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Norm conditions for uniform algebra isomorphisms

100%

Luttman A., Lambert S.

Open Mathematics

2008

tom 6

nr 2

272-280

In recent years much work has been done analyzing maps, not assumed to be linear, between uniform algebras that preserve the norm, spectrum, or subsets of the spectra of algebra elements, and it is shown that such maps must be linear and/or multiplicative. Letting A and B be uniform algebras on compact Hausdorff spaces X and Y, respectively, it is shown here that if λ ∈ ℂ / {0} and T: A → B is a surjective map, not assumed to be linear, satisfying $$ \left\| {T(f)T(g) + \lambda } \right\| = \left\| {fg + \lambda } \right\|\forall f,g \in A, $$ then T is an ℝ-linear isometry and there exist an idempotent e ∈ B, a function κ ∈ B with κ 2 = 1, and an isometric algebra isomorphism $$ \tilde T:{\rm A} \to Be \oplus \bar B(1 - e) $$ such that $$ T(f) = \kappa \left( {\tilde T(f)e + \gamma \overline {\tilde T(f)} (1 - e)} \right) $$ for all f ∈ A, where γ = λ / |λ|. Moreover, if T is unital, i.e. T(1) = 1, then T(i) = i implies that T is an isometric algebra isomorphism whereas T(i) = −i implies that T is a conjugate-isomorphism.

Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras

84%

Lykova Z.

Open Mathematics

2008

tom 6

nr 3

405-421

We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain $$ \hat \otimes $$ -algebras. We use well-developed homological techniques together with some niceties of the theory of locally convex spaces to generalize the results known in the case of Banach algebras and their inverse limits to wider classes of topological algebras. To this end we show that, for a continuous morphism ϕ: x → y of complexes of complete nuclear DF-spaces, the isomorphism of cohomology groups H n(ϕ): H n(x) → H n (y) is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective $$ \hat \otimes $$ -algebras: the tensor algebra E $$ \hat \otimes $$ F generated by the duality (E,F,<·,·>) for nuclear Fréchet spaces E and F or for nuclear DF-spaces E and F; nuclear biprojective Köthe algebras λ(P) which are Fréchet spaces or DF-spaces; the algebra of distributions ε*(G) on a compact Lie group G.

Ograniczanie wyników

2 Open Mathematics

1 Lambert S.

1 Luttman A.

1 Lykova Z.

2 2008

Wyniki wyszukiwania

Norm conditions for uniform algebra isomorphisms

Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras