Normal Structure in Dual Banach Spaces Associated With a Locally Compact Group

Lau, Anthony To-Ming; Mah, Peter F.

doi:10.2307/2001126

Cited by 3 publications

(2 citation statements)

References 6 publications

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“…We will use the characterization that

X

is scattered if and only if

M(X) = M_d(X)

, the space of discrete measures on

X

; cf. [23, Lemma 1]. (i)

\Rightarrow

(ii).…”

Section: Invariant Versus Natural Projectionsmentioning

confidence: 99%

On invariant and natural projections

Neufang

2022

Bulletin of London Math Soc

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We solve three open problems raised in Lau and Ülger, Trans. Amer. Math. Soc. 366 (2014), no. 8, 4151-4171 regarding invariant and natural projections. More precisely, we show that for a commutative 𝐶 * -algebra , every natural projection on  * is invariant under the canonical action of  if and only if  is scattered. We exhibit further classes of Banach algebras, comprising important examples from abstract harmonic analysis, for which the two types of projections differ, by establishing links of the latter property to almost periodicity, Arens regularity, and amenability. Moreover, for any commutative semisimple Banach algebra  with a bounded approximate identity (BAI), we characterize the 𝑤 * -continuous natural projections on  * as the adjoints of idempotent multipliers on . We also prove that ℂ is the only (non-zero) commutative semisimple Banach algebra  with a BAI such that every 𝑤 * -continuous projection on  * is natural.M S C 2 0 2 0 43A20, 46H25, 46J05, 46J40, 46L05 (primary) INTRODUCTION AND PRELIMINARIESInvariant projections play an important role in amenability theory, and in the study of ideals and multipliers in Banach algebras and abstract harmonic analysis; see, for example, [1,7,11,15,18,19,22,25,27]. Here, in the general setting, one considers a Banach algebra , and a projection on the dual  * (that is, a bounded linear map 𝑃 ∶  * →  * such that 𝑃 2 = 𝑃) which is invariant under

show abstract

“…We will use the characterization that

X

is scattered if and only if

M(X) = M_d(X)

, the space of discrete measures on

X

; cf. [23, Lemma 1]. (i)

\Rightarrow

(ii).…”

Section: Invariant Versus Natural Projectionsmentioning

confidence: 99%

On invariant and natural projections

Neufang

2022

Bulletin of London Math Soc

View full text Add to dashboard Cite

show abstract

“…This is a significant generalization of a classical result of Lim [34] for the circle group. This work is related to and was motivated by Tony's earlier work with Mah [31] andÜlger [27].…”

Section: Multipliers Of Commutative Banach Algebras With Applications Tomentioning

confidence: 99%