The Dineen problem for homogeneous orthogonally-additive polynomials. (English) Zbl 1310.06014
Let \(A,B\) be archimedean vector lattices. A subset \(D\) of \(A\times A\times\cdots\times A\) is called order bounded if there exist \((a_1,a_2,\ldots,a_n)\) and \((b_1,b_2,\ldots,b_n)\) in \(A\times A\times\cdots\times A\) with \((a_1,a_2,\ldots,a_n)\leq (x_1,x_2,\ldots,x_n)\leq (b_1,b_2,\ldots,b_n)\) for all \((x_1,x_2,\ldots,x_n)\in A\times A\times\cdots\times A\). An order bounded map \(P\colon A\to B\) is called a homogeneous polynomial of degree \(n\) if \(P(x)=\Psi(x_1,x_2,\ldots,x_n)\) where \(\Psi\) is an order bounded \(n\)-inear map from \(A\times A\times\cdots\times A\) into \(B\). A homogeneous polynomial of degree \(n\), \(P\colon A\to B\) is said to be orthogonally additive if \(P(x+y)=P(x)+P(y)\) for all \(x,y\in A\) with \(|x|\wedge |y|=0\).
Let \(B,C\) be vector spaces and \(A\) be a subspace of \(C\). The author studies whether a homogeneous polynomial \(P\colon A\to B\) can be extended to \(\overline P\colon C\to B\). He shows that the answer is affirmative if \(A,B\) are archimedean vector lattices, \(A^\delta\) is Dedekind completion of \(A\) and \((A')'_n\) is the order bidual of \(A\).
Let \(B,C\) be vector spaces and \(A\) be a subspace of \(C\). The author studies whether a homogeneous polynomial \(P\colon A\to B\) can be extended to \(\overline P\colon C\to B\). He shows that the answer is affirmative if \(A,B\) are archimedean vector lattices, \(A^\delta\) is Dedekind completion of \(A\) and \((A')'_n\) is the order bidual of \(A\).
Reviewer: Şafak Alpay (Ankara)
MSC:
| 06F25 | Ordered rings, algebras, modules |
| 46G25 | (Spaces of) multilinear mappings, polynomials |
| 47B65 | Positive linear operators and order-bounded operators |
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