A note on dual Banach spaces.

Kaijser, Sten

doi:10.7146/math.scand.a-11725

Cited by 19 publications

(15 citation statements)

References 5 publications

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“…Conversely, if M is dually complete, it is a dual space and the predual is a certain completion of A (see e.g. (21)). Furthermore M' = E A (M)' is a Banach algebra, and there exists a natural homomorphism of A into M'.…”

Section: \(Tr a (A P ®A' P )A}\ = \{A' P A-a P )\ = \(A P -A' P A)\mentioning

confidence: 99%

On Banach modules I

Kaijser

1981

Math. Proc. Camb. Phil. Soc.

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The purpose of this series of papers is to present a general theory of Banach modules and to give some applications of it. The applications of the theory arise from observations that certain important notions of functional analysis are very closely related to certain Banach algebras and thereby to a module structure. This holds in particular for Banach lattices (which are C(X)-modules), interpolation spaces, tensor products and operator ideals.

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Section: \(Tr a (A P ®A' P )A}\ = \{A' P A-a P )\ = \(A P -A' P A)\mentioning

confidence: 99%

On Banach modules I

Kaijser

1981

Math. Proc. Camb. Phil. Soc.

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“…Proof Kaijser proved in [15], that a Banach space Y is a dual space if there is a set of continuous linear functionals E on Y that separates the points of Y and the closed unit ball of Y is compact in the weak topology generated by E.…”

Section: Theorem 23mentioning

confidence: 99%

Energy Spaces, Dirichlet Forms and Capacities in a Nonlinear Setting

Claus

2021

Potential Anal

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In this article we study lower semicontinuous, convex functionals on real Hilbert spaces. In the first part of the article we construct a Banach space that serves as the energy space for such functionals. In the second part we study nonlinear Dirichlet forms, as defined by Cipriani and Grillo, and show, as it is well known in the bilinear case, that the energy space of such forms is a lattice. We define a capacity and introduce the notion quasicontinuity associated with these forms and prove several results, which are well known in the bilinear case.

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“…Using the matrix expressions of , a, and b, a direct computation shows that equation (42) poses no constraints on the factor A 2 in the matrix expression of a, or, equivalenty, we have that…”

Section: Positive Trace-class Operatorsmentioning

confidence: 99%

“…which is equivalent to the previous equation. Then, if we fix k ∈ [1, ..., dim(H ⊥ ) and take b = |f k f l |, we immediately see that equation (42) poses no constraints on a. Putted differently, if a is in g , then the factor A 4 in the matrix expression of a is arbitrary. Next, in order to characterize the matrix elements of a ∈ g with respect to the basis elements in the subspace H , we need to exploit the arbitrariness of b in equation (42) "twice": we first have to consider b = x e kl |e k e l | (for all k, l ∈ [1, ..., dim(H )]), and then we have to consider b = y e kl |e k e l | (for all k, l ∈ [1, ..., dim(H )]).…”

Section: Positive Trace-class Operatorsmentioning

confidence: 99%

Manifolds of classical probability distributions and quantum density operators in infinite dimensions

et al. 2019

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The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of C * -algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given C * -algebra A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states S of a possibly infinite-dimensional, unital C * -algebra A is partitioned into the disjoint union of the orbits of an action of the group G of invertible elements of A . Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space H are smooth, homogeneous Banach manifolds of G = GL(H), and, when A admits a faithful tracial state τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through τ is a smooth, homogeneous Banach manifold for G .

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A note on dual Banach spaces.

Cited by 19 publications

References 5 publications

On Banach modules I

On Banach modules I

Energy Spaces, Dirichlet Forms and Capacities in a Nonlinear Setting

Manifolds of classical probability distributions and quantum density operators in infinite dimensions

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