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Onesided ideals and approximate identities in operator algebras
 J. Australian Math. Soc
"... Abstract. A left ideal of any C ∗algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Indeed left ideals in C ∗algebras may be characterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here an ..."
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Cited by 13 (6 self)
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Abstract. A left ideal of any C ∗algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Indeed left ideals in C ∗algebras may be characterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper [9], that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a C ∗algebra. We study left ideals from the perspective of ‘Hamana theory ’ and using the multiplier algebras introduced by the author. More generally, we develop some general theory for operator algebras which have a 1sided identity or approximate identity, including a BanachStone theorem for these algebras, and an analysis of the ‘multiplier operator algebra’.
MORITA EQUIVALENCE OF DUAL OPERATOR ALGEBRAS
, 2008
"... We consider notions of Morita equivalence appropriate to weak* closed algebras of Hilbert space operators. We obtain new variants, appropriate to the dual algebra setting, of the basic theory of strong Morita equivalence, and new nonselfadjoint analogues of aspects of Rieffel’s W∗algebraic Morita ..."
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Cited by 9 (3 self)
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We consider notions of Morita equivalence appropriate to weak* closed algebras of Hilbert space operators. We obtain new variants, appropriate to the dual algebra setting, of the basic theory of strong Morita equivalence, and new nonselfadjoint analogues of aspects of Rieffel’s W∗algebraic Morita equivalence.
A MORITA THEOREM FOR DUAL OPERATOR ALGEBRAS
, 2008
"... We prove that two dual operator algebras are weak ∗ Morita equivalent in the sense of [4] if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak ∗continuous on appropriate morphism spaces. Moreover, in a fashion similar to th ..."
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Cited by 6 (1 self)
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We prove that two dual operator algebras are weak ∗ Morita equivalent in the sense of [4] if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak ∗continuous on appropriate morphism spaces. Moreover, in a fashion similar to the operator algebra case, we characterize such functors as the module normal Haagerup tensor product with an appropriate weak ∗ Morita equivalence bimodule. We also develop the theory of the W ∗dilation, which connects the nonselfadjoint dual operator algebra with the W ∗algebraic framework. In the case of weak ∗ Morita equivalence, this W ∗dilation is a W ∗module over a von Neumann algebra generated by the nonselfadjoint dual operator algebra. The theory of the W ∗dilation is a key part of the proof of our main theorem.
A Morita type equivalence for dual operator algebras, algebras
"... We generalize the main theorem of Rieffel for Morita equivalence of W ∗algebras to the case of unital dual operator algebras: two unital dual operator algebras A, B have completely isometric normal and β(B) = representations α,β such that α(A) = [M∗β(B)M] −w∗ [Mα(A)M∗] −w∗ for a ternary ring of op ..."
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Cited by 4 (2 self)
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We generalize the main theorem of Rieffel for Morita equivalence of W ∗algebras to the case of unital dual operator algebras: two unital dual operator algebras A, B have completely isometric normal and β(B) = representations α,β such that α(A) = [M∗β(B)M] −w∗ [Mα(A)M∗] −w∗ for a ternary ring of operators M (i.e. a linear space M such that MM∗M ⊂ M) if and only if there exists an equivalence functor F: AM → BM which “extends ” to a ∗−functor implementing an equivalence between the categories ADM and BDM.
DUAL OPERATOR SYSTEMS
, 807
"... Abstract. We characterize weak * closed unital vector spaces of operators on a Hilbert space H. More precisely, we first show that an operator system, which is the dual of an operator space, can be represented completely isometrically and weak * homeomorphically as a weak * closed operator subsystem ..."
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Cited by 4 (2 self)
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Abstract. We characterize weak * closed unital vector spaces of operators on a Hilbert space H. More precisely, we first show that an operator system, which is the dual of an operator space, can be represented completely isometrically and weak * homeomorphically as a weak * closed operator subsystem of B(H). An analogous result is proved for unital operator spaces. Finally, we give some somewhat surprising examples of dual unital operator spaces. 1.
A bicommutant theorem for dual Banach algebras
 Proc. Roy. Irish Acad. 111A
, 2011
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Tensor algebras of C * correspondences and their C * envelopes
, 2006
"... Abstract We show that the C * envelope of the tensor algebra of an arbitrary C * correspondence X coincides with the CuntzPimsner algebra O X , as defined by Katsura [T. Katsura, On C * algebras associated with C * correspondences, J. , who came to the same conclusion under the additional hypo ..."
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Abstract We show that the C * envelope of the tensor algebra of an arbitrary C * correspondence X coincides with the CuntzPimsner algebra O X , as defined by Katsura [T. Katsura, On C * algebras associated with C * correspondences, J. , who came to the same conclusion under the additional hypothesis that X is strict and faithful.