ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE XLV (1) (2005), 107-124
We show that the Banach space D(0, 1) of all scalar (real or complex) functions on [0, 1) that are right continuous at each point of [0, 1) with left-hand limit at each point of (0, 1] equipped with the uniform convergence topology is primary. 1991 Mathematics Subject Classification: 46B20, 46B25, 46E15. A Banach space X is said to be primary if for every continuous projection P : X → X, either P (X) or (I − P)(X) is isomorphic to X. Many classical C(K) spaces are known to be primary, for
... ce: C([0, 1]), C(βN \ N), C(α) for some ordinal numbers α (see , , ,  and ). Besides, the spaces c 0 and l ∞ = C(βN) are prime i.e. every infinite dimensional and complemented subspace of these spaces is isomorphic to the whole space (see ). We show that the Banach space D(0, 1) of all scalar (real or complex) functions on [0, 1) that are right continuous at each point of [0, 1) with left hands limit at each point of (0, 1] equipped with the uniform convergence topology is also a primary Banach space. The space D(0, 1) is isometrically isomorphic to the space of all continuous scalar functions on the two arrows space. This space has appeared in the literature (see e.g. , , ) as an example of a Banach space with some interesting properties. Thus, for example, H. Corson  showed that the quotient space D(0, 1)/C([0, 1]) is isomorphic to c 0 ([0, 1]) and also that the space D(0, 1) is not normal in its weak topology. There exists a countable set of evaluation functionals separating points of D(0, 1). Consequently, it contains no isomorphic copy of c 0 (Γ) for any uncountable set Γ. Every isometric copy of c 0 in D(0, 1) is complemented (see ). Moreover, D(0, 1) contains no isomorphic copy of l ∞ (see ). The present paper continues research on properties of the space D(0, 1) started in  and continued in . The space plays a crucial rule in the description of * The research was supported by Komitet Bada´nBada´n Naukowych (State Committee for Scientific Research), Poland, grant no. P 03A 022 25.