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13
Hereditary subalgebras of operator algebras
"... Abstract. In recent work of the second author, a technical result was proved establishing a bijective correspondence between certain open projections in a C ∗algebra containing an operator algebra A, and certain onesided ideals of A. Here we give several remarkable consequences of this result. The ..."
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Abstract. In recent work of the second author, a technical result was proved establishing a bijective correspondence between certain open projections in a C ∗algebra containing an operator algebra A, and certain onesided ideals of A. Here we give several remarkable consequences of this result. These include a generalization of the theory of hereditary subalgebras of a C ∗algebra, and the solution of a ten year old problem concerning the Morita equivalence of operator algebras. In particular, the latter gives a very clean generalization of the notion of Hilbert C ∗modules to nonselfadjoint algebras. We show that an ‘ideal ’ of a general operator space X is the intersection of X with an ‘ideal ’ in any containing C ∗algebra or C ∗module. Finally, we discuss the noncommutative variant of the classical theory of ‘peak sets’. 1.
Closed projections and peak interpolation for operator algebras
, 2005
"... Abstract. The closed onesided ideals of a C ∗algebra are exactly the closed subspaces supported by the orthogonal complement of a closed projection. Let A be a (not necessarily selfadjoint) subalgebra of a unital C ∗algebra B which contains the unit of B. Here we characterize the right ideals of ..."
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Cited by 15 (2 self)
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Abstract. The closed onesided ideals of a C ∗algebra are exactly the closed subspaces supported by the orthogonal complement of a closed projection. Let A be a (not necessarily selfadjoint) subalgebra of a unital C ∗algebra B which contains the unit of B. Here we characterize the right ideals of A with left contractive approximate identity as those subspaces of A supported by the orthogonal complement of a closed projection in B ∗ ∗ which also lies in A ⊥ ⊥. Although this seems quite natural, the proof requires a set of new techniques which may may be viewed as a noncommutative version of the subject of peak interpolation from the theory of function spaces. Thus, the right ideals with left approximate identity are closely related to a type of peaking phenomena in the algebra. In this direction, we introduce a class of closed projections which generalizes the notion of a peak set in the theory of uniform algebras to the world of operator algebras and operator spaces. 1.
Quasimultipliers of operator spaces
, 2003
"... Abstract. We use the injective envelope to study quasimultipliers of operator spaces. We prove that all representable operator algebra products that an operator space can be endowed with are induced by quasimultipliers. We obtain generalizations of the BanachStone theorem. 1. ..."
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Cited by 7 (0 self)
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Abstract. We use the injective envelope to study quasimultipliers of operator spaces. We prove that all representable operator algebra products that an operator space can be endowed with are induced by quasimultipliers. We obtain generalizations of the BanachStone theorem. 1.
IDEALS AND STRUCTURE OF OPERATOR ALGEBRAS
, 907
"... Abstract. We continue the study of rideals, ℓideals, and HSA’s in operator algebras. Some applications are made to the structure of operator algebras, including Wedderburn type theorems for a class of operator algebras. We also consider the onesided Mideal structure of certain tensor products of ..."
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Abstract. We continue the study of rideals, ℓideals, and HSA’s in operator algebras. Some applications are made to the structure of operator algebras, including Wedderburn type theorems for a class of operator algebras. We also consider the onesided Mideal structure of certain tensor products of operator algebras. 1.
The ideal envelope of an operator algebra
, 2001
"... A left ideal of any C ∗algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Conversely, we show here and in [6] that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a C ∗algebra. We study operator algebras and ..."
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A left ideal of any C ∗algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Conversely, we show here and in [6] that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a C ∗algebra. We study operator algebras and their multiplier algebras from the perspective of ‘Hamana theory’ and using the multiplier algebras introduced by the first author.
State spaces of JB*triples
, 2008
"... An atomic decomposition is proved for Banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process. This is then applied to yield, under appropriate assumptions, geometric characterizations, up to isometry, of the unit ball of the dua ..."
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Cited by 4 (1 self)
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An atomic decomposition is proved for Banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process. This is then applied to yield, under appropriate assumptions, geometric characterizations, up to isometry, of the unit ball of the dual space of a JB ∗triple, and up to complete isometry, of onesided ideals in C ∗algebras.
Closed projections and approximate identities for operator algebras
, 2005
"... Abstract. Let A be a (not necessarily selfadjoint) subalgebra of a unital C ∗algebra B which contains the unit of B. The right ideals of A with left contractive approximate identity are characterized as those subspaces of A supported by the orthogonal complement of a closed projection in B ∗ ∗ whic ..."
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Cited by 1 (1 self)
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Abstract. Let A be a (not necessarily selfadjoint) subalgebra of a unital C ∗algebra B which contains the unit of B. The right ideals of A with left contractive approximate identity are characterized as those subspaces of A supported by the orthogonal complement of a closed projection in B ∗ ∗ which also lies in A ⊥ ⊥. Although this seems quite natural, the nonselfadjointness requires us to develop some interpolation results for its proof. The right ideals with left approximate identity are closely related to a type of peaking phenomena in the algebra. In this direction we introduce a class of closed projections which generalizes the notion of a peak set in the theory of uniform algebras to the world of operator algebras and operator spaces. 1.
Extreme points of the unit ball of a quasimultiplier space
, 2004
"... Abstract. We study extreme points of the unit ball of the set of quasimultipliers of an operator space by introducing the new notion: (approximate) quasiidentities. We give a necessary and sucient condition for an operator space to become an operator algebra with a contractive approximate quasi ( ..."
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Abstract. We study extreme points of the unit ball of the set of quasimultipliers of an operator space by introducing the new notion: (approximate) quasiidentities. We give a necessary and sucient condition for an operator space to become an operator algebra with a contractive approximate quasi (respectively, left, right, twosided) identity in terms of extreme points of contractive quasimultipliers. We also give a necessary sucient condition for an operator space to become a Calgebra. Furthermore, we answer the open question about Properties (L) and (R) raised by D. P. Blecher. 1. Introduction. Quasimultipliers of operator spaces were introduced by V. I. Paulsen ([17] Denition 2.2) in late 2002 and their correspondence to operator algebra products was discovered by the author ([17] Theorem 2.6) in early 2003. That is, for a given operator space X, the possible operator algebra products X can be equipped with are precisely the bilinear mappings on
Multipliers, C∗modules, and algebraic structure in spaces of Hilbert space operators
 CONTEMPORARY MATHEMATICS
, 2004
"... Part I is a survey of the author’s work (on his own or with several coauthors) on onesided multipliers and their applications to algebraic structure in spaces of operators. The proofs given here are new for the most part, and the emphasis is on the connections with the theory of Hilbert C∗modules ..."
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Part I is a survey of the author’s work (on his own or with several coauthors) on onesided multipliers and their applications to algebraic structure in spaces of operators. The proofs given here are new for the most part, and the emphasis is on the connections with the theory of Hilbert C∗modules. In Part II we initiate a theory of onesided multipliers between two different operator spaces.