Isomorphic vector-valued Banach-Stone theorems for subspaces
Given a Banach space X, we define the number Ao(X) = inf d(X2,P(2)), where the infimum is taken over all two-dimensional subspaces X2 of X. Here, d(M, N) means the Banach-Mazur distance between Banach spaces M,N defined by d(M, N) = inf{||T||||T_1|| : T: M N is an isomorphism}. We establish some fac...
Elmentve itt :
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| Dokumentumtípus: | Cikk |
| Megjelent: |
Bolyai Institute, University of Szeged
Szeged
2015
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| Sorozat: | Acta scientiarum mathematicarum
81 No. 1-2 |
| Kulcsszavak: | Matematika |
| Tárgyszavak: | |
| doi: | 10.14232/actasm-014-255-x |
| Online Access: | http://acta.bibl.u-szeged.hu/35202 |
| Tartalmi kivonat: | Given a Banach space X, we define the number Ao(X) = inf d(X2,P(2)), where the infimum is taken over all two-dimensional subspaces X2 of X. Here, d(M, N) means the Banach-Mazur distance between Banach spaces M,N defined by d(M, N) = inf{||T||||T_1|| : T: M N is an isomorphism}. We establish some facts about Ao and then consider applications to Banach-Stone type theorems for isomorphisms on continuous, vector-valued function spaces. If Q,K are locally compact Hausdorff spaces, and X,Y are Banach spaces for which both Ao(X*) and AQ(Y*) are greater than one, it has been shown that if T is an isomorphism from CQ(Q. E) onto Co(K, Y) with ||T||||T~1 1| sufficiently small, then Q and K are homeomorphic, a generalization of the Banach-Stone Theorem for isometries. We examine such results for subspaces of these spaces. A closed subspace M of C'oiQ, X) is said to be a Co(Q)-module if it is closed under multiplication by functions in Co(<5). If M and N are Co(Q), Gb(X)-modules, respectively, then with assumptions similar to those mentioned above, we are able to obtain results in which the homeomorphism is between the strong boundaries of N and M. In this case, the strong boundaries are the subsets of K and Q, respectively, upon which the functions in N and M have nonzero values. We also obtain a new theorem concerning isometries. |
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| Terjedelem/Fizikai jellemzők: | 189-214 |
| ISSN: | 0001-6969 |