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 investigate the relationships between strongly extreme, complex extreme, and complex locally uniformly rotund points of the unit ball of a symmetric function space or a symmetric sequence space E, and of the unit ball of the space E(ℳ,τ) of τ-measurable operators associated to a semifinite von Neumann algebra (ℳ,τ) or of the unit ball in the unitary matrix space $C_E$. We prove that strongly extreme, complex extreme, and complex locally uniformly rotund points x of the unit ball of the symmetric...

The Banach operator ideal of (q,2)-summing operators plays a fundamental role within the theory of s-number and eigenvalue distribution of Riesz operators in Banach spaces. A key result in this context is a composition formula for such operators due to H. König, J. R. Retherford and N. Tomczak-Jaegermann. Based on abstract interpolation theory, we prove a variant of this result for (E,2)-summing operators, E a symmetric Banach sequence space.

The notion of a compressible operator on a Banach space, E, derives from automatic continuity arguments. It is related to the notion of a cartesian Banach space. The compressible operators on E form an ideal in ℬ(E) and the automatic continuity proofs depend on showing that this ideal is large. In particular, it is shown that each weakly compact operator on the James' space, J, is compressible, whence it follows that all homomorphisms from ℬ(J) are continuous.

We characterize some properties of a vector measure in terms of its associated Kluvánek conical measure. These characterizations are used to prove that the range of a vector measure determines these properties. So we give new proofs of the fact that the range determines the total variation, the σ-finiteness of the variation and the Bochner derivability, and we show that it also determines the (p,q)-summing and p-nuclear norm of the integration operator. Finally, we show that Pettis derivability...

We introduce and study the notions of w*-approximate Connes amenability and pseudo-Connes amenability for dual Banach algebras. We prove that the dual Banach sequence algebra ℓ¹ is not w*-approximately Connes amenable. We show that in general the concepts of pseudo-Connes amenability and Connes amenability are distinct. Moreover the relations between these new notions are also discussed.

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...

We study the presence of copies of $l_p^n$’s uniformly in the spaces ${\Pi }_2(C[0,1],X)$ and ${\Pi }_1(C[0,1],X)$. By using Dvoretzky’s theorem we deduce that if $X$ is an infinite-dimensional Banach space, then ${\Pi }_2(C[0,1],X)$ contains $\lambda \sqrt{2}$-uniformly copies of $l_{\infty }^n$’s and ${\Pi }_1(C[0,1],X)$ contains $\lambda$-uniformly copies of $l_2^n$’s for all $\lambda >1$. As an application, we show that if $X$ is an infinite-dimensional Banach space then the spaces ${\Pi }_2(C[0,1],X)$ and ${\Pi }_1(C[0,1],X)$ are distinct, extending the well-known result that the spaces ${\Pi }_2(C[0,1],X)$ and $𝒩\left(C\right[0,1],X)$ are distinct.

The main facts about unbounded C*-seminorms on partial *-algebras are reviewed and the link with the representation theory is discussed. In particular, starting from the more familiar case of *-algebras, we examine C*-seminorms that are defined from suitable families of positive linear or sesquilinear forms, mimicking the construction of the Gelfand seminorm for Banach *-algebras. The admissibility of these forms in terms of the (unbounded) C*-seminorms they generate is characterized.

A map φ: Mₘ(ℂ) → Mₙ(ℂ) is decomposable if it is of the form φ = φ₁ + φ₂ where φ₁ is a CP map while φ₂ is a co-CP map. It is known that if m = n = 2 then every positive map is decomposable. Given an extremal unital positive map φ: M₂(ℂ) → M₂(ℂ) we construct concrete maps (not necessarily unital) φ₁ and φ₂ which give a decomposition of φ. We also show that in most cases this decomposition is unique.