SYMMETRIC BANACH-KANTOROVICH SPACES - Zakirova, Gavhar; Chilin, Vladimir (Journal of Mathematics, Mechanics and Computer Science, 2025)
Let B be a complete Boolean algebra, let Q(B) be the Stone compact of B, let C_∞ (Q(B)) be the commutative unital algebra of all continuous functions x:Q(B) → [-∞,+∞], assuming possibly the values ±∞ on nowhere-dense subsets of Q(B). We consider Maharam measure m defined on B, which takes on value in the algebra L^0 of all real measurable functions. With the help of the property of equimeasurablity of elements from C_∞ (Q(B)), associated with such a measure m, the notion of a symmetric Banach-Kantorovich space (E,\|·|_E) over L^0 is introduced and studied in detail. Here EC_∞ (Q(B)), and \|·|_E – L^0-valued norm in E, endowing it with the structure of the space Banach-Kantorovich. Examples of symmetric Banach-Kantorovich spaces are given, which are vector-valued analogues of classical L^p-spaces, 1≤ p ≤ ∞, associated with a numerical σ-finite measure.