Uniqueness of unconditional basis of infinite direct sums of the 2-convexified Tsirelson space and its dual

Abstract The development of a substantial body of work on the subject of uniqueness of unconditional structure in Banach and p -Banach spaces sprang from the 1985 celebrated Memoir 12 by Bourgain et al. , where the authors aimed at classifying all Banach spaces with that property. One of the most striking results from that paper was that the 2-convexified Tsirelson space, T^ (2) T (2), had a unique unconditional basis (up to equivalence and permutation). Forty years later, many of the questions raised in the Memoir remain open but there has been a considerable effort in advancing a topic that had received relatively little attention until then. Continuing in the spirit of the program set in the Memoir, in this note we show that the direct sum of infinitely many copies of T^ (2) T (2) for 0 ℓ p (T (2) ), has a unique unconditional basis, and that the same property holds for ₏ ( (T^ (2) ) ^*) ℓ p ( (T (2) ) ∗). Our results and methods are relevant in applications since they permit us to reprove the uniqueness of the (discrete) lattice structure induced by an unconditional basis in other spaces.