Inverse-Closed Algebras of Integral Operators on Locally Compact Groups

Abstract

We construct some inverse-closed algebras of bounded integral operators with operator-valued kernels, acting in spaces of vector-valued functions on locally compact groups. To this end we make use of covariance algebras associated to C * -dynamical systems defined by the C * -algebras of right uniformly continuous functions with respect to the left regular representation.

On Convolution Dominated Operators

“…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].…”

Inverse-closedness of the set of integral operators with L1-continuously varying kernels

“…For convenience (without loss of generality), we assume that β is defined everywhere. Kernels of the class N 1 and the operators induced by them were considered in [8,11], [20, § 5.4], and [21]. In order to show that the notation used in [20, § 5.4] and [21] is equivalent to the notation used in the present paper, we note the following Proposition.…”

“…The idea of the proof consists of a combination of two results. The fist result [8,11,20,21] states that if estimate ( * ) holds for the kernel n of the operator N and 1 + N is invertible in L p (R c , E), then (1 + N ) −1 = 1 + M , where M is an integral operator with a kernel m satisfying estimate ( * ) as well. This easily implies that the fact of the invertibility of 1 + N and the kernel m do not depend on p. On the other hand, it was known [20] that (for a wide class of operators T ) if 'coefficients' of T : L p → L p vary continuously in an abstract sense, then T −1 has the same property (provided T −1 exists).…”

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

“…It is shown in [10] that a discrete group G is rigidly symmetric if and only if ℓ 1 α (G; A) is symmetric for every usual (untwisted) action α (see also [3]). In Section 2.2 we extend this equivalence to include twisted C * -dynamical systems, involving a 2-cocycle ω , as well as other types of decay than those expressed by the ℓ 1 -condition.…”

“…When developed starting with the action of G on an Abelian C * -algebra A, it has interesting Hilbert-space representations that are studied in 3.3. As a particular case, A could be a C * -algebra of bounded complex functions on G (see [3,10]) and then a natural class of twisted kernels is already isomorphic to the twisted crossed product. As a consequence of the results of the previous section, certain subfamilies of twisted kernels form symmetric Banach * -algebra.…”