On a duality for C*-crossed products by a locally compact group

Sho IMAI; Hiroshi TAKAI

doi:10.2969/jmsj/03030495

On a duality for C^*-crossed products by a locally compact group

Sho IMAI, Hiroshi TAKAI

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JOURNAL FREE ACCESS

1978 Volume 30 Issue 3 Pages 495-504

DOI https://doi.org/10.2969/jmsj/03030495

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Published: 1978 Received: March 12, 1977 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: ABSTRACT Details: Right : Let (_??_, G, α) be a C^*-dynamical system, and C^*_r(_??_; α) the reduced C^*-crossed product of _??_ by α. We construct a “dual” C^*-crossed product C^*_d(C^*_r(_??_; α); β) of C^*_r(_??_; α) by an isomorphism β from C^*_r(_??_; α) into the full operator algebra _??_(_??_) on a Hilbert space _??_. Then, it is isomorphic to the C^*-tensor product _??_ _??__*C(L²(G)) of _??_ and the C^*-algebra C(L²(G)) of all compact operators on L²(G).
In the abelian case, there exists a continuous action α of the dual group G of G on the C^*-crossed product C^*(_??_; α) of _??_ by α such that the C^*-crossed C^*(C^*(_??_; α); α) of C^*(_??_; α) by α is isomorphic to C^*_d (C^*_r(_??_; α); β).

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) J. Dixmier, Les C^*-algèbres et leurs représentations, Gauthier-Villars, Paris, 1969.
2) M. B. Landstad, Duality for dual covariance algebras, Preprint, Univ. of Trondheim, 1976.
3) Y. Nakagami, Dual action of a von Neumann algebra and Takesaki's duality for a locally compact group, to appear in Publ. Res. Inst. Math. Sci. Ser. A.
4) H. Takai, On a duality for crossed products of C^*-algebras, J. Functional Analysis. 19 (1975), 25-39.
5) H. Takai, The Quasi-orbit space of continuous C^*-dynamical systems, Trans. Amer. Math. Soc., 216 (1976), 105-113.
6) A. Wulfsohn, Le produit tensoriel de C^*-algèbres, Bull. Sci. Math., 87 (1963),13-27.

Right : [1] J. Dixmier, Les C^*-algèbres et leurs représentations, Gauthier-Villars, Paris, 1969.
[2] M. B. Landstad, Duality for dual covariance algebras, Preprint, Univ. of Trondheim, 1976.
[3] Y. Nakagami, Dual action of a von Neumann algebra and Takesaki's duality for a locally compact group, to appear in Publ. Res. Inst. Math. Sci. Ser. A.
[4] H. Takai, On a duality for crossed products of C^*-algebras, J. Functional Analysis. 19 (1975), 25-39.
[5] H. Takai, The Quasi-orbit space of continuous C^*-dynamical systems, Trans. Amer. Math. Soc., 216 (1976), 105-113.
[6] A. Wulfsohn, Le produit tensoriel de C^*-algèbres, Bull. Sci. Math., 87 (1963),13-27.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

Abstract

Let (_??_, G, α) be a C^*-dynamical system, and C^*_r(_??_; α) the reduced C^*-crossed product of _??_ by α. We construct a “dual” C^*-crossed product C^*_d(C^*_r(_??_; α); β) of C^*_r(_??_; α) by an isomorphism β from C^*_r(_??_; α) into the full operator algebra _??_(_??_) on a Hilbert space _??_. Then, it is isomorphic to the C^*-tensor product _??_ _??__*C(L²(G)) of _??_ and the C^*-algebra C(L²(G)) of all compact operators on L²(G).
In the abelian case, there exists a continuous action α of the dual group G of G on the C^*-crossed product C^*(_??_; α) of _??_ by α such that the C^*-crossed C^*(C^*(_??_; α); α) of C^*(_??_; α) by α is isomorphic to C^*_d (C^*_r(_??_; α); β).

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