Extending representations of Banach algebras to their biduals

For Banach spaces X and Y , we establish a natural bijection between preduals of Y and preduals of L(X, Y ) that respect the right L(X)module structure. If X is reflexive, it follows that there is a unique predual making L(X) into a dual Banach algebra. This removes the condition that X have the approximation property in a result of Daws.We further establish a natural bijection between projections that complement Y in its bidual and projections that complement L(X, Y ) in its bidual as a right L(X)-module. It follows that Y is complemented in its bidual if and only if L(X, Y ) is (either as a module or as a Banach space).Our results are new even in the well-studied case of isometric preduals.

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“…The results in this paper are the foundations for two other works of the authors; see [GT17] and [GT16].…”

Section: Introductionsupporting

confidence: 55%

Preduals and complementation of spaces of bounded linear operators

Gardella

Thiel

2020

Int. J. Math.

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“…So ∞ r=1 ψ t k (r) (a)P r − ψ 1 (a) has finite rank for every k ≥ 1. Hence to prove (iv), it suffices to show that The isometric version of this problem was solved by Gardella and Thiel [27]. A separable C * -algebra A is isometrically isomorphic to a subalgebra of B(l p ) if and only if A is commutative.…”

Section: Otherwisementioning

confidence: 99%

“…In 2012 and 2013, Phillips [51], [52], [53] introduced L p versions of some C * -algebra notions including UHF algebras and Cuntz algebras. See [25], [26], [32] [24], [27], [54] for some results in this direction. Most of the results about algebras of operators on L p that appear in the literature are in the "isometric theory," i.e., the "isomorphisms" (in the sense of category theory) between algebras of operators on L p are the isometric isomorphisms.…”

Section: Introductionmentioning

confidence: 99%

Similarity of operators on $l^{p}$

Boedihardjo

2017

Preprint

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We prove a version of each of the following results for operators on l p , 1 < p < ∞: (i) Voiculescu's absorption theorem (ii) that Ext(C(M )) is a group for every nonempty compact subset M of R d (iii) that Ext(A) −1 is homotopy invariant in A for separable unital C * -algebra A. Contents 1. Introduction 1 2. l p version of Voiculescu's theorem 8 3. Proof of the l p version of Voiculescu's theorem 9 4. Consequences of the l p version of Voiculescu's theorem 22 5. Isomorphic extensions of K(l p ) by C * -algebras 30 6. Ext ∼ s,r (C(M ), K(l p )) is a group 39 7. Homotopy invariance of Ext ∼ s,r (A, K(l p )) −1 52 8. Open problems 75 References 76 2010 Mathematics Subject Classification. 47A65.

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“…Subsequently others have followed him into this inquiry (for example, Gardella, Thiel, Lupini, and Viola; see e.g. [24,25,27,23,48]). However, as he has frequently stated, these investigations have been very largely focused on examples; one still lacks an abstract general theory in this setting.…”

Section: Introductionmentioning

confidence: 99%

L^p operator algebras with approximate identities I

Blecher,

Phillips

2018

Preprint

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We initiate an investigation into how much the existing theory of (nonselfadjoint) operator algebras on a Hilbert space generalizes to algebras acting on L p spaces. In particular we investigate the applicability of the theory of real positivity, which has recently been useful in the study of L 2 -operator algebras and Banach algebras, to algebras of bounded operators on L p spaces.In the process we answer some open questions on real positivity in Banach algebras from work of the first author and Ozawa.

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Extending representations of Banach algebras to their biduals

Cited by 11 publications

References 24 publications

Preduals and complementation of spaces of bounded linear operators

Preduals and complementation of spaces of bounded linear operators

Similarity of operators on $l^{p}$

L^p operator algebras with approximate identities I

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