Vector measure duality and tensor product representations of -spaces of vector measures

Author:
E. A. Sánchez Pérez

Journal:
Proc. Amer. Math. Soc. **132** (2004), 3319-3326

MSC (2000):
Primary 46E30; Secondary 46G10

DOI:
https://doi.org/10.1090/S0002-9939-04-07521-5

Published electronically:
June 2, 2004

MathSciNet review:
2073308

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a countably additive vector measure. In this paper we use the definition of vector measure duality to establish a tensor product representation theorem for the space of -integrable functions with respect to . In particular, we identify this space with the dual of a certain space of operators under reasonable restrictions for the vector measure .

**1.**R. G. Bartle, N. Dunford and J. Schwartz,*Weak compactness and vector measures*, Canad. J. Math.**7**(1955), 289-305. MR**16:1123c****2.**Andreas Defant and Klaus Floret,*Tensor norms and operator ideals*, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. MR**1209438****3.**Guillermo P. Curbera,*Operators into 𝐿¹ of a vector measure and applications to Banach lattices*, Math. Ann.**293**(1992), no. 2, 317–330. MR**1166123**, https://doi.org/10.1007/BF01444717**4.**Guillermo P. Curbera,*When 𝐿¹ of a vector measure is an AL-space*, Pacific J. Math.**162**(1994), no. 2, 287–303. MR**1251903****5.**Guillermo P. Curbera,*Banach space properties of 𝐿¹ of a vector measure*, Proc. Amer. Math. Soc.**123**(1995), no. 12, 3797–3806. MR**1285984**, https://doi.org/10.1090/S0002-9939-1995-1285984-2**6.**J. Diestel and J. J. Uhl Jr.,*Vector measures*, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR**0453964****7.**L. M. García-Raffi, D. Ginestar, and E. A. Sánchez-Pérez,*Integration with respect to a vector measure and function approximation*, Abstr. Appl. Anal.**5**(2000), no. 4, 207–226. MR**1885467**, https://doi.org/10.1155/S1085337501000227**8.**D. R. Lewis,*Integration with respect to vector measures*, Pacific J. Math.**33**(1970), 157–165. MR**0259064****9.**D. R. Lewis,*On integrability and summability in vector spaces*, Illinois J. Math.**16**(1972), 294–307. MR**0291409****10.**Joram Lindenstrauss and Lior Tzafriri,*Classical Banach spaces. I*, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR**0500056****11.**Joram Lindenstrauss and Lior Tzafriri,*Classical Banach spaces. II*, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR**540367****12.**Susumu Okada,*The dual space of ℒ¹(𝜇) for a vector measure 𝜇*, J. Math. Anal. Appl.**177**(1993), no. 2, 583–599. MR**1231503**, https://doi.org/10.1006/jmaa.1993.1279**13.**S. Oltra, E. A. Sánchez Pérez and O. Valero,*Spaces of a positive vector measure and generalized Fourier coefficients*, Rocky Mountain Math. J., to appear.**14.**E. A. Sánchez Pérez,*Compactness arguments for spaces of 𝑝-integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces*, Illinois J. Math.**45**(2001), no. 3, 907–923. MR**1879243****15.**E. A. Sánchez Pérez,*Spaces of integrable functions with respect to vector measures of convex range and factorization of operators from -spaces*, Pacific J. Math.**207**(2002), 489-495.**16.**E. A. Sánchez Pérez,*Vector measure orthonormal functions and best approximation for the 4-norm*, Arch. Math. (Basel)**80**(2003), no. 2, 177–190. MR**1979033**, https://doi.org/10.1007/s00013-003-0450-8

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Additional Information

**E. A. Sánchez Pérez**

Affiliation:
Departamento de Matemática Aplicada, E.T.S. Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Valencia, Camino de Vera, 46071 Valencia, Spain

Email:
easancpe@mat.upv.es

DOI:
https://doi.org/10.1090/S0002-9939-04-07521-5

Keywords:
Vector measures,
$p$-integrable functions,
tensor products

Received by editor(s):
October 23, 2002

Received by editor(s) in revised form:
August 21, 2003

Published electronically:
June 2, 2004

Dedicated:
The author dedicates this paper to the memory of Professor Klaus Floret.

Communicated by:
N. Tomczak-Jaegermann

Article copyright:
© Copyright 2004
American Mathematical Society