Complemented subspaces of Banach spaces C(K× L)

2024/05/29 by Plebanek, Grzegorz, Rondoš, Jakub, Sobota, Damian
#46B26 #46E15. Secondary: 28A33 #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 46B20

paper · doi:10.48550/arxiv.2405.19120

Abstract

We prove that, for every compact spaces K₁,K₂ and compact group G, if both K₁ and K₂ map continuously onto G, then the Banach space C(K₁ × K₂) contains a complemented subspace isometric to the Banach space C(G). Consequently, C(K₁× K₂) contains a complemented copy of C([0,1]) for every non-scattered K₁,K₂. Also, answering a question of Alspach and Galego, we get that C(βω×βω) contains a complemented copy of C([0,1]^κ) for every cardinal number 1≤κ≤\mathfrak c and hence a complemented copy of C(K) for every metric compact space K. On the other hand, for the pointwise topology, we show that C_p(βω×βω) contains no complemented copy of C_p(2^ω).

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