Multidimensional operator multipliers

Juschenko, K.; Todorov, I.; Turowska, L.

doi:10.1090/S0002-9947-09-04771-0

Multidimensional operator multipliers
HTML articles powered by AMS MathViewer

by K. Juschenko, I. G. Todorov and L. Turowska PDF

Trans. Amer. Math. Soc. 361 (2009), 4683-4720 Request permission

Abstract:

We introduce multidimensional Schur multipliers and characterise them, generalising well-known results by Grothendieck and Peller. We define a multidimensional version of the two-dimensional operator multipliers studied recently by Kissin and Shulman. The multidimensional operator multipliers are defined as elements of the minimal tensor product of several $C^*$-algebras satisfying certain boundedness conditions. In the case of commutative $C^*$-algebras, the multidimensional operator multipliers reduce to continuous multidimensional Schur multipliers. We show that the multipliers with respect to some given representations of the corresponding $C^*$-algebras do not change if the representations are replaced by approximately equivalent ones. We establish a non-commutative and multidimensional version of the characterisations by Grothendieck and Peller which shows that universal operator multipliers can be obtained as certain weak limits of elements of the algebraic tensor product of the corresponding $C^*$-algebras.

References

William Arveson, Operator algebras and invariant subspaces, Ann. of Math. (2) 100 (1974), 433–532. MR 365167, DOI 10.2307/1970956
C. Badea and V. I. Paulsen, Schur multipliers and operator-valued Foguel-Hankel operators, Indiana Univ. Math. J. 50 (2001), no. 4, 1509–1522. MR 1888651, DOI 10.1512/iumj.2001.50.2121
M.S. Birman and M.Z. Solomyak, Stieltjes double-integral operators. II, (Russian) Prob. Mat. Fiz. 2 (1967), 26-60
M.S. Birman and M.Z. Solomyak, Stieltjes double-integral operators, III (Passage to the limit under the integral sign), (Russian) Prob. Mat. Fiz. 6 (1973), 27-53
M. Sh. Birman and M. Z. Solomyak, Operator integration, perturbations and commutators, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 170 (1989), no. Issled. Lineĭn. Oper. Teorii Funktsiĭ. 17, 34–66, 321 (Russian, with English summary); English transl., J. Soviet Math. 63 (1993), no. 2, 129–148. MR 1039572, DOI 10.1007/BF01099305
Mikhail Sh. Birman and Michael Solomyak, Double operator integrals in a Hilbert space, Integral Equations Operator Theory 47 (2003), no. 2, 131–168. MR 2002663, DOI 10.1007/s00020-003-1157-8
David P. Blecher and Christian Le Merdy, Operator algebras and their modules—an operator space approach, London Mathematical Society Monographs. New Series, vol. 30, The Clarendon Press, Oxford University Press, Oxford, 2004. Oxford Science Publications. MR 2111973, DOI 10.1093/acprof:oso/9780198526599.001.0001
David P. Blecher and Roger R. Smith, The dual of the Haagerup tensor product, J. London Math. Soc. (2) 45 (1992), no. 1, 126–144. MR 1157556, DOI 10.1112/jlms/s2-45.1.126
Erik Christensen and Allan M. Sinclair, Representations of completely bounded multilinear operators, J. Funct. Anal. 72 (1987), no. 1, 151–181. MR 883506, DOI 10.1016/0022-1236(87)90084-X
Kenneth R. Davidson and Vern I. Paulsen, Polynomially bounded operators, J. Reine Angew. Math. 487 (1997), 153–170. MR 1454263
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
Edward G. Effros, Advances in quantized functional analysis, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 906–916. MR 934293
Edward G. Effros and Zhong-Jin Ruan, Multivariable multipliers for groups and their operator algebras, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 197–218. MR 1077387
Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
Edward G. Effros and Zhong-Jin Ruan, Operator space tensor products and Hopf convolution algebras, J. Operator Theory 50 (2003), no. 1, 131–156. MR 2015023
A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1–79 (French). MR 94682
Donald W. Hadwin, Nonseparable approximate equivalence, Trans. Amer. Math. Soc. 266 (1981), no. 1, 203–231. MR 613792, DOI 10.1090/S0002-9947-1981-0613792-6
Lawrence A. Harris, A generalization of $C^{\ast }$-algebras, Proc. London Math. Soc. (3) 42 (1981), no. 2, 331–361. MR 607306, DOI 10.1112/plms/s3-42.2.331
Fumio Hiai and Hideki Kosaki, Means of Hilbert space operators, Lecture Notes in Mathematics, vol. 1820, Springer-Verlag, Berlin, 2003. MR 2005250, DOI 10.1007/b13213
Takashi Itoh, The Haagerup type cross norm on $C^*$-algebras, Proc. Amer. Math. Soc. 109 (1990), no. 3, 689–695. MR 1014645, DOI 10.1090/S0002-9939-1990-1014645-5
Edward Kissin and Victor S. Shulman, Operator multipliers, Pacific J. Math. 227 (2006), no. 1, 109–141. MR 2247875, DOI 10.2140/pjm.2006.227.109
Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR 1976867
B. S. Pavlov, Multidimensional operator integrals, Problems of Math. Anal., No. 2: Linear Operators and Operator Equations (Russian), Izdat. Leningrad. Univ., Leningrad, 1969, pp. 99–122 (Russian). MR 0415371
V. V. Peller, Hankel operators in the theory of perturbations of unitary and selfadjoint operators, Funktsional. Anal. i Prilozhen. 19 (1985), no. 2, 37–51, 96 (Russian). MR 800919
V. V. Peller, Multiple operator integrals and higher operator derivatives, J. Funct. Anal. 233 (2006), no. 2, 515–544. MR 2214586, DOI 10.1016/j.jfa.2005.09.003
Gilles Pisier, Similarity problems and completely bounded maps, Second, expanded edition, Lecture Notes in Mathematics, vol. 1618, Springer-Verlag, Berlin, 2001. Includes the solution to “The Halmos problem”. MR 1818047, DOI 10.1007/b55674
Gilles Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003. MR 2006539, DOI 10.1017/CBO9781107360235
R. R. Smith, Completely bounded module maps and the Haagerup tensor product, J. Funct. Anal. 102 (1991), no. 1, 156–175. MR 1138841, DOI 10.1016/0022-1236(91)90139-V
Nico Spronk, Measurable Schur multipliers and completely bounded multipliers of the Fourier algebras, Proc. London Math. Soc. (3) 89 (2004), no. 1, 161–192. MR 2063663, DOI 10.1112/S0024611504014650
M. Z. Solomjak and V. V. Sten′kin, A certain class of multiple operator Stieltjes integrals, Problems of Math. Anal., No. 2: Linear Operators and Operator Equations (Russian), Izdat. Leningrad. Univ., Leningrad, 1969, pp. 122–134 (Russian). MR 0438161
V. V. Sten′kin, Multiple operator integrals, Izv. Vysš. Učebn. Zaved. Matematika 4 (179) (1977), 102–115 (Russian). MR 0460588
Dan Voiculescu, A non-commutative Weyl-von Neumann theorem, Rev. Roumaine Math. Pures Appl. 21 (1976), no. 1, 97–113. MR 415338