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.)

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.)

Reviewer: Hermann Pfitzner (Orléans)

##### 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.) |

##### Keywords:

symmetric spaces of measurable operators; noncommutative function spaces; unitary matrix spaces; Banach envelopes; Mackey completion
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\textit{M. M. Czerwińska} and \textit{A. Kamińska}, Positivity 21, No. 1, 473--492 (2017; Zbl 1386.46049)

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##### References:

[1] | Arazy, J, On the geometry of the unit ball of unitary matrix spaces, Integral Equ. Oper. Theory, 4, 151-171, (1981) · Zbl 0459.47029 |

[2] | Bennett, C., Sharpley, R.: Interpolation of operators, pure and applied mathematics, vol. 129. Academic Press Inc., Boston (1988) · Zbl 0647.46057 |

[3] | Cwikel, M; Fefferman, C, Maximal seminorms in weak \(L^1\), Studia Mathematica, 69, 149-154, (1980) · Zbl 0459.46022 |

[4] | Cwikel, M; Fefferman, C, The canonical seminorm on weak \(L^1\), Studia Mathematica, 78, 275-278, (1984) · Zbl 0574.46020 |

[5] | Czerwińska, MM; Kamińska, A, \(k\)-extreme points in symmetric spaces of measurable operators, Integral Equ. Oper. Theory, 82, 189-222, (2015) · Zbl 1343.46060 |

[6] | Dodds, PG; Dodds, TK; Pagter, B, Fully symmetric operator spaces, Integral Equ. Oper. Theory, 15, 942-972, (1992) · Zbl 0807.46028 |

[7] | Dodds, PG; Dodds, TK; Pagter, B, Noncommutative Köthe duality, Trans. Am. Math. Soc., 339, 717-750, (1993) · Zbl 0801.46074 |

[8] | Dodds, P. G. , De Pagter, B., Sukochev, F. A.: Theory of noncommutative integration, unpublished monograph; to appear |

[9] | Drewnowski, L, Compact operators on Musielak-Orlicz spaces, Comment. Math. Prace Math., 27, 225-232, (1988) · Zbl 0676.46024 |

[10] | Drewnowski, L; Nawrocki, M, On the MacKey topology of Orlicz sequence spaces, Arch. Math., 39, 59-68, (1982) · Zbl 0502.46004 |

[11] | Fack, T; Kosaki, H, Generalized \(s\)-numbers of \(τ \)-measurable operators, Pacific J. Math., 123, 269-300, (1986) · Zbl 0617.46063 |

[12] | Kalton, NJ, Compact and strictly singular operators on Orlicz spaces, Israel J. Math., 26, 126-136, (1977) · Zbl 0348.47016 |

[13] | Kalton, NJ, Orlicz sequence spaces without local convexity, Math. Proc. Camb. Phil. Soc., 81, 253-277, (1977) · Zbl 0345.46013 |

[14] | Kalton, NJ, The three space problem for locally bounded \(F\)-spaces, Compositio Math., 37, 243-276, (1978) · Zbl 0395.46003 |

[15] | Kalton, NJ, Banach envelopes of nonlocally convex spaces, Can. J. Math., 38, 65-86, (1986) · Zbl 0577.46016 |

[16] | Kalton, N.J., Peck, N.T., Roberts, J.W.: An \(F\)-space Sampler, London Mathematical Society Lecture Note Series 89. Cambridge University Press, Cambridge (1984) · Zbl 0556.46002 |

[17] | Kalton, NJ; Sukochev, FA, Symmetric norms and spaces of operators, J. Reine Angew. Math., 621, 81-121, (2008) · Zbl 1152.47014 |

[18] | Kamińska, A., Lin, P.K.: Banach envelopes of \(p\)-Banach lattices, \(0<p<1\), and Cesáro spaces. Funct. Approx. Comment. Math. 50(2), 297-306 (2014) |

[19] | Kamińska, A; Mastyło, M, Abstract duality sawyers formula and its applications, Monatsh. Math., 151, 223-245, (2007) · Zbl 1132.46020 |

[20] | Kamińska, A; Raynaud, Y, New formulas for decreasing rearrangements and a class of Orlicz-Lorentz spaces, Rev. Mat. Complut., 27, 587-621, (2014) · Zbl 1314.46035 |

[21] | Kamińska, A; Raynaud, Y, Isomorphic copies in the lattice \(E\) and its symmetrization \(E^{(*)}\) with applications to Orlicz-Lorentz spaces, J. Funct. Anal., 257, 271-331, (2009) · Zbl 1183.46030 |

[22] | S. G. Kreĭn, Yu. I. Petunīn, and E. M. Semënov, Interpolation of Linear Operators, Transl. Math. Monogr. 54, Amer. Math. Soc., Providence, RI, 1982 |

[23] | Lord, S., Sukochev, F., Zanin, D.: Singular traces, theory and applications, De Gruyter Studies in Mathematics 46. De Gruyter, Berlin (2013) · Zbl 1275.47002 |

[24] | Nawrocki, M, Remarks on duals of Schatten ideals \(S_X\), Funct. Approx. Comment. Math., 20, 3-7, (1992) · Zbl 0801.47030 |

[25] | de Pagter, B.: Non-commutative Banach function spaces, Positivity, pp. 197-227. Birkhäuser, Basel, Trends Math. (2007) · Zbl 1162.46034 |

[26] | Peetre, J, Remark on the dual of an interpolation space, Math. Scand., 34, 124-128, (1974) · Zbl 0285.46029 |

[27] | Pietsch, A, About the Banach envelope of \(ℓ _{1,∞ }\), Rev. Mat. Complut., 22, 209-226, (2009) · Zbl 1175.46004 |

[28] | J. H. Shapiro, Remarks on \(F\)-spaces of analytic functions, Banach spaces of analytic functions (Proc. Pelczynski Conf., Kent State Univ., Kent, Ohio, 1976), 107-124. Lecture Notes in Math. 604, Springer, Berlin, (1977) · Zbl 1152.47014 |

[29] | Sukochev, F, Completeness of quasi-normed symmetric operator spaces, Indag. Math. (N.S.), 25, 376-388, (2014) · Zbl 1298.46051 |

[30] | Takesaki, M.: Theory of operator algebras I. Springer, New York (1979) · Zbl 0436.46043 |

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