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When weak and local measure convergence implies norm convergence. (English) Zbl 1423.46086
The paper addresses generalizations of the basic fact that, in $$L^1[0,1]$$, a sequence converges in norm if (and only if) it converges simultaneously in the weak and in the measure topology. If the underlying measure $$\mu$$ is only $$\sigma$$-finite, then the statement remains valid if convergence in measure is replaced by local measure convergence (i.e., convergence in measure on every set of finite measure).
The setting for the generalizations is as follows. Let $$\mathcal{M}$$ be a von Neumann algebra equipped with a faithful normal semifinite trace $$\tau$$ and consider $$S(\mathcal{M},\tau)$$, the set of $$\tau$$-measurable operators affiliated with $$\mathcal{M}$$. (If $$\mathcal{M}=L^\infty(\mu)$$, then $$S(\mathcal{M},\tau)$$ consists of all measurable functions which are bounded except on a set of finite measure.) On $$S(\mathcal{M},\tau)$$, the (by now classical) complete metrizable measure topology $$t_\tau$$ is considered: a sequence $$(X_n)$$ of $$S(\mathcal{M},\tau)$$ converges in measure to $$0$$ if $$\tau(\chi_{]\varepsilon,\infty[}(|X_n|))\to0$$ for every $$\varepsilon>0$$. Also, there is the weaker topology of local convergence in measure on $$S(\mathcal{M},\tau)$$ which is defined naturally such that it corresponds to convergence in measure on sets of finite measure in the case $$\mathcal{M}=L^\infty(\mu)$$. Elements $$X$$ of $$\mathcal{M}$$ with range projection of finite trace are said to be of $$\tau$$-finite range, their closure $$S_0(\mathcal{M},\tau)$$ in $$S(\mathcal{M},\tau)$$ with respect to the measure topology forms the $$\tau$$-compact operators.
In this setting, the authors consider the sets $$I_B=\{A\in S(\mathcal{M},\tau): A=A^*, -B\le A\le B\}$$ and $$K_B=\{A\in S(\mathcal{M},\tau): A^*A\le B\}$$ for positive $$B\in S(\mathcal{M},\tau)$$.
They show that both $$I_B$$ and $$K_B$$ are convex and closed in measure and can be written in the form $$I_B=\{\sqrt{B}T\sqrt{B}: T\in\mathcal{M},\,T=T^*, \,\|T\|_\infty\leq1\}$$ and $$K_B=\{T\sqrt{B}: T\in\mathcal{M},\, \|T\|_\infty\leq1\}$$. The authors show that, if $$\mathcal{M}$$ is atomic, then $$I_B$$ is $$t_\tau$$-compact if and only if $$B$$ is and that this characterizes atomic von Neumann algebras $$(\mathcal{M},\tau)$$; under a mild additional hypothesis, a Krein-Milman theorem with respect to $$t_\tau$$ holds for $$I_B$$. If $$I_B$$ is considered in certain symmetrically normed operator spaces $$E(\mathcal{M},\tau)$$ (which include the predual $$S_1(H)$$ of $$B(H)$$), then a sequence in $$I_B$$, converging locally in measure, converges in norm. To put this in perspective, note that, as the authors point out, in general in $$S_1(H))$$, a sequence converging weakly and locally in measure need not converge in norm, contrary to what happens in the commutative case.

##### MSC:
 46L51 Noncommutative measure and integration 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.)
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##### References:
 [1] Aliprantis, C. D.; Burkinshaw, O., Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs, vol. 105, (2003), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1043.46003 [2] Arazy, Jonathan; Lin, Pei-Kee, On p-convexity and q-concavity of unitary matrix spaces, Integral Equations Operator Theory, 8, 3, 295-313, (1985) · Zbl 0572.46006 [3] Astashkin, S. V.; Kalton, N.; Sukochev, F. A., Cesaro mean convergence of martingale differences in rearrangement invariant spaces, Positivity, 12, 3, 387-406, (2008) · Zbl 1160.46020 [4] Bekjan, T. N.; Dauitbek, D., Submajorization inequalities of τ-measurable operators for concave and convex functions, Positivity, 19, 2, 341-345, (2015) · Zbl 1432.46048 [5] Ber, A. F.; de Pagter, B.; Sukochev, F. A., Derivations in algebras of operator-valued functions, J. Operator Theory, 66, 2, 261-300, (2011) · Zbl 1247.47011 [6] Bikchentaev, A. M., On a lemma of F.A. Berezin, Math. Notes, 87, 5-6, 768-773, (2010) · Zbl 1273.47033 [7] Bikchentaev, A. M., Block projection operator in normed solid spaces of measurable operators, Russian Math. (Izv. VUZ), 56, 2, 75-79, (2012) · Zbl 1266.46050 [8] Bikchentaev, A. M., On Hermitian operators X and Y meeting the condition $$- Y \leq X \leq Y$$, Lobachevskii J. Math., 34, 3, 227-233, (2013) · Zbl 1408.47008 [9] Bikchentaev, A. M., On operator monotone and operator convex functions, Russian Math. (Izv. VUZ), 60, 5, 61-65, (2016) · Zbl 1360.46056 [10] Bikchentaev, A. M., Ideal spaces of measurable operators affiliated to a semifinite von Neumann algebra, Sib. Math. J., 59, 2, 243-251, (2018) · Zbl 06908372 [11] Bikchentaev, A. M., On sets of measurable operators convex and closed in topology of convergence in measure, Dokl. Math., 98, 3, 545-548, (2018) · Zbl 1417.46044 [12] Bikchentaev, A. M.; Sabirova, A. A., Dominated convergence in measure on semifinite von Neumann algebras and arithmetic averages of measurable operators, Sib. Math. J., 53, 2, 207-216, (2012) · Zbl 1257.46034 [13] Bogachev, V. I.; Smolyanov, O. G., Real and Functional Analysis: A University Course, (2009), Regular and Chaotic Dynamics: Regular and Chaotic Dynamics Moscow, (Russian) [14] Chilin, V. I.; Sukochev, F. A., Weak convergence in non-commutative symmetric spaces, J. Operator Theory, 31, 1, 35-65, (1994) · Zbl 0836.46057 [15] Chilin, V. I.; Dodds, P. G.; Sedaev, A. A.; Sukochev, F. A., Characterisations of Kadec-Klee properties in symmetric spaces of measurable functions, Trans. Amer. Math. Soc., 348, 12, 4895-4918, (1996) · Zbl 0862.46015 [16] Długosz, J., A connection between spectral radius and trace, Colloq. Math., 44, 2, 323-326, (1988) · Zbl 0486.47023 [17] Dodds, P. G., Sets of uniformly absolutely continuous norm in symmetric spaces of measurable operators, Trans. Amer. Math. Soc., 368, 6, 4315-4355, (2016) · Zbl 1362.46063 [18] Dodds, P. G.; Ben Pagter, de, The non-commutative Yosida-Hewitt decomposition revisited, Trans. Amer. Math. Soc., 364, 12, 6425-6457, (2012) · Zbl 1303.46054 [19] Dodds, P.; de Pagter, B., Normed Köthe spaces: a non-commutative viewpoint, Indag. Math. (N.S.), 25, 2, 206-249, (2014) · Zbl 1318.46041 [20] Dodds, P. G.; Dodds, T. K.-Y., Lipschitz continuity of the absolute value and Riesz projections in symmetric operator spaces, J. Funct. Anal., 148, 1, 28-69, (1997) · Zbl 0899.46052 [21] Dodds, P. G.; Dodds, T. K.-Y.; de Pagter, Ben, Noncommutative Köthe duality, Trans. Amer. Math. Soc., 339, 2, 717-750, (1993) · Zbl 0801.46074 [22] Dodds, P.; Dodds, T.; Dowling, P.; Lennard, C.; Sukochev, F., A uniform Kadec-Klee property for symmetric operator spaces, Math. Proc. Cambridge Philos. Soc., 118, 3, 487-502, (1995) · Zbl 0847.46033 [23] Dodds, P. G.; Dodds, T. K.; Sukochev, F. A.; Tikhonov, O. Ye., A non-commutative Yosida-Hewitt theorem and convex sets of measurable operators closed locally in measure, Positivity, 9, 3, 457-484, (2005) · Zbl 1123.46044 [24] Dodds, P.; Dodds, T.; Sukochev, F., On p-convexity and q-concavity in non-commutative symmetric spaces, Integral Equations Operator Theory, 78, 91-114, (2014) · Zbl 1298.46052 [25] P. Dodds, B. de Pagter, F. Sukochev, Theory of noncommutative integration, unpublished manuscript. · Zbl 1362.46063 [26] Dunford, N.; Schwartz, J. T., Linear Operators. I. General Theory, Pure and Applied Mathematics, vol. 7, (1958), Interscience Publishers, Inc., With the assistance of W.G. Bade and R.G. Bartle [27] Fack, T.; Kosaki, H., Generalized s-numbers of τ-measurable operators, Pacific J. Math., 123, 2, 269-300, (1986) · Zbl 0617.46063 [28] Gohberg, I. C.; Kreı̌n, M. G., Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs, vol. 18, (1969), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0181.13504 [29] Green, W. L.; Morley, T. D., The extreme points of order intervals of positive operators, Adv. in Appl. Math., 15, 3, 360-370, (1994) · Zbl 0805.47015 [30] Hansen, F., An operator inequality, Math. Ann., 246, 3, 249-250, (1980) · Zbl 0407.47012 [31] J. Huang, F. Sukochev, Interpolation between $$L_0(\mathcal{M}, \tau)$$ and $$L_\infty(\mathcal{M}, \tau)$$, submitted for publication. [32] Huang, J.; Levitina, G.; Sukochev, F., Completeness of symmetric Δ-normed spaces of τ-measurable operators, Studia Math., 237, 3, 201-219, (2017) · Zbl 1385.46046 [33] Huang, J.; Sukochev, F.; Zanin, D., Logarithmic submajorization and order-preserving isometries, submitted for publication [34] Hudzik, H.; Maligranda, L., An interpolation theorem in symmetric function F-spaces, Proc. Amer. Math. Soc., 110, 1, 89-96, (1990) · Zbl 0704.46019 [35] Kadison, R. V.; Ringrose, J. R., Fundamentals of the Theory of Operator Algebras. Vol. II. Advanced Theory, Pure Appl. Math., vol. 100, (1986), Academic Press: Academic Press New York and London · Zbl 0601.46054 [36] Kalton, N. J.; Sukochev, F. A., Symmetric norms and spaces of operators, J. Reine Angew. Math., 621, 81-121, (2008) · Zbl 1152.47014 [37] Krygin, A. V.; Sheremet’ev, E. M.; Sukochev, F., Conjugation of weak and measure convergence in noncommutative symmetric spaces, Dokl. Akad. Nauk UzSSR, 2, 8-9, (1993), (in Russian) [38] A.V. Krygin, E.M. Sheremet’ev, F. Sukochev, Convergence in measure, weak convergence and structure of subspaces in symmetric spaces of measurable operators, 1993, unpublished manuscript. [39] Lord, S.; Sukochev, F.; Zanin, D., Singular Traces. Theory and Applications, de Gruyter Studies in Mathematics, vol. 46, (2013) · Zbl 1275.47002 [40] Lozanovsky, G. Ya., Lozanovsky’s Notebooks. Part II. Problems 610-1469 (Notebooks IV-XII), (2012), Wydawnictwo Uniwersytetu Kazimierza Wielkiego: Wydawnictwo Uniwersytetu Kazimierza Wielkiego Bydgoszcz, Translated from the Russian and edited by Marek Wójtowicz [41] Meyr-Nieberg, P., Banach Lattices, (1991), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York [42] Narcowich, F. J., R-operators. II. On the approximation of certain operator-valued analytic functions and the Hermitian moment problem, Indiana Univ. Math. J., 26, 3, 483-513, (1977) · Zbl 0351.41012 [43] Narcowich, F. J., On the extreme points of the interval between two operators, Proc. Amer. Math. Soc., 67, 1, 84-86, (1977) · Zbl 0369.47020 [44] Nelson, E., Notes on non-commutative integration, J. Funct. Anal., 15, 2, 103-116, (1974) · Zbl 0292.46030 [45] Novikov, Andrej, $$L_1$$-space for a positive operator affiliated with von Neumann algebra, Positivity, 21, 1, 359-375, (2017) · Zbl 1386.46048 [46] Rosenblum, M.; Rovnyak, J., Hardy Classes and Operator Theory, (1985), Oxford University Press: Oxford University Press New York · Zbl 0586.47020 [47] Segal, I. E., A non-commutative extension of abstract integration, Ann. Math., 57, 3, 401-457, (1953) · Zbl 0051.34201 [48] Strătilă, Ş.; Zsidó, L., Lectures on von Neumann Algebras, (1979), Abacus Press: Abacus Press England · Zbl 0391.46048 [49] Sukochev, F., Completeness of quasi-normed symmetric operator spaces, Indag. Math. (N.S.), 25, 2, 376-388, (2014) · Zbl 1298.46051 [50] Takesaki, M., Theory of Operator Algebras I, (1979), Springer-Verlag: Springer-Verlag New York · Zbl 0990.46034 [51] Tikhonov, O. E., Continuity of operator functions in topologies connected with a trace on a von Neumann algebra, Sov. Mat. (Izv. VUZ), 31, 1, 110-114, (1987) · Zbl 0639.46057 [52] Tikhonov, O. E., Subadditivity inequalities in von Neumann algebras and characterization of tracial functionals, Positivity, 9, 2, 259-264, (2005) · Zbl 1102.46040 [53] Yeadon, F. J., Convergence of measurable operators, Proc. Camb. Philos. Soc., 74, 2, 257-268, (1973) · Zbl 0272.46043 [54] Yosida, K., Functional Analysis, Fundamental Principles of Mathematical Sciences, vol. 123, (1980), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0152.32102
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