Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces. (English) Zbl 1027.46054
An \(s\)-tensor norm of order \(n\) is an assignment to each normed space \(E\) of a norm \(\alpha\) on the symmetric \(n\)-fold tensor product of \(E\), \(\bigotimes_{n,s}E\), such that \(\varepsilon_s\leq \alpha\leq \pi_s\) and for all continuous linear mappings \(T: E\to F\), \(\|\bigotimes_{n,s}T\|\leq \|T\|^n\) where \(\varepsilon_s\) and \(\pi_s\) are respectively the injective symmetric and projective symmetric norms on \(\bigotimes_{n,s}E\) and \((\bigotimes_{n,s} T)(\sum_i\lambda_i x_i\otimes\ldots\otimes x_i)=\sum_i\lambda_i T(x_i)\otimes\ldots\otimes T(x_i)\).
The authors introduce an ideal \(\mathcal Q\) of \(n\)-homogeneous polynomials as a subclass of \({\mathcal P}^n\), the class of all \(n\)-homogeneous polynomials, which is closed under addition and scalar multiplication, which is closed under composition on the right with linear operators and which contains all \(n\)-powers of linear forms. A normed ideal of \(n\)-homogeneous polynomials is a natural generalisation of the concept of operator ideal. A normed ideal \(({\mathcal Q}, \|\cdot\|_Q)\) is maximal if \(q\in {\mathcal P}(^nE)\) and \(\|q\|_{Q\max}:=\sup\{\|q|_M\|_Q :M\in \text{FIN}(E)\}<\infty\) implies that \(q\in {\mathcal Q}(^nE)\) and \(\|q\|_Q=\|q\|_{Q\max}\). A normed ideal \(({\mathcal Q}, \|\cdot\|_Q)\) is ultrastable if \(q_i\in {\mathcal Q}(^nE_i)\) and \(\sup_i \|q_i\|_Q<\infty\) implies that \(\lim_{\mathcal U}q_i\in {\mathcal Q}(^n(E_i)_ {\mathcal U})\) and \(\|\lim_{\mathcal U}q_i\|_Q\leq \sup\|q_i\|_Q\).
The authors show that for a normed ideal \(({\mathcal Q}, \|\cdot\|_Q)\) the following are equivalent: (i) \(({\mathcal Q}, \|\cdot\|_Q)\) is maximal; (ii) \(({\mathcal Q}, \|\cdot\|_Q)\) is ultrastable; (iii) there is a finitely generated \(s\)-tensor norm of order \(n\), \(\alpha\), such that \({\mathcal Q}(^nE)\) is isometrically isomorphic to \((\bigotimes_{n,s,\alpha} E)'\) for all \(E\). Using this result, it is shown that if \(({\mathcal Q}, \|\cdot\|_Q)\) is a maximal normed ideal of \(n\)-homogeneous polynomials, \(\mathcal U\) is a local ultrafilter on a Banach space \(E\) and \(q\in {\mathcal P}(^nE)\) then \(q\in {\mathcal Q}(^nE)\) if and only if the ultraiterated Aron-Berner extension, \(\overline{q}^{\mathcal U}\), of \(q\) belongs to \({\mathcal Q}(^nE'')\). When this occurs one also has that \(\|\overline{q}^{\mathcal U}\|_Q=\|q\|_Q\).
Given a finitely generated \(s\)-tensor norm of order \(n\), \(\alpha\), and a finitely generated tensor norm of order \(2\), \(\beta\), the authors say that \(q\in{\mathcal P}_{\alpha,\beta}(^nE;F)\) if \(\chi_F\circ q\) is in \((\bigotimes_ {n,s}E\otimes_\beta F)'\). (Here \(\chi_F\) is the natural embedding of \(F\) into \(F''\).) It is shown that the vector-valued ideal \({\mathcal P}_{\alpha,\beta}\) is ultrastable. From this it follows that if \(q\in {\mathcal P}(^nE;F)\) and \(\mathcal U\) is a local ultrafilter then \(q\in {\mathcal P}_{\alpha,\beta} (^nE,F)\) if and only if \(\overline{q}^{\mathcal U}\in {\mathcal P}_{\alpha,\beta}(^nE'',F'')\). Analogous results to some of those mentioned above are also obtained for ideals of \(n\)-linear continuous mappings.
The authors introduce an ideal \(\mathcal Q\) of \(n\)-homogeneous polynomials as a subclass of \({\mathcal P}^n\), the class of all \(n\)-homogeneous polynomials, which is closed under addition and scalar multiplication, which is closed under composition on the right with linear operators and which contains all \(n\)-powers of linear forms. A normed ideal of \(n\)-homogeneous polynomials is a natural generalisation of the concept of operator ideal. A normed ideal \(({\mathcal Q}, \|\cdot\|_Q)\) is maximal if \(q\in {\mathcal P}(^nE)\) and \(\|q\|_{Q\max}:=\sup\{\|q|_M\|_Q :M\in \text{FIN}(E)\}<\infty\) implies that \(q\in {\mathcal Q}(^nE)\) and \(\|q\|_Q=\|q\|_{Q\max}\). A normed ideal \(({\mathcal Q}, \|\cdot\|_Q)\) is ultrastable if \(q_i\in {\mathcal Q}(^nE_i)\) and \(\sup_i \|q_i\|_Q<\infty\) implies that \(\lim_{\mathcal U}q_i\in {\mathcal Q}(^n(E_i)_ {\mathcal U})\) and \(\|\lim_{\mathcal U}q_i\|_Q\leq \sup\|q_i\|_Q\).
The authors show that for a normed ideal \(({\mathcal Q}, \|\cdot\|_Q)\) the following are equivalent: (i) \(({\mathcal Q}, \|\cdot\|_Q)\) is maximal; (ii) \(({\mathcal Q}, \|\cdot\|_Q)\) is ultrastable; (iii) there is a finitely generated \(s\)-tensor norm of order \(n\), \(\alpha\), such that \({\mathcal Q}(^nE)\) is isometrically isomorphic to \((\bigotimes_{n,s,\alpha} E)'\) for all \(E\). Using this result, it is shown that if \(({\mathcal Q}, \|\cdot\|_Q)\) is a maximal normed ideal of \(n\)-homogeneous polynomials, \(\mathcal U\) is a local ultrafilter on a Banach space \(E\) and \(q\in {\mathcal P}(^nE)\) then \(q\in {\mathcal Q}(^nE)\) if and only if the ultraiterated Aron-Berner extension, \(\overline{q}^{\mathcal U}\), of \(q\) belongs to \({\mathcal Q}(^nE'')\). When this occurs one also has that \(\|\overline{q}^{\mathcal U}\|_Q=\|q\|_Q\).
Given a finitely generated \(s\)-tensor norm of order \(n\), \(\alpha\), and a finitely generated tensor norm of order \(2\), \(\beta\), the authors say that \(q\in{\mathcal P}_{\alpha,\beta}(^nE;F)\) if \(\chi_F\circ q\) is in \((\bigotimes_ {n,s}E\otimes_\beta F)'\). (Here \(\chi_F\) is the natural embedding of \(F\) into \(F''\).) It is shown that the vector-valued ideal \({\mathcal P}_{\alpha,\beta}\) is ultrastable. From this it follows that if \(q\in {\mathcal P}(^nE;F)\) and \(\mathcal U\) is a local ultrafilter then \(q\in {\mathcal P}_{\alpha,\beta} (^nE,F)\) if and only if \(\overline{q}^{\mathcal U}\in {\mathcal P}_{\alpha,\beta}(^nE'',F'')\). Analogous results to some of those mentioned above are also obtained for ideals of \(n\)-linear continuous mappings.
Reviewer: Christopher Boyd (Dublin)
MSC:
| 46G25 | (Spaces of) multilinear mappings, polynomials |
| 46B28 | Spaces of operators; tensor products; approximation properties |
| 46B08 | Ultraproduct techniques in Banach space theory |
Keywords:
tensor products; symmetric tensor products; ideals of polynomials; ideals of \(n\)-linear mappings; ultraproductsReferences:
| [1] | Raymundo Alencar, On reflexivity and basis for \?(^{\?}\?), Proc. Roy. Irish Acad. Sect. A 85 (1985), no. 2, 131 – 138. · Zbl 0594.46043 |
| [2] | Tsutomu Okada, On a representation of the projective connections of Finsler manifolds, Publ. Math. Debrecen 42 (1993), no. 1-2, 11 – 27. · Zbl 0795.53068 |
| [3] | Richard M. Aron and Paul D. Berner, A Hahn-Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), no. 1, 3 – 24 (English, with French summary). · Zbl 0378.46043 |
| [4] | Daniel Carando and Verónica Dimant, Duality in spaces of nuclear and integral polynomials, J. Math. Anal. Appl. 241 (2000), no. 1, 107 – 121. · Zbl 0993.46007 · doi:10.1006/jmaa.1999.6626 |
| [5] | Daniel Carando and Ignacio Zalduendo, A Hahn-Banach theorem for integral polynomials, Proc. Amer. Math. Soc. 127 (1999), no. 1, 241 – 250. · Zbl 0908.46031 |
| [6] | Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. · Zbl 0774.46018 |
| [7] | A. M. Davie and T. W. Gamelin, A theorem on polynomial-star approximation, Proc. Amer. Math. Soc. 106 (1989), no. 2, 351 – 356. · Zbl 0683.46037 |
| [8] | Seán Dineen and Richard M. Timoney, Complex geodesics on convex domains, Progress in functional analysis (Peñíscola, 1990) North-Holland Math. Stud., vol. 170, North-Holland, Amsterdam, 1992, pp. 333 – 365. · Zbl 0785.46044 · doi:10.1016/S0304-0208(08)70330-X |
| [9] | Klaus Floret, Natural norms on symmetric tensor products of normed spaces, Proceedings of the Second International Workshop on Functional Analysis (Trier, 1997), 1997, pp. 153 – 188 (1999). · Zbl 0961.46013 |
| [10] | K. Floret, The metric theory of symmetric tensor products of normed spaces, in preparation. · Zbl 0961.46013 |
| [11] | Stefan Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72 – 104. · Zbl 0412.46017 · doi:10.1515/crll.1980.313.72 |
| [12] | J. A. Jaramillo and L. A. Moraes, Duality and reflexivity in spaces of polynomials, Arch. Math. (Basel) 74 (2000), no. 4, 282 – 293. · Zbl 0959.46038 · doi:10.1007/s000130050444 |
| [13] | K.D. Kürsten, \(s\)-Zahlen und Ultraprodukte von Operatoren in Banachräumen, Doctoral Thesis, Leipzig, 1976. |
| [14] | Pádraig Kirwan and Raymond A. Ryan, Extendibility of homogeneous polynomials on Banach spaces, Proc. Amer. Math. Soc. 126 (1998), no. 4, 1023 – 1029. · Zbl 0890.46032 |
| [15] | Mikael Lindström and Raymond A. Ryan, Applications of ultraproducts to infinite-dimensional holomorphy, Math. Scand. 71 (1992), no. 2, 229 – 242. · Zbl 0845.46022 · doi:10.7146/math.scand.a-12424 |
| [16] | Hellmut Baumgärtel, Gerd Lassner, Albrecht Pietsch, and Armin Uhlmann , Proceedings of the second international conference on operator algebras, ideals, and their applications in theoretical physics, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 67, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1984. Held in Leipzig, September 25 – October 2, 1983. · Zbl 0535.00014 |
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