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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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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.
An abstract characterization of unital operator spaces
, 2008
"... In this article, we give an abstract characterization of the “identity ” of an operator space V by looking at a quantity ncb(V, u) which is defined in analogue to a wellknown quantity in Banach space theory. More precisely, we show that there exists a complete isometry from V to some L(H) sending u ..."
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In this article, we give an abstract characterization of the “identity ” of an operator space V by looking at a quantity ncb(V, u) which is defined in analogue to a wellknown quantity in Banach space theory. More precisely, we show that there exists a complete isometry from V to some L(H) sending u to idH if and only if ncb(V, u) = 1. We will use it to give an abstract characterization of operator systems. Moreover, we will show that if V is a unital operator space and W is a proper complete Mideal, then V/W is also a unital operator space. As a consequece, the quotient of an operator system by a proper complete Mideal is again an operator system. In the appendix, we will also give an abstract characterisation of “nonunital operator systems ” using an idea arose from the definition of ncb(V, u). 1
WEAK* CONTINUOUS STATES ON BANACH ALGEBRAS
, 2008
"... We prove that if a unital Banach algebra A is the dual of a Banach space A♯, then the set of weak* continuous states is weak* dense in the set of all states on A. Further, weak* continuous states linearly span A♯. ..."
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We prove that if a unital Banach algebra A is the dual of a Banach space A♯, then the set of weak* continuous states is weak* dense in the set of all states on A. Further, weak* continuous states linearly span A♯.
PARTIAL ISOMETRIES: A SURVEY
, 2018
"... Abstract. We survey the main results characterizing partial isometries in C * algebras and tripotents in JB * triples obtained in terms of regularity, conorm, quadraticconorm, and the geometric structure of the underlying Banach spaces. ..."
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Abstract. We survey the main results characterizing partial isometries in C * algebras and tripotents in JB * triples obtained in terms of regularity, conorm, quadraticconorm, and the geometric structure of the underlying Banach spaces.
THE CLASSIFICATION PROBLEM FOR FINITELY GENERATED OPERATOR SYSTEMS AND SPACES
"... Abstract. The classification of separable operator systems and spaces is commonly believed to be intractable. We analyze this belief from the point of view of Borel complexity theory. On one hand we confirm that the classification problems for arbitrary separable operator systems and spaces are intr ..."
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Abstract. The classification of separable operator systems and spaces is commonly believed to be intractable. We analyze this belief from the point of view of Borel complexity theory. On one hand we confirm that the classification problems for arbitrary separable operator systems and spaces are intractable. On the other hand we show that the finitely generated operator systems and spaces are completely classifiable (or smooth); in fact a finitely generated operator system is classified by its complete theory when regarded as a structure in continuous logic. In the particular case of operator systems generated by a single unitary, a complete invariant is given by the spectrum of the unitary up to a rigid motion of the circle, provided that the spectrum contains at least 5 points. As a consequence of these results we show that the relation on compact subsets of Cn, given by homeomorphism via a degree 1 map, is smooth. Contents