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## Studia Mathematica

1998 | 128 | 3 | 273-285
Tytuł artykułu

### Factorization of operators on C*-algebras

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let A be a C*-algebra. We prove that every absolutely summing operator from A into $ℓ_2$ factors through a Hilbert space operator that belongs to the 4-Schatten-von Neumann class. We also provide finite-dimensional examples that show that one cannot replace the 4-Schatten-von Neumann class by the p-Schatten-von Neumann class for any p < 4. As an application, we show that there exists a modulus of capacity ε → N(ε) so that if A is a C*-algebra and $T ∈ Π_1(A,ℓ_2)$ with $π_1(T) ≤ 1$, then for every ε >0, the ε-capacity of the image of the unit ball of A under T does not exceed N(ε). This answers positively a question raised by Pełczyński.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
273-285
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-02-17
poprawiono
1997-06-24
Twórcy
• Department of Mathematics and Statistics, Miami University, Oxford, Ohio 45056, U.S.A., randrin@muohio.edu
Bibliografia
• [1] C. H. Chu and B. Iochum, Complementation of Jordan triples in von Neumann algebras, Proc. Amer. Math. Soc. 108 (1990), 19-24.
• [2] J. Diestel, Sequences and Series in Banach Spaces, Grad. Texts in Math. 92, Springer, New York, 1984.
• [3] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math. 43, Cambridge Univ. Press, 1995.
• [4] Y. Gordon and D. R. Lewis, Absolutely summing operators and local unconditional structures, Acta Math. 133 (1974), 27-48.
• [5] S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72-104.
• [6] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. II, Pure Appl. Math. 100, Academic Press, Orlando, Fla., 1986.
• [7] A. N. Kolmogorov and V. M. Tikhomirov, ε-Entropy and ε-capacity of sets in function spaces, Uspekhi Mat. Nauk 86 (1959), 3-86 (in Russian); English transl.: Amer. Math. Soc. Trans. Ser. 2 17 (1961), 277-364.
• [8] S. Kwapień, Some remarks on (p,q)-summing operators on $ℓ_p$-spaces, Studia Math. 29 (1968), 327-337.
• [9] B. S. Mitjagin [B. S. Mityagin] and A. Pełczyński, Nuclear operators and approximative dimension, in: Proc. ICM (Moscow, 1966), Mir, Moscow, 1968, 366-372.
• [10] A. Pełczyński, Compactness of absolutely summing operators, in: Linear and Complex Analysis Problem Book 3, Part I, V. P. Havin and N. K. Nikolski (eds.), Lecture Notes in Math. 1573, Springer, 1994, 19-20.
• [11] A. Pełczyński and C. Schütt, Factoring the natural injection $ι^(n):L_n^∞ → L_n^1$ through finite dimensional Banach spaces and geometry of finite dimensional unitary ideals, in: Mathematical Analysis and Applications, Part B, L. Nachbin (ed.), Adv. Math. Suppl. Stud. 7B, Academic Press, 1981, 653-683.
• [12] A. Pietsch, Operator Ideals, North-Holland Math. Library 20, North-Holland, 1980.
• [13] G. Pisier, Grothendieck's theorem for non-commutative C*-algebras with appendix on Grothendieck's constants, J. Funct. Anal. 29 (1978), 397-415.
• [14] G. Pisier, Factorization of operators through $L_{p∞}$ or $L_{1∞}$ and non-commutative generalizations, Math. Ann. 276 (1986), 105-136.
• [15] N. Randrianantoanina, Absolutely summing operators on non-commutative C*-algebras and applications, Houston J. Math., to appear.
• [16] A. I. Singh and N. Mittal, Complete finite representability in C*-algebras, Yokohama Math. J. 38 (1991), 83-94.
• [17] M. Takesaki, Theory of Operator Algebras I, Springer, New York, 1979.
• [18] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Pitman Monographs Surveys Pure Appl. Math. 38, Longman Sci. Tech., 1989.
• [19] H. Upmeier, Symmetric Banach Manifolds and Jordan C*-Algebras, North-Holland Math. Stud. 104, North-Holland, Amsterdam, 1985.
• [20] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Univ. Press, 1991.
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Bibliografia
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