Multilinear isometries on function algebras

Abstract

Abstract. Let A 1 , ..., A k be function algebras (or more generally, dense subspaces of uniformly closed function algebras) on locally compact Hausdorff spaces X 1 , ..., X k , respectively, and let Z be a locally compact Hausdorff space. A k-linear map T :Based on a new version of the additive Bishop's Lemma, we provide a weighted composition characterization of such maps. These results generalize the well-known Holsztyński's theorem ([9]) and the bilinear version of this theorem provided in [10] by a different approach.

Multilinear isometries on spaces of vector-valued continuous functions

“…We note that for the case where E 1 = ... = E k = C or R, from the k-linearity of T it follows easily that for each (x 1 , ..., x k ) ∈ X 1 × ... × X k and (e 1 , ..., e k ) ∈ T k (or {1, −1} k ), I x1,...,x k = I e1,...,e k x1,...,x k , which is a non-empty set. The following result, which is an easy consequence of Theorem 4.1, is a generalization of the main theorems in [7] and [9] for certain function spaces. for all (f 1 , ..., f k ) ∈ A 1 × ... × A k and y ∈ Y 0 , where π i is the ith projection map.…”

“…In [7] (see also [4]), based on a new version of the additive Bishop's Lemma, the authors extended the above results to multilinear isometries of function algebras on locally compact Hausdorff spaces, a context where the Stone-Weierstrass theorem is not applicable.…”

Real-Multilinear Isometries on Function Algebras

“…So we can obtain immediately the main result in [8] as follows: for all (f 1 , ..., f k ) ∈ A 1 × ... × A k and y ∈ Y 0 , where π j is the jth projection map. Remark 4.3.…”

“…More precisely, as seen in [8] and in all previous papers dealing with 1-complex-linear (not necessarily surjective) isometries starting with Holsztyński's seminal paper…”