is a semifinite von Neumann algebra and \(2 \leq p < \infty\). Then \[ \text{Ave} \biggl\| \sum_i \pm a_i \biggr\| \sim \max \biggl\{\biggl\|\Bigl(\sum_i a_i a_i^*\Bigr)^{1/2}\biggr\| ,\biggl\|\Bigl(\sum_i a_i^* a_i\Bigr)^{1/2}\biggr\|\biggr\} \] (the average is taken over all choices of signs). The inequality looks somewhat different for \(1 \leq p < 2\). Nevertheless, many classical martingale results (such as the inequalities of Burkholder–Gundy and Doob) have non-commutative analogues.

(3) The isomorphic classification of the spaces \(L^p(N)\) (where \(N\) is a hyperfinite semifinite von Neumann algebra) is also presented. Moreover, embeddability of non-commutative \(L^p\) spaces into each other is discussed. In particular, the authors state a generalization of a classical result by Kadec and Pelczynski: suppose \(N\) is a finite von Neumann algebra, and \(X\) is a subspace of \(L^p(N)\). Then either \(X\) is isomorphic to a Hilbert space, or \(X\) contains a copy of \(\ell_p\).
In addition, the authors discuss topics such as non-commutative Hardy spaces and BMO, completely bounded Schur multipliers on Schatten classes, and non-commutative analogues of \(\Lambda_p\) sets. Several open problems are also stated in the paper.
For the entire collection see [Zbl 1013.46001].

46L52 Noncommutative function spaces
46B07 Local theory of Banach spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46L07 Operator spaces and completely bounded maps
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis