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Convolution dominated operators on compact extensions of abelian groups. (English) Zbl 06946445

Summary: If \(G\) is a locally compact group, \(CD(G)\) the algebra of convolution dominated operators on \(L^2(G)\), then an important question is: Is \(\mathbb {C}1+CD(G)\) (or \(CD(G)\) if \(G\) is discrete) inverse-closed in the algebra of bounded operators on \(L^2(G)\)?
In this note we answer this question in the affirmative, provided \(G\) is such that one of the following properties is satisfied.
(1)
There is a discrete, rigidly symmetric, and amenable subgroup \(H\subset G\) and a (measurable) relatively compact neighbourhood of the identity \(U\), invariant under conjugation by elements of \(H\), such that \(\{hU\;:\;h\in H\}\) is a partition of \(G\).
(2)
The commutator subgroup of \(G\) is relatively compact. (If \(G\) is connected, this just means that \(G\) is an IN group.)
All known examples where \(CD(G)\) is inverse-closed in \(B(L^2(G))\) are covered by this.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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References:

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