Inner product modules over -algebras

Author:
William L. Paschke

Journal:
Trans. Amer. Math. Soc. **182** (1973), 443-468

MSC:
Primary 46K05; Secondary 46H25

DOI:
https://doi.org/10.1090/S0002-9947-1973-0355613-0

MathSciNet review:
0355613

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is an investigation of right modules over a -algebra *B* which posses a *B*-valued ``inner product'' respecting the module action. Elementary properties of these objects, including their normability and a characterization of the bounded module maps between two such, are established at the beginning of the exposition. The case in which *B* is a -algebra is of especial interest, since in this setting one finds an abundance of inner product modules which satisfy an analog of the self-duality property of Hilbert space. It is shown that such self-dual modules have important properties in common with both Hilbert spaces and -algebras. The extension of an inner product module over *B* by a -algebra *A* containing *B* as a -subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.

**[1]**W. B. Arveson,*Subalgebras of*-*algebras*, Acta Math.**123**(1969), 141-224. MR**40**#6274. MR**0253059 (40:6274)****[2]**K. H. Hofmann,*Representation of algebras by continuous sections*, Bull. Amer. Math. Soc.**78**(1972), 291-373. MR**0347915 (50:415)****[3]**B. E. Johnson,*Centralisers and operators reduced by maximal ideals*, J. London Math. Soc.**43**(1968), 231-233. MR**36**#6937. MR**0223890 (36:6937)****[4]**I. Kaplansky,*Modules over operator algebras*. Amer. J. Math.**75**(1953), 839-858. MR**15**, 327. MR**0058137 (15:327f)****[5]**T. W. Palmer, -*representations of*-*algebras*, Indiana Univ. Math. J.**20**(1971), 929-933. MR**0410396 (53:14146)****[6]**-,*The Gel' fand-Naímark pseudo-norm on Banach*-*algebras*, J. London Math. Soc. (2)**3**(1971), 59-66. MR**43**#7932. MR**0282219 (43:7932)****[7]**M. A. Rieffel,*Induced representations of*-*algebras*(preprint).**[8]**S. Sakai, -*algebras and*-*algebras*, Springer, New York, 1971. MR**0442701 (56:1082)****[9]**W. F. Stinespring,*Positive functions on*-*algebras*, Proc. Amer. Math. Soc.**6**(1955), 211-216. MR**16**, 1033. MR**0069403 (16:1033b)****[10]**A. Takahashi,*Fields of Hilbert modules*, Dissertation, Tulane University, New Orleans, La., 1971.**[11]**M. Takesaki,*A note on the cross-norm of the direct product of operator algebra*, Kōdai Math. Sem. Rep.**10**(1958), 137-140. MR**20**#6666. MR**0100233 (20:6666)****[12]**H. Widom,*Embedding in algebras of type*I, Duke Math. J.**23**(1956), 309-324. MR**17**, 1228. MR**0078668 (17:1228c)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46K05,
46H25

Retrieve articles in all journals with MSC: 46K05, 46H25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0355613-0

Article copyright:
© Copyright 1973
American Mathematical Society