The Phillips properties

Freedman, Walden; Ülger, A.

doi:10.1090/s0002-9939-00-05703-8

Cited by 7 publications

(3 citation statements)

References 15 publications

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“…Proof. This follows from our Theorem 4.1 combined with [10,Corollary 2.14], which states that a dual Banach space with the Phillips property is finite-dimensional.…”

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confidence: 80%

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Geometry of C⁎-algebras, and the bidual of their projective tensor product

Neufang

2020

Journal of Functional Analysis

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Let A be a C * -algebra, and consider the Banach algebra A⊗ γ A, where ⊗ γ denotes the projective Banach space tensor product; if A is commutative, this is the Varopoulos algebra V A . It has been an open problem for more than 35 years to determine precisely when A ⊗ γ A is Arens regular; cf. [26], [40], [41]. Even the situation for commutative A, in particular the case A = ℓ ∞ , has remained unsolved. We solve this classical question for arbitrary C * -algebras by using von Neumann algebra and operator space methods, mainly relying on versions of the (commutative and non-commutative) Grothendieck Theorem, and the structure of completely bounded module maps. Establishing these links allows us to show that A ⊗ γ A is Arens regular if and only if A has the Phillips property; equivalently, A is scattered and has the Dunford-Pettis Property. A further equivalent condition is that A * has the Schur property, or, again equivalently, the enveloping von Neumann algebra A * * is finite atomic, i.e., a direct sum of matrix algebras. Hence, Arens regularity of A ⊗ γ A is encoded in the geometry of the C * -algebra A. In case A is a von Neumann algebra, we conclude that A ⊗ γ A is Arens regular (if and) only if A is finite-dimensional. We also show that this does not generalize to the class of non-selfadjoint dual (even commutative) operator algebras. Specializing to commutative C * -algebras A, we obtain that V A is Arens regular if and only if A is scattered. In fact, we determine precisely the centre of the bidual, namely, Z(V A * * ) is Banach algebra isomorphic to A * * ⊗ eh A * * , where ⊗ eh denotes the extended Haagerup tensor product. We deduce that V A is strongly Arens irregular (if and) only if A is finite-dimensional. Hence, V A is neither Arens regular nor strongly Arens irregular, if and only if A is non-scattered (as mentioned above, this is new even for the case A = ℓ ∞ ).

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“…Proof. This follows from our Theorem 4.1 combined with [10,Corollary 2.14], which states that a dual Banach space with the Phillips property is finite-dimensional.…”

mentioning

confidence: 80%

“…• E has the Phillips property if the Dixmier projection E * * * → E * is sequentially w * -norm continuous (cf. [10]).…”

Section: Preliminariesmentioning

confidence: 99%

Geometry of C⁎-algebras, and the bidual of their projective tensor product

Neufang

2020

Journal of Functional Analysis

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“…4 shows that the space c 0 and C K spaces have the D property. But, we give the following example to show that the converse of Corollary 3.4 does not hold.…”

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confidence: 97%

The (D) Property in Banach Spaces

Soybaş

2012

Abstract and Applied Analysis

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A Banach space is said to have (D) property if every bounded linear operator is weakly compact for every Banach space whose dual does not contain an isomorphic copy of . Studying this property in connection with other geometric properties, we show that every Banach space whose dual has (V∗) property of Pełczyński (and hence every Banach space with (V) property) has (D) property. We show that the space of real functions, which are integrable with respect to a measure with values in a Banach space , has (D) property. We give some other results concerning Banach spaces with (D) property.

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Arens regularity of projective tensor products

Rajpal

Kumar

2016

Arch. Math.

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For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product A ⊗B (respectively the Haagerup tensor product A ⊗ h B) to be Arens regular are obtained. Using the non-commutative Grothendieck's inequality, we show that, for C * -algebras A and B, the Arens regularity of Banach algebras A ⊗ h B, A ⊗ γ B, A ⊗ s B and A ⊗B are equivalent, where ⊗ h , ⊗ γ , ⊗ s and ⊗ are the Haagerup, the Banach space projective tensor norm, the Schur tensor norm and the operator space projective tensor norm, respectively.

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The Phillips properties

Cited by 7 publications

References 15 publications

Geometry of C⁎-algebras, and the bidual of their projective tensor product

Geometry of C⁎-algebras, and the bidual of their projective tensor product

The (D) Property in Banach Spaces

Arens regularity of projective tensor products

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