Quantized orthonormal systems: A non-commutative Kwapień theorem

J. García-Cuerva¹ ; J. Parcet¹

¹ Departamento de Matemáticas, C-XV Universidad Autónoma de Madrid 28049 Madrid, Spain

Studia Mathematica, Tome 155 (2003) no. 3, pp. 273-294

Résumé

The concepts of Riesz type and cotype of a given Banach space are extended to a non-commutative setting. First, the Banach space is replaced by an operator space. The notion of quantized orthonormal system, which plays the role of an orthonormal system in the classical setting, is then defined. The Fourier type and cotype of an operator space with respect to a non-commutative compact group fit in this context. Also, the quantized analogs of Rademacher and Gaussian systems are treated. All this is used to obtain an operator space version of the classical theorem of Kwapień characterizing Hilbert spaces by means of vector-valued orthogonal series. Several approaches to this result with different consequences are given.

Détail

DOI : 10.4064/sm155-3-6

Mots-clés : concepts riesz type cotype given banach space extended non commutative setting first banach space replaced operator space notion quantized orthonormal system which plays role orthonormal system classical setting defined fourier type cotype operator space respect non commutative compact group fit context quantized analogs rademacher gaussian systems treated obtain operator space version classical theorem kwapie characterizing hilbert spaces means vector valued orthogonal series several approaches result different consequences given