Mathematics Colloquia and Seminars

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One is often compelled to study operators and their inverses under conditions for which polynomial decay is too slow, and exponential decay is too fast or not exactly preserved. In my talk I establish that a broad class of involutive Banach algebras (Banach *-algebras) of infinite integral operators, defined by the property that the kernels of the elements of the algebras possess subexponential off-diagonal decay, is inverse closed in $\B(\Ltwo).$ This means that each subalgebra of $\B(\Ltwo)$ under consideration contains the inverse of each of its elements. A second result, concerning symmetry of Banach algebras, is demonstrated en route while proving the above facts about decay of kernels of integral operator inverses. An involutive algebra is {\em symmetric} if the spectrum of positive elements is positive. Historically it has often been quite difficult to verify that a given involutive algebra is symmetric, but using techniques discovered by Gr\"ochenig and Leinert I will show that the algebras under consideration are symmetric.