arXiv subjects
Publications and source records attributed to Javier Parcet.
Sharp Fourier type and cotype of Lebesgue spaces and Schatten classes with respect to an arbitrary compact semisimple Lie group are investigated. In the process, a local variant of the Hausdorff-Young inequality on such groups is given.
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 the 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.
The notion of B-convexity for operator spaces, which a priori depends on a set of parameters indexed by $Σ$, is defined. Some of the classical characterizations of this geometric notion for Banach spaces are studied in this new context. For instance, an operator space is $B_Σ$-convex if and only if it has $Σ$-subtype. The class of uniformly non-$L^1(Σ)$ operator spaces, which is also the class of $B_Σ$-convex operator spaces, is introduced. Moreover, an operator space having non-trivial $Σ$-type is $B_Σ$-convex. However, the converse is false. The row and column operator spaces are nice counterexamples of this fact, since both are Hilbertian. In particular, this result shows that a version of the Maurey-Pisier theorem does not hold in our context. Some other examples of Hilbertian operator spaces will be treated. In the last part of this paper, the independence of $B_Σ$-convexity with respect to $Σ$ is studied. This provides some interesting problems which will be posed.