Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces

Authors:
Klaus Floret and Stephan Hunfeld

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1425-1435

MSC (2000):
Primary 46B08; Secondary 46B28, 46G25

Published electronically:
December 27, 2001

MathSciNet review:
1879966

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Using the theory of full and symmetric tensor norms on normed spaces, a theorem of Kürsten and Heinrich on ultrastability and maximality of normed operator ideals is extended to ideals of -homogeneous polynomials and -linear mappings--scalar-valued and vector-valued. The motivation for these results is the following important special case: the ``uniterated'' Aron-Berner extension : of an -homogeneous polynomial to the bidual remains in certain ideals under preservation of the norm. Moreover, Lotz's characterization of maximal normed ideals of linear mappings through appropriate tensor norms is proved for ideals of -homogeneous scalar-valued polynomials and ideals of -linear mappings.

**[A1]**Raymundo Alencar,*On reflexivity and basis for 𝑃(^{𝑚}𝐸)*, Proc. Roy. Irish Acad. Sect. A**85**(1985), no. 2, 131–138. MR**845536****[A2]**Tsutomu Okada,*On a representation of the projective connections of Finsler manifolds*, Publ. Math. Debrecen**42**(1993), no. 1-2, 11–27. MR**1208849****[AB]**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). MR**508947****[CD]**Daniel Carando and Verónica Dimant,*Duality in spaces of nuclear and integral polynomials*, J. Math. Anal. Appl.**241**(2000), no. 1, 107–121. MR**1738337**, 10.1006/jmaa.1999.6626**[CZ]**Daniel Carando and Ignacio Zalduendo,*A Hahn-Banach theorem for integral polynomials*, Proc. Amer. Math. Soc.**127**(1999), no. 1, 241–250. MR**1458865**, 10.1090/S0002-9939-99-04485-8**[DF]**Andreas Defant and Klaus Floret,*Tensor norms and operator ideals*, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. MR**1209438****[DG]**A. M. Davie and T. W. Gamelin,*A theorem on polynomial-star approximation*, Proc. Amer. Math. Soc.**106**(1989), no. 2, 351–356. MR**947313**, 10.1090/S0002-9939-1989-0947313-8**[DT]**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. MR**1150757**, 10.1016/S0304-0208(08)70330-X**[F1]**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). MR**1749787****[F2]**K. Floret,*The metric theory of symmetric tensor products of normed spaces*, in preparation.**[H]**Stefan Heinrich,*Ultraproducts in Banach space theory*, J. Reine Angew. Math.**313**(1980), 72–104. MR**552464**, 10.1515/crll.1980.313.72**[JM]**J. A. Jaramillo and L. A. Moraes,*Duality and reflexivity in spaces of polynomials*, Arch. Math. (Basel)**74**(2000), no. 4, 282–293. MR**1742640**, 10.1007/s000130050444**[K]**K.D. Kürsten,*-Zahlen und Ultraprodukte von Operatoren in Banachräumen*, Doctoral Thesis, Leipzig, 1976.**[KR]**Pádraig Kirwan and Raymond A. Ryan,*Extendibility of homogeneous polynomials on Banach spaces*, Proc. Amer. Math. Soc.**126**(1998), no. 4, 1023–1029. MR**1415346**, 10.1090/S0002-9939-98-04009-X**[LR]**Mikael Lindström and Raymond A. Ryan,*Applications of ultraproducts to infinite-dimensional holomorphy*, Math. Scand.**71**(1992), no. 2, 229–242. MR**1212706****[P]**Hellmut Baumgärtel, Gerd Lassner, Albrecht Pietsch, and Armin Uhlmann (eds.),*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. MR**763516**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46B08,
46B28,
46G25

Retrieve articles in all journals with MSC (2000): 46B08, 46B28, 46G25

Additional Information

**Klaus Floret**

Affiliation:
Department of Mathematics, University of Oldenburg, D-26111 Oldenburg, Germany

Email:
floret@mathematik.uni-oldenburg.de

**Stephan Hunfeld**

Affiliation:
Werstener Dorfstrasse 209, D-40591 Düsseldorf, Germany

DOI:
https://doi.org/10.1090/S0002-9939-01-06228-1

Keywords:
Tensor products,
symmetric tensor products,
ideals of polynomials,
ideals of $n$-linear mappings,
ultraproducts

Received by editor(s):
February 9, 1999

Received by editor(s) in revised form:
November 20, 2000

Published electronically:
December 27, 2001

Communicated by:
Dale Alspach

Article copyright:
© Copyright 2001
American Mathematical Society