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Factorization through matrix spaces for finite rank operators between $$C^*$$-algebras. (English) Zbl 0947.46053
The authors present factorization results involving the completely bounded norm $$\|.\|_{cb}$$ and the Haagerup decomposable norm $$\|.\|_{dec}$$. Here is the main theorem:
Let $$A$$ and $$B$$ be two $$C^*$$-algebras and let $$M_n$$ be the $$C^*$$-algebra of all $$n\times n$$ complex matrices. Suppose $$u:A\to B$$ is a finite rank bounded operator. Then:
(a) For every $$\varepsilon> 0$$ there exists an integer $$n\geq 1$$ and two linear maps $$\alpha: A\to M_n$$, $$\beta: M_n\to B$$ such that $$u= \beta\alpha$$ and $$\|\alpha\|_{cb}\|\beta\|_{dec}\leq (1+\varepsilon)\|u\|_{dec}$$.
(b) If $$A$$ is a von Neumann algebra and $$u$$ is $$w^*$$-continuous, then $$\alpha$$ in (a) can be taken $$w^*$$-continuous, too.
The paper contains also some related results and comments on the properties of the Haagerup decomposable norm.

##### MSC:
 46L99 Selfadjoint operator algebras ($$C^*$$-algebras, von Neumann ($$W^*$$-) algebras, etc.) 47L99 Linear spaces and algebras of operators 47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators 47B49 Transformers, preservers (linear operators on spaces of linear operators) 46A32 Spaces of linear operators; topological tensor products; approximation properties 46L06 Tensor products of $$C^*$$-algebras
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##### References:
 [1] David P. Blecher, The standard dual of an operator space , Pacific J. Math. 153 (1992), no. 1, 15-30. · Zbl 0726.47030 · doi:10.2140/pjm.1992.153.15 [2] David P. Blecher and Vern I. Paulsen, Tensor products of operator spaces , J. Funct. Anal. 99 (1991), no. 2, 262-292. · Zbl 0786.46056 · doi:10.1016/0022-1236(91)90042-4 [3] Man Duen Choi and Edward G. Effros, Nuclear $$C^*$$-algebras and injectivity: the general case , Indiana Univ. Math. J. 26 (1977), no. 3, 443-446. · Zbl 0378.46052 · doi:10.1512/iumj.1977.26.26034 [4] Man Duen Choi and Edward G. Effros, Nuclear $$C^*$$-algebras and the approximation property , Amer. J. Math. 100 (1978), no. 1, 61-79. · Zbl 0397.46054 · doi:10.2307/2373876 [5] Edward G. Effros and Uffe Haagerup, Lifting problems and local reflexivity for $$C^ \ast$$-algebras , Duke Math. J. 52 (1985), no. 1, 103-128. · Zbl 0613.46047 · doi:10.1215/S0012-7094-85-05207-X [6] E. Effros, M. Junge, and Z.-J. Ruan, Integral mappings and the principal of local reflexivity for noncommutative $$L^{1}$$-spaces , to appear in Ann. of Math. (2). · Zbl 0957.47051 · doi:10.2307/121112 · www.math.princeton.edu · eudml:120860 [7] Edward G. Effros and Zhong-Jin Ruan, On approximation properties for operator spaces , Internat. J. Math. 1 (1990), no. 2, 163-187. · Zbl 0747.46014 · doi:10.1142/S0129167X90000113 [8] Edward G. Effros and Zhong-Jin Ruan, A new approach to operator spaces , Canad. Math. Bull. 34 (1991), no. 3, 329-337. · Zbl 0769.46037 · doi:10.4153/CMB-1991-053-x [9] E. Effros and Z.-J. Ruan, The Grothendieck-Pietsch and Dvoretzky-Rogers theorems for operator spaces , J. Funct. Anal. 122 (1994), no. 2, 428-450. · Zbl 0802.46014 · doi:10.1006/jfan.1994.1075 [10] Edward G. Effros and Zhong-Jin Ruan, Mapping spaces and liftings for operator spaces , Proc. London Math. Soc. (3) 69 (1994), no. 1, 171-197. · Zbl 0814.47053 · doi:10.1112/plms/s3-69.1.171 [11] Uffe Haagerup, Injectivity and decomposition of completely bounded maps , Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983), Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 170-222. · Zbl 0591.46050 · doi:10.1007/BFb0074885 [12] Uffe Haagerup and Gilles Pisier, Factorization of analytic functions with values in noncommutative $$L_ 1$$-spaces and applications , Canad. J. Math. 41 (1989), no. 5, 882-906. · Zbl 0821.46074 · doi:10.4153/CJM-1989-041-6 [13] Uffe Haagerup and Gilles Pisier, Bounded linear operators between $$C^ *$$-algebras , Duke Math. J. 71 (1993), no. 3, 889-925. · Zbl 0803.46064 · doi:10.1215/S0012-7094-93-07134-7 [14] M. Junge, Factorization theory for spaces of operators , Habilitationsschrift, Universitat Kiel, 1996. [15] M. Junge and G. Pisier, Bilinear forms on exact operator spaces and $$B(H)\otimes B(H)$$ , Geom. Funct. Anal. 5 (1995), no. 2, 329-363. · Zbl 0832.46052 · doi:10.1007/BF01895670 · eudml:58192 [16] Eberhard Kirchberg, $$C^*$$-nuclearity implies CPAP , Math. Nachr. 76 (1977), 203-212. · Zbl 0383.46011 · doi:10.1002/mana.19770760115 [17] Eberhard Kirchberg, Commutants of unitaries in UHF algebras and functorial properties of exactness , J. Reine Angew. Math. 452 (1994), 39-77. · Zbl 0796.46043 · doi:10.1515/crll.1994.452.39 · crelle:GDZPPN002211750 · eudml:153631 [18] Christopher Lance, On nuclear $$C^{\ast}$$-algebras , J. Functional Analysis 12 (1973), 157-176. · Zbl 0252.46065 · doi:10.1016/0022-1236(73)90021-9 [19] D. R. Lewis, Spaces on which each absolutely summing map is nuclear , Proc. Amer. Math. Soc. 31 (1972), 195-198. · Zbl 0256.47014 · doi:10.2307/2038541 [20] Zeev Nehari, On bounded bilinear forms , Ann. of Math. (2) 65 (1957), 153-162. · Zbl 0077.10605 · doi:10.2307/1969670 [21] Lavon B. Page, Bounded and compact vectorial Hankel operators , Trans. Amer. Math. Soc. 150 (1970), 529-539. · Zbl 0203.45701 · doi:10.2307/1995535 [22] Stephen Parrott, On a quotient norm and the Sz.-Nagy-Foiaş lifting theorem , J. Funct. Anal. 30 (1978), no. 3, 311-328. · Zbl 0409.47004 · doi:10.1016/0022-1236(78)90060-5 [23] Vern I. Paulsen, Every completely polynomially bounded operator is similar to a contraction , J. Funct. Anal. 55 (1984), no. 1, 1-17. · Zbl 0557.46035 · doi:10.1016/0022-1236(84)90014-4 [24] Vern I. Paulsen, Completely bounded maps and dilations , Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow, 1986. · Zbl 0614.47006 [25] Vern I. Paulsen, The maximal operator space of a normed space , Proc. Edinburgh Math. Soc. (2) 39 (1996), no. 2, 309-323. · Zbl 0857.46038 · doi:10.1017/S0013091500023038 [26] Gilles Pisier, Exact operator spaces , Astérisque (1995), no. 232, 159-186, in Recent Advances in Operator Algebras (Orléans, 1992), Soc. Math. France, Montrouge. · Zbl 0844.46031 [27] Gilles Pisier, The operator Hilbert space $${\mathrm OH}$$, complex interpolation and tensor norms , Mem. Amer. Math. Soc. 122 (1996), no. 585, viii+103. · Zbl 0932.46046 [28] Gilles Pisier, A simple proof of a theorem of Kirchberg and related results on $$C^ *$$-norms , J. Operator Theory 35 (1996), no. 2, 317-335. · Zbl 0858.46045 [29] G. Pisier, An introduction to the theory of operator spaces , preprint, 1997. · Zbl 0884.46036 [30] Gilles Pisier, Non-commutative vector valued $$L_ p$$-spaces and completely $$p$$-summing maps , Astérisque (1998), no. 247, vi+131, Soc. Math. France, Montrouge. · Zbl 0937.46056 [31] Zhong-Jin Ruan, Subspaces of $$C^ *$$-algebras , J. Funct. Anal. 76 (1988), no. 1, 217-230. · Zbl 0646.46055 · doi:10.1016/0022-1236(88)90057-2 [32] Masamichi Takesaki, Theory of operator algebras. I , Springer-Verlag, New York, 1979. · Zbl 0990.46034 [33] Gerd Wittstock, Ein operatorwertiger Hahn-Banach Satz , J. Funct. Anal. 40 (1981), no. 2, 127-150. · Zbl 0495.46005 · doi:10.1016/0022-1236(81)90064-1
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