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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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