Weak compactness in the dual space of a C∗-algebra

Akemann, Charles A.; Dodds, P. G.; Gamlen, John L. B.

doi:10.1016/0022-1236(72)90040-7

Cited by 42 publications

(35 citation statements)

References 10 publications

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“…This fact together with the above lemma and comments show that, for each 2 < p < +∞, s * ( p , * p ) coincides with σ( p , * p ) (respectively, s * (c 0 , c * 0 ) coincides with σ(c 0 , c * 0 )) on bounded sets of p (respectively, c 0 ). The strong* topology in a von Neumann algebra or in a JBW*-triple has proved to be a very good tool to characterise relatively weakly compact subsets in their respective preduals (see for example [24], [1], [2], [22], and [15]). The following proposition generalises the above cited results to the class of Banach spaces Y in which the topologies s * (Y * * , Y * ) and ρ(Y ) coincide on bounded subsets; the proof is very similar to the one given in [15, Theorem 2.1].…”

Section: Unconditionally Converging and Quasi Completely Continuous Omentioning

confidence: 99%

Weakly compact operators and the strong* topology for a Banach space

Peralta¹,

Villanueva²,

Wright³

et al. 2010

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Abstract. The strong* topology s * (X) of a Banach space X is defined as the locally convex topology generated by the seminorms x → Sx for bounded linear maps S from X into Hilbert spaces. The w-right topology for X, ρ(X), is a stronger locally convex topology, which may be analogously characterised by taking reflexive Banach spaces in place of Hilbert spaces. For any Banach space Y , a linear map T : X → Y is known to be weakly compact precisely when T is continuous from the w-right topology to the norm topology of Y . The main results deal with conditions for, and consequences of, the coincidence of these two topologies on norm bounded sets. A large class of Banach spaces, including all C * -algebras, and more generally, all JB * -triples, exhibit this behaviour.

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Section: Unconditionally Converging and Quasi Completely Continuous Omentioning

confidence: 99%

Weakly compact operators and the strong* topology for a Banach space

Peralta¹,

Villanueva²,

Wright³

et al. 2010

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…We also extend an elegant result of Akemann, Dodds and Gamlen [2] by showing that a continuous linear map T from a C * -algebra into a complete locally convex space which contains no copy of c 0 must be weakly compact.…”

Section: µ(Dt) For Each F ∈ C(k)mentioning

confidence: 59%

Non-Commutative Locally Convex Measures

Bonet

Wright

2009

The Quarterly Journal of Mathematics

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We study weakly compact operators from a C * -algebra with values in a complete locally convex space. They constitute a natural non-commutative generalization of finitely additive vector measures with values in a locally convex space. Several results of Brooks, Saîto and Wright are extended to this more general setting. Building on an approach due to Saîto and Wright, we obtain our theorems on non-commutative finitely additive measures with values in a locally convex space, from more general results on weakly compact operators defined on Banach spaces X whose strong dual X is weakly sequentially complete. Weakly compact operators are also characterized by a continuity property for a certain "Right topology" as in joint work by Peralta, Villanueva, Wright and Ylinen.

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“…This proves (1). (2) This follows from (1) and part (4) To prove (b) implies (a), suppose a e sA ut * and x e X.…”

Section: Associated With the Module Multiplicationmentioning

confidence: 80%

“…In the other direction, Gillespie [13] showed that if X does contain a copy of c 0 , then there is a Banach c-module structure on X that does not have weakly compact action. Pelczynski's result was extended to C*-algebras by Akemann, Dodds and Gamlen [1], who proved that if X does not contain a copy of c 0 , then, for every C*-algebra si and every Banach j^-module structure on X, si has weakly compact action. Gillespie's result shows that the converse of the latter result is true.…”

Section: Ii) If Bep and (P-b)x^o Then There Is A B' In 9> Such That Bmentioning

confidence: 99%

A noncommutative theory of Bade functionals

Hadwin

Orhon

1991

Glasgow Math. J.

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24]). Via the Stone representation space of the Boolean algebra, the theory can be studied through Banach modules over C(K), where K is a compact Hausdorff space. One of the key concepts in the theory is the notion of Bade functionals. If X is a Banach C(^)-module and x e X, then a Bade functional of x with respect to C{K) is a continuous linear functional a-on A" such that, for each a in C(K) with a > 0, we haveIt is clear that the definition of Bade functionals makes sense when C(K) is replaced by an arbitrary C*-algebra. In this paper we show, using elementary C*-algebraic techniques, that most of the known results on Bade functionals for C(K) carry over to the C*-algebra setting. Moreover, we prove an existence theorem in the C*-algebraic case that is new even in the C(K) setting. This result shows that Bade functionals always exist in many important cases, e.g. when the C*-algebra is separable or when the Banach space is either separable or the dual of a separable space. In [14], T. A. Gillespie showed that, for a Banach C(K)-modu\e X, Bade functionals always exist if X does not contain a copy of c 0 , and he asked if the converse is true. However Bade functionals always exist when X = c 0) since c 0 is separable.Although the C*-algebraic point of view clarifies many of the results in [11, XVII.3, XVIII.3], this paper is written for a readership that is not assumed to be expert in C*-algebras. We therefore provide a brief account of the properties of C*-algebras and of Arens extensions that we need.Throughout, si will denote a C*-algebra with 1. The Gelfand-Naimark theorem says that we can assume that si is a C*-subalgebra of the algebra L(H) of all (bounded linear) operators on a Hilbert space H. We will choose H to have the additional properties listed in the following lemma. The interested reader can consult [22] for details. If AT is a Banach space, we follow the notation of [15] and let X** denote the normed dual of X. We will use si' to denote the commutant of si, and we use * to denote the involution in a C*-algebra. However, we still use w* to denote the weak-star topology.

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Weak compactness in the dual space of a C∗-algebra

Cited by 42 publications

References 10 publications

Weakly compact operators and the strong* topology for a Banach space

Weakly compact operators and the strong* topology for a Banach space

Non-Commutative Locally Convex Measures

A noncommutative theory of Bade functionals

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