Primariness of some spaces of continuous functions. (English) Zbl 0734.46012

Drewnowski, Lech

Rev. Mat. Univ. Complutense Madr. 2, Suppl., 119-127 (1989).

Summary: J. Roberts and the author have recently shown that, under the continuum hypothesis, the Banach space \(\ell_{\infty}/c_ 0\) is primary. Since this space is isometrically isomorphic to the space \(C(\omega^*)\) of continuous scalar functions on \(\omega^*=\beta \omega -\omega\), it is quite natural to consider the question of primariness also for the spaces of continuous vector functions on \(\omega^*\). The present paper contains some partial results in that direction. In particular, from our results it follows that \(C(\omega^*,C(K))\) is primary for any infinite metrizable compact space K (without assuming the CH).

MSC:

46B25	Classical Banach spaces in the general theory
46B45	Banach sequence spaces
46E40	Spaces of vector- and operator-valued functions
46E15	Banach spaces of continuous, differentiable or analytic functions
03E50	Continuum hypothesis and Martin’s axiom

Keywords:

continuum hypothesis; primariness; spaces of continuous vector functions