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Standard ideals in convolution Sobolev algebras on the half-line. (English) Zbl 1238.46045
The authors study the closed ideals of the convolution Sobolev algebras \(\mathcal T^{(n)}_+(t^n)\), which is defined as the completion of the space \(C^{(\infty)}_c[0,\infty)\) under the norm \[ \|f\|:=\int_0^\infty|f^{(n)}(t)|t^n dt\,. \] The authors show that the closed ideals of \(\mathcal T^{(n)}_+(t^n)\) are in a one-to-one correspondence with closed ideals of the Beurling convolution algebra \(L^1(\omega_n)\), where \(\omega_n(t)=(1+t)^n\), that are included in \[ \mathfrak M_n:=\left\{\varphi\in L^1(\omega_n):\;\int_0^\infty \varphi(t)t^k dt=0\;(0\leq k\leq n-1)\right\}\,. \] Moreover, a closed ideal is standard in \(\mathcal T^{(n)}_+(t^n)\) if and only if its image via this correspondence is standard in \(\mathfrak M_n\). As an application of this connection, they show that every closed ideal in \(\mathcal T^{(n)}_+(t^n)\) with a compact countable hull is standard. Finally, the authors characterize those closed ideals in \(\mathcal T^{(n)}_+(t^n)\) that have empty hull.

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
46J20 Ideals, maximal ideals, boundaries
26A46 Absolutely continuous real functions in one variable
42A85 Convolution, factorization for one variable harmonic analysis
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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