Counterexamples in isometric theory of symmetric and greedy bases
We continue the study initiated in [F. Albiac and P. Wojtaszczyk, Characterization of $1$-greedy bases, J. Approx. Theory 138 (2006), no. 1, 65-86] of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to $1$. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be $1$-suppression unconditional, thus settling Problem 4.4 from [F. Albiac and J.L. Ansorena, Characterization of $1$-almost greedy bases, Rev. Mat. Complut. 30 (2017), no. 1, 13-24]. In particular, our construction demonstrates that bases with Property (A) need not be $1$-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional.