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Banach envelopes in symmetric spaces of measurable operators. (English) Zbl 1386.46049
Positivity 21, No. 1, 473-492 (2017); erratum ibid. 21, No. 1, 493 (2017).
Many papers in the literature concern the question which properties pass from a symmetric (function or sequence) space to its Banach envelope and its noncommutative counterparts. The paper under review gives some natural and satisfactory answers for the property of being symmetric when \(E\) is quasi-normed.
Recall that a map \(x\mapsto\|x\|_X\) on a vector space \(X\) is called a quasi-norm if it is like a norm except for the triangle inequality which is replaced by the existence of a \(K\geq1\) such that \(\|x+y\|_X\leq K(\|x\|_X+\|y\|_X)\) for \(x,y\in X\). Classical examples are the sequence spaces \(\ell_p\) and the function spaces \(L_p\) where (as throughout below) \(0 < p < 1\). In the two-dimensional \(\mathbb{R}^2\) with the \(\ell_p\)-norm, the convex hull of the unit ball is the unit ball of \((\mathbb{R}^2, \|\cdot\|_1)\), hence, intuitively, \(\ell_1\) is the smallest Banach space containing \(\ell_p\). This is a simple motivating example from N. J. Kalton et al. [An F-space sampler. Cambridge: Cambridge University Press. London: London Mathematical Society (1984; Zbl 0556.46002)], a reference for basics on Banach envelopes (another reference is the work of N. Kalton [in: Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland. 1099–1130 (2003; Zbl 1059.46004)]): If \((X, \|\cdot\|_X)\) is quasi-normed, then the Minkowski functional of the convex hull of its unit ball gives rise to a semi-norm \(\|\cdot\|_{\widehat{X}}\); the completion of the quotient of \(X\) by the null space of this semi-norm is called the Banach envelope \(\widehat{X}\) of \(X\). If the dual of \((X,\|\cdot\|_X)\) separates the points of \(X\), then \(\|\cdot\|_{\widehat{X}}\) is actually a norm and \(\widehat{X}\) is just the completion of \((X, \|\cdot\|_{\widehat{X}})\); further, \(\|\cdot\|_{\widehat{X}}\leq\|\cdot\|_X\) on \(X\), \(X\) and \(\widehat{X}\) have the same duals with equal norms. The dual of \(\ell_p\) separates the points of \(\ell_p\), whereas the dual of \(L_p\) with a non-atomic measure is trivial in the sense that it consists only of the zero functional. Further, \(\widehat{\ell_p}=\ell_1\), \(\ell_p^*=\ell_\infty\), \(\widehat{L_p}=\{0\}\), \(L_p^*=\{0\}\).
Let \(\tau\) be a faithful normal \(\sigma\)-finite trace on a non-atomic semi-finite von Neumann algebra \(\mathcal{M}\). Let \(\mu(x)\) be the generalized singular value function of a \(\tau\)-measurable operator \(x\) affiliated to \(\mathcal{M}\) which is the decreasing rearrangement of \(x\) in the case when \(\mathcal{M}\) is commutative.
Let \(E\subset L_0(I)\) be a quasi-normed symmetric (function or sequence) space with separating dual, where \(I\) is either the interval \([0,\tau(1))\) (possibly \(\tau(1)=\infty\)) with the Lebesgue measure or \(\mathbb{N}\) with the counting measure; if \(I=\mathbb{N}\), then \(E\subset c_0\) is supposed. (‘Symmetric’ means that, if \(f\in L_0(I)\), \(g\in E\), \(\mu(f)\leq\mu(g)\), then \(f\in E\) and \(\|f\|_E\leq\|g\|_E\).) If \(I=[0,\tau(1))\), then the non-commutative counterpart of \(E\), the symmetric space of measurable operators \(E(\mathcal{M},\tau)\), is defined as the set of all \(\tau\)-measurable operators \(x\) affiliated to \(\mathcal{M}\) such that \(\mu(x)\in E\); if \(I=\mathbb{N}\), then the non-commutative counterpart of \(E\), the unitary matrix space \(C_E\), is defined to be the set of all compact operators \(x\) whose sequence of singular values \(S(x)\) belongs to \(E\), for example, the Schatten classes \(C_p=C_{\ell_p}\). They are quasi-normed by \(\|x\|_{E(\mathcal{M},\tau)}=\|\mu(x)\|_E\), respectively by \(\|x\|_{C_E}=\|S(x)\|_E\).
Finally, \(E\) is defined to satisfy the condition (HC) if any Cauchy sequence in \((E,\|\cdot\|_{\widehat{E}})\) converging to \(0\) in the measure topology converges to \(0\) in the norm \(\|\cdot\|_{\widehat{E}}\). This condition is related to the extendability of the inclusion \(E\hookrightarrow L_1+L_\infty\) from \(E\) to \(\widehat{E}\).
Now we can state some of the main results.
1. (HC) is shown to be satisfied if \(E\) is order continuous, but the authors point out that there are \(E\) satisfying (HC) without being order continuous.
2. It is partly known, and partly shown in the paper under review that, in general, \((E,\|\cdot\|_{\widehat{E}})\) is symmetric.
Moreover, it is shown that, if \(E\) satisfies (HC), then \((\widehat{E},\|\cdot\|_{\widehat{E}})\) is a Banach symmetric space.
3. Calculation of the norm of the Banach envelope: In general, \[ \|x\|_{\widehat{E(\mathcal{M},\tau)}}=\|\mu(x)\|_{\widehat{E}} \quad\text{ for } x\in E(\mathcal{M},\tau) \] and \[ \|x\|_{\widehat{C_E}}=\|S(x)\|_{\widehat{E}} \quad\text{ for } x\in C_E. \] Moreover, if \(E\) satisfies (HC), then \[ (\widehat{E}(\mathcal{M},\tau),\|\cdot\|_{\widehat{E}(\mathcal{M},\tau)}) =(\widehat{E(\mathcal{M},\tau)},\|\cdot\|_{\widehat{E(\mathcal{M},\tau)}})\eqno(*) \] and \[ (\widehat{C_E},\|\cdot\|_{\widehat{C_E}})=(C_{\widehat{E}},\|\cdot\|_{C_{\widehat{E}}}), \] in both cases with equal norms. In particular, since \(\ell_p\) is order continuous, \(\widehat{C_p}=C_1\) with \(\|x\|_{\widehat{C_p}}=\|S(x)\|_1\). (Also, the special case \(E=L_p\) is considered to the effect that \(\widehat{L_p(\mathcal{M},\tau)}=\{0\}\), but in the paper \(E\) is supposed to have separating dual and I don’t know to which extent \((*)\) remains valid also in the case of a trivial dual.)

MSC:
46L52 Noncommutative function spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47L20 Operator ideals
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
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