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Automorphisms and derivations of a Fréchet algebra of locally integrable functions

100%

Ghahramani F., McClure J. P.

Studia Mathematica

1992

tom 103

nr 1

51-69

We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra $L¹_{loc}$ of locally integrable functions on the half-line $ℝ^+$. We show, among other things, that every automorphism θ of $L¹_{loc}$ is of the form $θ = φ _ae^{λX}e^D$, where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and $φ_a$ is the dilation operator $(φ_af)(x) = af(ax)$ ($f ∈ L¹_{loc}$, $x ∈ ℝ^+$). It is also shown that the automorphism group is a topological group with the topology of uniform convergence on bounded sets and is the semidirect product of a connected subgroup and a discrete group which is isomorphic to the discrete group of real numbers.

Derivations into iterated duals of Banach algebras

81%

Dales H. G., Ghahramani F., Grønbæek N.

Studia Mathematica

1998

tom 128

nr 1

19-54

We introduce two new notions of amenability for a Banach algebra A. The algebra A is n-weakly amenable (for n ∈ ℕ) if the first continuous cohomology group of A with coefficients in the n th dual space $A^{(n)}$ is zero; i.e., $ℋ^1(A,A^{(n)}) = {0}$. Further, A is permanently weakly amenable if A is n-weakly amenable for each n ∈ ℕ. We begin by examining the relations between m-weak amenability and n-weak amenability for distinct m,n ∈ ℕ. We then examine when Banach algebras in various classes are n-weakly amenable; we study group algebras, C*-algebras, Banach function algebras, and algebras of operators. Our results are summarized and some open questions are raised in the final section.

Ograniczanie wyników

2 Studia Mathematica

2 Ghahramani F.

1 Dales H. G.

1 Grønbæek N.

1 McClure J. P.

1 1998

1 1992

Wyniki wyszukiwania

Automorphisms and derivations of a Fréchet algebra of locally integrable functions

Derivations into iterated duals of Banach algebras