10
Nov

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Title: Nonexpansive Mappings in Fixed-Point Theory and Entropy

Abstract: TheoryFixed-point theory studies the structures on a space X that provide every sufficiently nice self-mapping f : C → C, on a sufficiently nice subset C ⊂ X, with a fixed point. Here we present two counterexamples. The positive face of the unit sphere of C(K)* fails the fixed-point-property with a contractive map, if K is an infinite, compact Hausdorff space. We also show that if the continuum hypothesis is assumed, then the unit ball of C(N*) fails the fixed-point-property for nonexpansive maps, where N*=βN \N, the Stone-˘Cech remainder space. We then consider an ℓ1-extension of the classical Shannon entropy functional for finite, discrete probability spaces, and we present an L1([0, 1])-analogue. In each case, the set of elements with finite entropy can be equipped with a natural topological vector space structure, and we show that entropy can be useful in characterizing the closed subspaces of ℓ1 and L1([0, 1]) that fail the fixed-point-property.

Advisor: Dr. Christopher Lennard

Friday, November 10 at 2:30 p.m. to 4:00 p.m.

Thackeray Hall, 427

139 University Place, Pittsburgh, 15260

Title: Nonexpansive Mappings in Fixed-Point Theory and Entropy

Abstract: TheoryFixed-point theory studies the structures on a space X that provide every sufficiently nice self-mapping f : C → C, on a sufficiently nice subset C ⊂ X, with a fixed point. Here we present two counterexamples. The positive face of the unit sphere of C(K)* fails the fixed-point-property with a contractive map, if K is an infinite, compact Hausdorff space. We also show that if the continuum hypothesis is assumed, then the unit ball of C(N*) fails the fixed-point-property for nonexpansive maps, where N*=βN \N, the Stone-˘Cech remainder space. We then consider an ℓ1-extension of the classical Shannon entropy functional for finite, discrete probability spaces, and we present an L1([0, 1])-analogue. In each case, the set of elements with finite entropy can be equipped with a natural topological vector space structure, and we show that entropy can be useful in characterizing the closed subspaces of ℓ1 and L1([0, 1]) that fail the fixed-point-property.

Advisor: Dr. Christopher Lennard

Friday, November 10 at 2:30 p.m. to 4:00 p.m.

Thackeray Hall, 427

139 University Place, Pittsburgh, 15260

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- Department of Mathematics
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