On the algebraic foundations of bounded cohomology

Bühler, Theo

doi:10.1090/s0065-9266-2011-00618-0

Cited by 25 publications

(35 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Ban is an additive category possessing only finite product and coproduct. However, we remark that there is no infinite product and coproducts in Ban (See [1]). …”

Section: The Category Of Banach Spacesmentioning

confidence: 99%

“…The proof can be found in the section IV.2.1. in [1]. Recall that an additive category such that every morphism has a kernel and cokernel is called preabelian and so Ban is preabelian.…”

Section: The Category Of Banach Spacesmentioning

confidence: 99%

“…In [1], it is shown that an exact structure on Ban is not unique and the following: Proposition 2.2. Let E max be the class of all kernel-cokernel pais in Ban.…”

Section: The Category Of Banach Spacesmentioning

confidence: 99%

“…Since the spaces L * and H * (U) are flat, by applying Corollary 2.5.3 (ii) in [1] to the sequence (3.4.2), the space Z * is flat. Then, by applying it to the sequence (3.4.1), the space U p is flat from the first sequence (3.4.1).…”

Section: Künneth Type Formulamentioning

confidence: 99%

“…So, contrary to the usual case, there are no short exact sequences of Banach spaces induced from the boundary operators. To make up for these defect, in Section 2, we review some proper conditions for the category of Banach spaces following [1]. In Section 3, we investigate Künneth formula on Banach complex and so on Bounded cohomology.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Some Remarks for Künneth Formula on Bounded Cohomology

Park¹

2015

Honam Mathematical Journal

View full text Add to dashboard Cite

Abstract. Künneth formula is to compute (co)-homology of A⊗B for known (co)-homology of the complexes A and B. In the ordinary case, this is done by using elementary homological methods in an abelian category. However, when we consider the bounded cochain complex with values in R and its structure as a real Banach space, the techniques of homological algebra for constructing Künneth type formulas on it are not effective. The most notable facts are the image of a morphism of Banach spaces is not necessarily closed, and also the closed summand of a Banach space need not be a topological direct summand. The main goal of this paper is to construct the abstract theory of Künneth type formula on bounded cohomology with real coefficients in the suitable category of Banach spaces with some restricted conditions.

show abstract

“…Ban is an additive category possessing only finite product and coproduct. However, we remark that there is no infinite product and coproducts in Ban (See [1]). …”

Section: The Category Of Banach Spacesmentioning

confidence: 99%

“…The proof can be found in the section IV.2.1. in [1]. Recall that an additive category such that every morphism has a kernel and cokernel is called preabelian and so Ban is preabelian.…”

Section: The Category Of Banach Spacesmentioning

confidence: 99%

“…In [1], it is shown that an exact structure on Ban is not unique and the following: Proposition 2.2. Let E max be the class of all kernel-cokernel pais in Ban.…”

Section: The Category Of Banach Spacesmentioning

confidence: 99%

Section: Künneth Type Formulamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Some Remarks for Künneth Formula on Bounded Cohomology

Park¹

2015

Honam Mathematical Journal

View full text Add to dashboard Cite

show abstract

An Application of a Functional Inequality to Quasi-Invariance in Infinite Dimensions

Gordina

2017

Convexity and Concentration

View full text Add to dashboard Cite

One way to interpret smoothness of a measure in infinite dimensions is quasi-invariance of the measure under a class of transformations. Usually such settings lack a reference measure such as the Lebesgue or Haar measure, and therefore we can not use smoothness of a density with respect to such a measure. We describe how a functional inequality can be used to prove quasi-invariance results in several settings. In particular, this gives a different proof of the classical Cameron-Martin (Girsanov) theorem for an abstract Wiener space. In addition, we revisit several more geometric examples, even though the main abstract result concerns quasi-invariance of a measure under a group action on a measure space.1991 Mathematics Subject Classification. Primary 58G32 58J35; Secondary 22E65 22E30 22E45 58J65 60B15 60H05.1 2 GORDINA moreover we do not refer to the extensive literature on the subject, as it is beyond the scope of our paper.We start by describing an abstract setting of how finite-dimensional approximations can be used to prove such a quasi-invariance. In [9] this method was applied to projective and inductive limits of finite-dimensional Lie groups acting on themselves by left or right multiplication. In that setting a functional inequality (integrated Harnack inequality) on the finite-dimensional approximations leads to a quasi-invariance theorem on the infinite-dimensional group space. Similar methods were used in the elliptic setting on infinite-dimensional Heisenberg-like groups in [8], and on semi-infinite Lie groups in [14]. Note that the assumptions we make below in Section 3 have been verified in these settings, including the sub-elliptic case for infinite-dimensional Heisenberg group in [1]. Even though the integrated Harnack inequality we use in these situations have a distinctly geometric flavor, we show in this paper that it does not have to be.The paper is organized as follows. The general setting is described in Section 2 and 3, where Theorem 3.2 is the main result. One of the ingredients for this result is quasi-invariance for finite-dimensional approximations which is described in Section 3. We review the connection between an integrated Harnack inequality and Wang's Harnack inequality in Section 4. Finally, Section 5 gives several examples of how one can use Theorem 3.2. We describe in detail the case of an abstract Wiener space, where the group in question is identified with the Cameron-Martin subspace acting by translation on the Wiener space. In addition we discuss elliptic (Riemannian) and sub-elliptic (sub-Riemannian) infinite-dimensional groups which are examples of a subgroup acting on the group by multiplication.Acknowledgement. The author is grateful to Sasha Teplyaev and Tom Laetsch for useful discussions and helpful comments.

show abstract

Change of base for operator space modules

Rosbotham¹

2020

Arch. Math.

View full text Add to dashboard Cite

show abstract

On the algebraic foundations of bounded cohomology

Cited by 25 publications

References 0 publications

Some Remarks for Künneth Formula on Bounded Cohomology

Some Remarks for Künneth Formula on Bounded Cohomology

An Application of a Functional Inequality to Quasi-Invariance in Infinite Dimensions

Change of base for operator space modules

Contact Info

Product

Resources

About