The inclusion of the Schur algebra in B(l2) is not inverse-closed

Romain Tessera

doi:10.1007/s00605-010-0216-x

The inclusion of the Schur algebra in B(l²) is not inverse-closed

Romain Tessera

Ecole Normale Supérieure de Lyon

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

The Schur algebra is the algebra of operators which are bounded on ℓ¹ and on ℓ^∞. In this note, we exhibit an element of the group algebra of the free group with two generators, which, as a convolution operator, is invertible in ℓ², and whose inverse is not bounded on ℓ¹ nor on ℓ^∞. In particular, this shows that the Schur algebra is not inverse-closed.

Original language	English
Pages (from-to)	115-118
Number of pages	4
Journal	Monatshefte fur Mathematik
Volume	164
Issue number	1
DOIs	https://doi.org/10.1007/s00605-010-0216-x
Publication status	Published - 1 Sept 2011
Externally published	Yes

Keywords

Convolution operators on groups
Inverse-closed subalgebras of B(H)
Schur algebra
Symmetric Banach algebras

Access to Document

10.1007/s00605-010-0216-x

Cite this

@article{3d74001e81c149f9b5d1985acc2c0c9c,

title = "The inclusion of the Schur algebra in B(l2) is not inverse-closed",

abstract = "The Schur algebra is the algebra of operators which are bounded on ℓ1 and on ℓ∞. In this note, we exhibit an element of the group algebra of the free group with two generators, which, as a convolution operator, is invertible in ℓ2, and whose inverse is not bounded on ℓ1 nor on ℓ∞. In particular, this shows that the Schur algebra is not inverse-closed.",

keywords = "Convolution operators on groups, Inverse-closed subalgebras of B(H), Schur algebra, Symmetric Banach algebras",

author = "Romain Tessera",

year = "2011",

month = sep,

day = "1",

doi = "10.1007/s00605-010-0216-x",

language = "English",

volume = "164",

pages = "115--118",

journal = "Monatshefte fur Mathematik",

issn = "0026-9255",

publisher = "Springer",

number = "1",

}

TY - JOUR

T1 - The inclusion of the Schur algebra in B(l2) is not inverse-closed

AU - Tessera, Romain

PY - 2011/9/1

Y1 - 2011/9/1

N2 - The Schur algebra is the algebra of operators which are bounded on ℓ1 and on ℓ∞. In this note, we exhibit an element of the group algebra of the free group with two generators, which, as a convolution operator, is invertible in ℓ2, and whose inverse is not bounded on ℓ1 nor on ℓ∞. In particular, this shows that the Schur algebra is not inverse-closed.

AB - The Schur algebra is the algebra of operators which are bounded on ℓ1 and on ℓ∞. In this note, we exhibit an element of the group algebra of the free group with two generators, which, as a convolution operator, is invertible in ℓ2, and whose inverse is not bounded on ℓ1 nor on ℓ∞. In particular, this shows that the Schur algebra is not inverse-closed.

KW - Convolution operators on groups

KW - Inverse-closed subalgebras of B(H)

KW - Schur algebra

KW - Symmetric Banach algebras

U2 - 10.1007/s00605-010-0216-x

DO - 10.1007/s00605-010-0216-x

M3 - Article

AN - SCOPUS:80052039316

SN - 0026-9255

VL - 164

SP - 115

EP - 118

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

IS - 1