For a completely contractive Banach algebra $B$, we find conditions under which the completely bounded multiplier algebra ${ℳ}_{cb}\left(B\right)$ is a dual Banach algebra and the operator amenability of $B$ is equivalent to the operator Connes-amenability of ${ℳ}_{cb}\left(B\right)$. We also show that, in this case, these are equivalent to the existence of a normal virtual operator diagonal.

We investigate a notion of relative operator entropy, which develops the theory started by J. I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341-348]. For two finite sequences A = (A₁,...,Aₙ) and B = (B₁,...,Bₙ) of positive operators acting on a Hilbert space, a real number q and an operator monotone function f we extend the concept of entropy by setting $S_q^f\left(A\right|B):={\sum }_{j=1}^n{A}_j^{1/2}{\left(A_j^{-1/2}{B}_j{A}_j^{-1/2}\right)}^{q}f\left(A_j^{-1/2}{B}_j{A}_j^{-1/2}\right)A_j^{1/2}$, and then give upper and lower bounds for $S_q^f\left(A\right|B)$ as an extension of an inequality due to T. Furuta [Linear Algebra Appl. 381 (2004), 219-235] under...

Let G be a locally compact group. We use the canonical operator space structure on the spaces $L^p\left(G\right)$ for p ∈ [1,∞] introduced by G. Pisier to define operator space analogues $O{A}_p\left(G\right)$ of the classical Figà-Talamanca-Herz algebras $A_p\left(G\right)$. If p ∈ (1,∞) is arbitrary, then $A_p\left(G\right)\subset O{A}_p\left(G\right)$ and the inclusion is a contraction; if p = 2, then OA₂(G) ≅ A(G) as Banach spaces, but not necessarily as operator spaces. We show that $O{A}_p\left(G\right)$ is a completely contractive Banach algebra for each p ∈ (1,∞), and that $O{A}_q\left(G\right)\subset O{A}_p\left(G\right)$ completely contractively for amenable...

Given a vector measure m with values in a Banach space X, a desirable property (when available) of the associated Banach function space L¹(m) of all m-integrable functions is that L¹(m) = L¹(|m|), where |m| is the [0,∞]-valued variation measure of m. Closely connected to m is its X-valued integration map Iₘ: f ↦ ∫f dm for f ∈ L¹(m). Many traditional operators from analysis arise as integration maps in this way. A detailed study is made of the connection between the property L¹(m) = L¹(|m|) and the...

We show that the Moore-Penrose inverse of an operator T is idempotent if and only if it is a product of two projections. Furthermore, if P and Q are two projections, we find a relation between the entries of the associated operator matrix of PQ and the entries of associated operator matrix of the Moore-Penrose inverse of PQ in a certain orthogonal decomposition of Hilbert C*-modules.

Let G be a locally compact group, A(G) its Fourier algebra and L¹(G) the space of Haar integrable functions on G. We study the Segal algebra S¹A(G) = A(G) ∩ L¹(G) in A(G). It admits an operator space structure which makes it a completely contractive Banach algebra. We compute the dual space of S¹A(G). We use it to show that the restriction operator ${u\mapsto u|}_H:S¹A\left(G\right)\to A\left(H\right)$, for some non-open closed subgroups H, is a surjective complete quotient map. We also show that if N is a non-compact closed subgroup, then the averaging...

In questo lavoro, motivati dalla teoria di Fredholm in spazi di Banach e dalla cosiddetta teoria degli ideali di operatori nel senso di Pietsch, viene definito un nuovo concetto di semigruppo di operatori. Questa nuova definizione include quella di molte classi di operatori già studiate in letteratura, come la classe degli operatori di semi-Fredholm, quella degli operatori tauberiani ed altre ancora. Inoltre permette un nuovo ed unificante approccio ad una serie di problemi in teoria degli operatori...

We generalize an important class of Banach spaces, the M-embedded Banach spaces, to the non-commutative setting of operator spaces. The one-sided M-embedded operator spaces are the operator spaces which are one-sided M-ideals in their second dual. We show that several properties from the classical setting, like the stability under taking subspaces and quotients, unique extension property, Radon-Nikodým property and many more, are retained in the non-commutative setting. We also discuss the dual...