Isometries of certain operator spaces

Approximation properties of certain operator-induced norms on Hilbert spaces

We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the L2 norm, and study their approximation properties over Hilbert subspaces of L2. The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in a reproducing kernel Hilbert space (RKHS). Our results have implications to the analysi...

Representation of certain homogeneous Hilbertian operator spaces and applications

Following Grothendieck’s characterization of Hilbert spaces we consider operator spaces F such that both F and F ∗ completely embed into the dual of a C*-algebra. Due to Haagerup/Musat’s improved version of Pisier/Shlyakhtenko’s Grothendieck inequality for operator spaces, these spaces are quotients of subspaces of the direct sum C ⊕ R of the column and row spaces (the corresponding class being...

On Isometries of Euclidean Spaces

Completely Bounded Maps into Certain Hilbertian Operator Spaces

We prove a factorization of completely bounded maps from a C *-algebra A (or an exact operator space E ⊂ A) to ℓ 2 equipped with the operator space structure of (C, R) θ (0 < θ < 1) obtained by complex interpolation between the column and row Hilbert spaces. More precisely, if F denotes ℓ 2 equipped with the operator space structure of (C, R) θ , then u : A → F is completely bounded iff there a...

Metric Characterizations of Isometries and of Unital Operator Spaces and Systems

We give some new characterizations, of unitaries, isometries, unital operator spaces, unital function spaces, operator systems, C-algebras, and related objects. Several characterizations of these objects are already known; see e.g. [15, Theorem 9.5.16], [1], the discussion on p. 316 of [4], [14], and [13]. One difference between our paper and these cited references, is that our results only use...