Idempotent measures: absolute retracts and soft maps (Q2027803)

This paper concerns the space \(IX\) of all idempotent measures on a compactum \(X\), where the notion of idempotent measure is in the sense of \textit{M. M. Zarichnyi} [Izv. Math. 74, No. 3, 481--499 (2010; Zbl 1220.18002); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 74, No. 3, 45--64 (2010)]. It mainly consists of two parts. In the first half of the paper, the author investigates the problem when the space \(IX\) is an absolute retract. The main result states: The compactum \(IX\) is an absolute retract if and only if \(X\) is an openly generated compactum of weight \(\leq \omega_1\), the first uncountable cardinal number. Here a compactum being openly generated means that it is the limit of a continuous \(\omega\)-complete system consisting of compacta of weight \(\leq \omega\) and open bonding maps, where \(\omega\) is the countable cardinal number. In the second half of the paper, motivated by the softness problem for barycenter maps of probability measures (Fedorchuk), the author studies the softness of idempotent barycenter maps. Here, the notion of softness is in the sense of \textit{E. V. Shchepin} [Russ. Math. Surv. 31, No. 5, 155--191 (1976; Zbl 0356.54026)], and the notion of the idempotent barycenter map is in the sense of \textit{M. M. Zarichnyi} [loc. cit.]. The main result states: If \(X\) is a compactum such that the idempotent barycenter map \(\beta_{IX}: I(I(X)) \to IX\) is \(0\)-soft, \(X\) is metrizable; and the idempotent barycenter map \(\beta_{[0,1]^{\omega_1}}: I[0,1]^{\omega_1} \to [0,1]^{\omega_1}\) is not \(0\)-soft.