Faithful representations of crossed products by endomorphisms
Authors:
Sarah Boyd, Navin Keswani and Iain Raeburn
Journal:
Proc. Amer. Math. Soc. 118 (1993), 427436
MSC:
Primary 46L55
MathSciNet review:
1126190
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Abstract: Stacey has recently characterised the crossed product of a algebra by an endomorphism as a algebra whose representations are given by covariant representations of the system . Following work of O'Donovan for automorphisms, we give conditions on a covariant representation of which ensure that the corresponding representation of is faithful. We then use this result to improve a theorem of Paschke on the simplicity of .
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 [1]
 W. B. Arveson, Continuous analogues of Fock space, Mem. Amer. Math. Soc., vol. 409, Amer. Math. Soc., Providence, RI, 1989. MR 987590 (90f:47061)
 [2]
 J. Cuntz, Simple algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173185. MR 0467330 (57:7189)
 [3]
 , The internal structure of simple algebras, Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, RI, 1982, pp. 85115.
 [4]
 S. Doplicher and J. E. Roberts, Endomorphisms of algebras, cross products and duality for compact groups, Ann. of Math. (2) 130 (1989), 75119. MR 1005608 (90j:46058)
 [5]
 R. G. Douglas, On the algebra of a oneparameter semigroup of isometries, Acta Math. 128 (1972), 143152. MR 0394296 (52:15099)
 [6]
 B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., vol. 127, Amer. Math. Soc., Providence, RI, 1972. MR 0374934 (51:11130)
 [7]
 G. J. Murphy, algebras and operator theory, Academic Press, San Diego and London, 1990. MR 1074574 (91m:46084)
 [8]
 D. P. O'Donovan, Weighted shifts and covariance algebras, Trans. Amer. Math. Soc. 208 (1975), 125. MR 0385632 (52:6492)
 [9]
 W. L. Paschke, The crossed product by an endomorphism, Proc. Amer. Math. Soc. 80 (1980), 113118. MR 574518 (81m:46081)
 [10]
 , theory for actions of the circle group, J. Operator Theory 6 (1981), 125133.
 [11]
 I. Raeburn, On crossed products and Takai duality, Proc. Edinburgh Math. Soc. (2) 31 (1988), 321330. MR 989764 (90d:46093)
 [12]
 P. J. Stacey, Crossed products of algebras by endomorphisms, J. Austral. Math. Soc. (to appear).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919931126190X
PII:
S 00029939(1993)1126190X
Keywords:
algebra,
endomorphism,
covariant representation,
crossed product,
simple algebra
Article copyright:
© Copyright 1993
American Mathematical Society
