On projective Banach lattices of the form C(K) and FBL[E]

Abstract

We show that if a Banach lattice is projective, then every bounded sequence that can be mapped by a homomorphism onto the basis of c 0 must contain an ℓ 1 -subsequence. As a consequence, if Banach lattices ℓ p or F BL[E] are projective, then p = 1 or E has the Schur property, respectively. On the other hand, we show that C(K) is projective whenever K is an absolute neighbourhood retract, answering a question by de Pagter and Wickstead.2010 Mathematics Subject Classification. 46B43, 06BXX.

Free Banach lattices under convexity conditions

“…To conclude this section, we find a Banach space property of E which characterizes when FBL[E] is projective. Note in this connection that it was shown in [4] that if FBL[E] is projective, then necessarily E has the Schur property. Since we now also have the spaces FBL D [E] -and our characterizations extend to these spaces in the appropriate way -we introduce two new definitions.…”

“…As a consequence of the fact that 1 (A) is a projective Banach space for any nonempty set A, it was shown in [17] that FBL 1 (A) is a projective Banach lattice. Other examples of projective Banach lattices include all finite dimensional Banach lattices [17] and C(K ) spaces for every compact Hausdorff space K which is an absolute neighbourhood retract; see [4,17], as well as [5] for more recent related results.…”

“…Theorem 5. 4 Let E be a Banach space. For every compact Hausdorff space K and every norm one operator T : E → C(K ), there exists a unique lattice homomorphism T : C(B E * ) → C(K ) such that T •φ E = T and T 1 = 1, where 1 denotes the constant function 1.…”

“…Proposition 7. 4 Let E be a Banach space whose dual has finite cotype r ≥ 2, and suppose that 1 ≤ q < p < ∞ satisfy 1 q − 1 p ≥ 1 − 1 r . Then H p,q [E] = H ∞ [E] with equivalence of norms.…”

“…Despite this, the universal properties of these spaces are powerful enough to establish several results, which we do in Sect. 4. Section 5 is devoted to free AM-spaces and free C(K )-spaces.…”