Finite-dimensional left ideals in the duals of introverted spaces

Filali, M.; Monfared, M. Sangani

doi:10.1090/s0002-9939-2011-10784-6

Cited by 7 publications

(7 citation statements)

References 36 publications

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“…The proofs that are similar to those given above are omitted. We remark that using the same methods given in Filali-Monfared [5], right π-invariant elements of l ∞ (n, A * * ) can be used to characterize finite-dimensional right ideals in A * * equipped with the right Arens product 3. More generally, if X is a faithful, right introverted subspace of A * for which X * is equipped with the induced right Arens product 3, then [5, Lemma 2.2] remains true without any change, and the analogues of [5, Lemma 2.4, Theorems 2.7 and 2.8] can be readily formulated and proved for right π-invariants and finite-dimensional right ideals in X * .…”

Section: Remark 29mentioning

confidence: 99%

“…Arens [1] or Dales [3]). For continuous finitedimensional representations π : A → M n (C), the (left) π-invariant elements of l ∞ (n, A * * ) were recently studied by the authors in connection with the characterization of finite-dimensional left ideals in the dual of left introverted subspaces of A * (Filali-Monfared [5]). In this paper we extend the concept of π-invariance to continuous representations on Hilbert spaces, and we prove an interesting link between the existence of π-invariant elements and the vanishing of certain Hochschild cohomology groups of A.…”

Section: Introductionmentioning

confidence: 99%

“…We denote the canonical extension of π to A * * by π, so that π is a w * -continuous representation of A * * on H and for every Φ ∈ A * * , ( π(Φ)e j |e i ) = Φ, π i j (cf. Filali-Monfared [5]). The projection map on the (i, j)-th coordinate is defined by pr i j : L (H) → C, T → (Te j |e i ).…”

Section: Introductionmentioning

confidence: 99%

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A Cohomological Property of π-invariant Elements

Filali¹,

Monfared²

2013

Can. math. bull.

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Abstract. Let A be a Banach algebra and π : A −→ L (H) be a continuous representation of A on a separable Hilbert space H with dim H = m. Let π i j be the coordinate functions of π with respect to an orthonormal basis and suppose that for eachIn this paper we prove a link between the existence of left π-invariant elements and the vanishing of certain Hochschild cohomology groups of A. Our results extend an earlier result by Lau on F-algebras and recent results of Kaniuth-Lau-Pym and the second named author in the special case that π : A −→ C is a non-zero character on A.

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Section: Remark 29mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Cohomological Property of π-invariant Elements

Filali¹,

Monfared²

2013

Can. math. bull.

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“…First we prove that i∈I d i tr(ψ(i)π i ) ∈ AP(A). Since for every i ∈ I, π i is a continuous finite-dimensional representation of A, we have tr(ψ(i)π i ) ∈ AP(A) [15,Lemma 2.3]. For every b ∈ A we have…”

Section: Rfd Transform and Almost Periodicitymentioning

confidence: 99%

Almost periodic functionals and finite-dimensional representations

Filali

Monfared

2018

Studia Math.

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We show that if A is a C *-algebra and λ ∈ A * is a nonzero almost periodic functional which is a coordinate functional of a topologically irreducible involutive representation π, then dim π < ∞. We introduce the RFD transform αA : A → U (A) of a Banach algebra A and establish its universal property. We show that if A has a bounded two-sided approximate identity, then almost periodic functionals on A which are limits of coordinate functionals of finite-dimensional representations have lifts to almost periodic functionals on U (A). Other connections with almost periodicity and harmonic analysis are also discussed.

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“…It is known that for an involutive Banach algebra A with bounded approximate identity, every positive linear functional on A is a coordinate function of an involutive representation on some Hilbert space (see for example, Dixmier [14,Proposition 2.4.4]). Being a coordinate function of a representation on some Hilbert space, a positive linear functional is in fact weakly almost periodic by Young [38, page 102] or Filali-Monfared [18,Lemma 2.3].…”

Section: Introductionmentioning

confidence: 99%

Representations of Banach algebras subordinate to topologically introverted spaces

Filali

Neufang

Monfared

2015

Trans. Amer. Math. Soc.

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Let A be a Banach algebra, X a closed subspace of A * , Y a dual Banach space with predual Y * , and π a continuous representation of A on Y . We call π subordinate to X if each coordinate function π y,λ ∈ X, for all y ∈ Y, λ ∈ Y * . If X is topologically left (right) introverted and Y is reflexive, we show the existence of a natural bijection between continuous representations of A on Y subordinate to X, and normal representations of X * on Y . We show that if A has a bounded approximate identity, then every weakly almost periodic functional on A is a coordinate function of a continuous representation of A subordinate to W AP (A). We show that a function f on a locally compact group G is left uniformly continuous if and only if it is the coordinate function of the conjugate representation of L 1 (G), associated to some unitary representation of G. We generalize the latter result to an arbitrary Banach algebra with bounded right approximate identity. We prove the functionals in LU C(A) are all coordinate functions of some norm continuous representation of A on a dual Banach space.

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Finite-dimensional left ideals in the duals of introverted spaces

Cited by 7 publications

References 36 publications

A Cohomological Property of π-invariant Elements

A Cohomological Property of π-invariant Elements

Almost periodic functionals and finite-dimensional representations

Representations of Banach algebras subordinate to topologically introverted spaces

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