On the sum of two closed algebras of continuous functions on a compactum

Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras

“…These objects were first introduced by Diliberto and Straus [5] (in [5], they are called "permissible lines"). They appeared further in a number of papers with several different names such as "paths" (see, e.g., [17,18]), "trips" (see [20,21]), "links" (see, e.g., [3,15,16]), etc. The term "bolt of lightning" was due to Arnold [1].…”

“…For a given compact space X, the closedness of A 1 + A 2 in C(X) strictly depends on the internal structure of X. There are several results on closedness of a sum of two algebras (see, e.g., [14,20,21]). The most explicit and practical result is due to Medvedev (see [21]).…”

“…There are several results on closedness of a sum of two algebras (see, e.g., [14,20,21]). The most explicit and practical result is due to Medvedev (see [21]). He showed that the sum A 1 + A 2 is closed in C(X) if and only if the lengths of all irreducible bolts of X are uniformly bounded.…”

On the representation by sums of algebras of continuous functions

“…For k = 2, the condition τ n (X) = ∅ can be expressed in terms of sets of points in X that are geometrically explicit. In the special case of the algebras U and V considered above, these points were introduced in the literature under different names such as "permissible lines" [4] "bolts of lightning" [1,6,7,13,14,21,22], "trips" [20], "paths" [5,8,10,18,19], "links" [3,15], etc. The term bolt of lightning is the most common one and is due to Arnold [1].…”

Geometry of good sets inn-fold Cartesian product

“…A necessary and sufficient condition for a subset S of X 1 × X 2 × • • • × X n to be good was derived in our paper [7] and some consequences for simplicial measures and sums of algebras were discussed. For n = 2 these questions are well-discussed in [1,2,3,5,6,7,10,11,12,13,14,17]. The notion of a link or path between two points plays a crucial role in all these papers.…”