A product space with the fixed point property

Abstract

We show that the l 1 sum of the van Dulst space with itself possesses the fixed point property, although it fails most of the known conditions that imply (FPP). Mathematics Subject Classification: Primary, 46B20; 47H09.

The fixed point property for renormings of ℓ2

“…This space is UNC and hence it satisfies the PS condition (see [52]). Notice that this space is isometric to Proof (See [16]) Indeed, ( 2 , • 2 ) is P-convex, and from Theorem 1.5 in [6] P-convexity is preserved under ∞ -direct sums. Then, 2 ⊕ 1 2 = ( 2 ⊕ 1 2 ) * is P-convex and hence O-convex.…”

“…Then, it seems even more unusual to fail to have each one of the above referred properties ANS, OC, PSz and WORTH. But very recently in [16] it has been shown that the space W := V D ⊕ 1 V D enjoys the FPP but it lacks these four geometrical sufficient conditions for FPP. In the same way, in [34] where m = max{a, b √ 2, 2c, d}.…”

“…5, we will give a renorming of 2 lacking of many of them, which still has the FPP. Most of the results presented here are taken from [33,34] and [16].…”