Let X be a closed subspace of B(H) for some Hilbert space H. In [9], Pisier introduced Sp [X] (1 ≤ p ≤ +∞) by setting Sp [X] = (S∞ [X] , S1 [X])θ , (where θ =1/p , S∞ [X] = S∞ ⊗min X and S1 [X] = S1 ⊗∧ X) and showed that there are p−matricially normed spaces. In this paper we prove that conversely, if X is a p−matricially normed space with p = 1, then there is an operator structure on X, such that M1,n (X) = S1 [X] where Sn,1 [X] is the finite dimentional version of S1 [X]. For p...

In this paper, we introduce and study the notion of completely bounded ${\Lambda }_p$ sets (${\Lambda }_p^{cb}$ for short) for compact, non-abelian groups G. We characterize ${\Lambda }_p^{cb}$ sets in terms of completely bounded $L^p\left(G\right)$ multipliers. We prove that when G is an infinite product of special unitary groups of arbitrarily large dimension, there are sets consisting of representations of unbounded degree that are ${\Lambda }_p$ sets for all p < ∞, but are not ${\Lambda }_p^{cb}$ for any p ≥ 4. This is done by showing that the space of completely bounded $L^p\left(G\right)$ multipliers...

We study two related questions. (1) For a compact group G, what are the ranges of the convolution maps on A(G × G) given for u,v in A(G) by u × v ↦ u*v̌ (v̌(s) = v(s^-1)) and u × v ↦ u*v? (2) For a locally compact group G and a compact subgroup K, what are the amenability properties of the Fourier algebra of the coset space A(G/K)? The algebra A(G/K) was defined and studied by the first named author. In answering the first question, we obtain, for compact groups which do not...