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

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
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