Linear Isometries of Banach-Kantorovich $L_p$-spaces

V. I. Chilin^a, G. B. Zakirova<sup>b

a</sup> Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, 9, Universitet, Tashkent, 100174, Uzbekistan
^b Tashkent State Transport University, 1, Odilxodjaev, Tashkent, 100167, Uzbekistan

Abstract: Let $B$ be a complete Boolean algebra, $Q(B)$ be the Stone compact of $B$, and $C_\infty (Q(B))$ be the commutative unital algebra of all continuous functions $x: Q(B) \to [-\infty, +\infty]$, assuming possibly the values $\pm\infty$ on nowhere-dense subsets of $Q(B)$. We consider the Banach-Kantorovich spaces $L_p(B,m)\subset C_\infty (Q(B)),$ associated with a measure $m$ defined on $B$ with the values in the algebra of measurable real functions. It is shown that in the case when the measure $m$ has the Maharam property, for any linear isometry $U: L_p(B,m) \to L_p(B,m), 1\leq p < \infty, p \neq 2,$ there exist an injective normal homomorphisms $T : C_\infty (Q(B)) \to C_\infty (Q(B))$ and an element $y \in L_p(B,m)$ such that $U(x ) =y\cdot T(x)$ for all $x\in L_p(B,m)$.

Keywords: Banach-Kantorovich space, Maharam measure, vector integration, linear isometry.

UDC: 517.98

MSC: 46B04, 46B42, 46E30, 46G10

Language: English