Representation of a completely bounded bimodule map

Author:
Qi Yuan Na

Journal:
Proc. Amer. Math. Soc. **123** (1995), 1137-1143

MSC:
Primary 46L05; Secondary 46L10, 47D25

DOI:
https://doi.org/10.1090/S0002-9939-1995-1223518-9

MathSciNet review:
1223518

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we give a representation for a completely bounded bimodule map into , where *A* and *B* are unital operator subalgebras of . When *A* and *B* are -subalgebras we give a new proof of the Wittstock's theorem by using this representation. We also prove that a von Neumann algebra is an injective operator bimodule over its unital operator algebras if and only if it is a finitely injective operator bimodule.

**[1]**William Arvesion,*Subalgebras of*-*algebra*, Acta Math.**123**(1969), 141-224. MR**0253059 (40:6274)****[2]**David P. Blecher, Zhong-Jin Ruan, and Allian M. Sinclair,*A characterization of operator algebras*, J. Funct. Anal.**89**(1990), 188-175. MR**1040962 (91b:47098)****[3]**Erik Christensen, Edward G. Effros, and Allan Sinclair,*Completely bounded multilinear maps and*-*algebras cohomology*, Invent. Math.**90**(1987), 279-296. MR**910202 (89k:46084)****[4]**Erik Christensen and Allan Sinclair,*A survey of completely bounded operators*, Bull. London Math. Soc.**21**(1989), 417-448. MR**1005819 (91b:46051)****[5]**Paul S. Muhly and Qiyuan Na,*Extension of completely bounded**bimodule maps*, preprint, 1992.**[6]**-,*Dilation of operator bimodules*, preprint, 1992.**[7]**Vern I. Paulsen,*Completely bounded maps and dilations*, Pitman Res. Notes in Math. Ser., vol. 146, Longman, New York, 1986. MR**868472 (88h:46111)****[8]**-,*Every completely polynomially bounded operator is similar to a contraction*, J. Funct. Anal.**55**(1984), 1-17. MR**733029 (86c:47021)****[9]**Vern I. Paulsen and R. R. Smith,*Multilinear maps and tensor norm on operator system*, J. Funct. Anal.**73**(1987), 258-276. MR**899651 (89m:46099)****[10]**Vern I. Paulsen and Ching Yun Suen,*Commutant representation of completely bounded maps*, J. Operator Theory**131**(1985), 87-101. MR**768304 (86d:46052)****[11]**A Guyan Robertson,*Injective matrical Hilbert spaces*, Math. Proc. Cambridge Philos. Soc.**110**(1991), 183-190. MR**1104613 (93d:46101)****[12]**Zhong-Jin Ruan,*Subspaces of*-*algebras*, J. Funct. Anal.**76**(1988), 217-230. MR**923053 (89h:46082)****[13]**-,*Injectivity of operator spaces*, Trans. Amer. Math. Soc.**315**(1989), 89-104. MR**929239 (91d:46078)****[14]**R. R. Smith,*Completely bounded module maps and the Haagerup tensor product*, J. Funct. Anal.**102**(1991), 156-175. MR**1138841 (93a:46115)****[15]**G. Wittstock,*Extension of completely bounded*-*module homomorphisms*, Proc. Conf. On Operator Algebras and Group Representations (Neptun, 1980), Pitman, New York, 1983, pp. 238-250. MR**733321 (85i:46080)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1223518-9

Keywords:
Operator bimodules,
representation,
dilation,
injectivity,
finite injectivity,
finitely generated operator bimodule

Article copyright:
© Copyright 1995
American Mathematical Society