Symmetry and inverse closedness for Banach $^*$-algebras associated to discrete groups

Măntoiu, Marius

doi:10.15352/bjma/09-2-19

Cited by 9 publications

(13 citation statements)

References 21 publications

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“…We extend the results presented in the exposition [14] and provide proofs for them. Meanwhile some of our results have been reproved by I. Beltiţȃ and D. Beltiţȃ [7,6] and by Mȃntoiu [36] in the discrete case. After archiving (on arxiv) this paper A. R. Schep kindly informed us that our proposition 2.3 is a special case of a theorem in his dissertation [40].…”

Section: Introductionmentioning

confidence: 74%

On Convolution Dominated Operators

Fendler

Leinert

2016

Integr. Equ. Oper. Theory

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Abstract. For a locally compact group G we consider the algebra CD(G)In the case of discrete groups those operators can be dealt with quite sufficiently if the group in question is rigidly symmetric. For non-discrete groups we investigate the subalgebra of regular convolution dominated operators CDreg(G). For amenable G which is rigidly symmetric as a discrete group we show that any element of CDreg(G) is invertible in CDreg(G) if it is invertible as a bounded operator on L 2 (G). We give an example of a symmetric group E for which the convolution dominated operators are not inverse-closed in the bounded operators on L 2 (E).Mathematics Subject Classification. Primary 47B35; Secondary 43A20.

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Section: Introductionmentioning

confidence: 74%

On Convolution Dominated Operators

Fendler

Leinert

2016

Integr. Equ. Oper. Theory

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“…Looking at the proof of Theorem 3.1 we might hope in light of results on symmetric (Banach) * -algebras in e.g. [6,19,20,29,45] that it is possible to obtain similar results for the algebras considered in these papers. However, considering the crucial role C * -uniqueness plays in order to get spectral invariance for all * -representations for the * -algebra in Theorem 3.1, it would look like a key ingredient in a proof of such a result should be analogous C * -uniqueness results for these algebras, and for the time being these remain elusive.…”

Section: Remark 39mentioning

confidence: 87%

“…Moreover, we note that in recent years quite a bit of work has been done on spectral invariance of various algebras motivated by a plethora of different problems, see e.g. [6,29,45,46].…”

Section: Introductionmentioning

confidence: 99%

Spectral Invariance of $$*$$-Representations of Twisted Convolution Algebras with Applications in Gabor Analysis

Austad

2021

J Fourier Anal Appl

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We show spectral invariance for faithful $$*$$ ∗ -representations for a class of twisted convolution algebras. More precisely, if G is a locally compact group with a continuous 2-cocycle c for which the corresponding Mackey group $$G_c$$ G c is $$C^*$$ C ∗ -unique and symmetric, then the twisted convolution algebra $$L^1 (G,c)$$ L 1 ( G , c ) is spectrally invariant in $${\mathbb {B}}({\mathcal {H}})$$ B ( H ) for any faithful $$*$$ ∗ -representation of $$L^1 (G,c)$$ L 1 ( G , c ) as bounded operators on a Hilbert space $${\mathcal {H}}$$ H . As an application of this result we give a proof of the statement that if $$\Delta $$ Δ is a closed cocompact subgroup of the phase space of a locally compact abelian group $$G'$$ G ′ , and if g is some function in the Feichtinger algebra $$S_0 (G')$$ S 0 ( G ′ ) that generates a Gabor frame for $$L^2 (G')$$ L 2 ( G ′ ) over $$\Delta $$ Δ , then both the canonical dual atom and the canonical tight atom associated to g are also in $$S_0 (G')$$ S 0 ( G ′ ) . We do this without the use of periodization techniques from Gabor analysis.

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“…The simplest case of a trivial action leads to the projective tensor product between L 1 (G) and a C * -algebra. Some references are: [20,4,18,23,22,14,15,3,12,13,5,21,11].…”

Section: Introductionmentioning

confidence: 99%