Preduals and complementation of spaces of bounded linear operators

For Banach spaces [Formula: see text] and [Formula: see text], we establish a natural bijection between preduals of [Formula: see text] and preduals of [Formula: see text] that respect the right [Formula: see text]-module structure. If [Formula: see text] is reflexive, it follows that there is a unique predual making [Formula: see text] into a dual Banach algebra. This removes the condition that [Formula: see text] have the approximation property in a result of Daws. We further establish a natural bijection between projections that complement [Formula: see text] in its bidual and [Formula: see text]-linear projections that complement [Formula: see text] in its bidual. It follows that [Formula: see text] is complemented in its bidual if and only if [Formula: see text] is (either as a module or as a Banach space). Our results are new even in the well-studied case of isometric preduals.

A generalization on character Connes-amenability

In the current paper, we introduce the concepts of left ?-approximate Connes-amenability and left character approximate Connes-amenability of a dual Banach algebra A that ? is a ?*-continuous homomorphism fromAto C. We also characterize left ?-approximate Connes-amenability ofAin terms of certain derivations and study some hereditary properties for such Banach algebras. Some examples show that these new notions are different from approximate Connes-amenability and left character Connesamenability for dual Banach algebras.

Extension of Derivations, and Connes- Amenability of the Enveloping Dual Banach Algebra

If $D:A \to X$ is a derivation from a Banach algebra to a contractive, Banach $A$-bimodule, then one can equip $X^{**}$ with an $A^{**}$-bimodule structure, such that the second transpose $D^{**}: A^{**} \to X^{**}$ is again a derivation. We prove an analogous extension result, where $A^{**}$ is replaced by $\mathsf{F}(A)$, the enveloping dual Banach algebra of $A$, and $X^{**}$ by an appropriate kind of universal, enveloping, normal dual bimodule of $X$. Using this, we obtain some new characterizations of Connes-amenability of $\mathsf{F}(A)$. In particular we show that $\mathsf{F}(A)$ is Connes-amenable if and only if $A$ admits a so-called $\operatorname{WAP}$-virtual diagonal. We show that when $A=L^1(G)$, existence of a $\operatorname{WAP}$-virtual diagonal is equivalent to the existence of a virtual diagonal in the usual sense. Our approach does not involve invariant means for $G$.

On bounded dual-valued derivations on certain Banach algebras

We consider the class D(U) of bounded derivations Ud?U*defined on a Banach algebra U with values in its dual space U*so that ?x,d(x)? = 0 for all x?U U. The existence of such derivations is shown, but lacking the simplest structure of an inner one. We characterize the elements of D(U) if span(U2) is dense in U or if U is a unitary dual Banach algebra.

Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule

Let $\mathcal A$ be a dual Banach algebra with predual $\mathcal A_*$ and consider the following assertions: (A) $\mathcal A$ is Connes-amenable; (B) $\mathcal A$ has a normal, virtual diagonal; (C) $\mathcal A_*$ is an injective $\mathcal A$-bimodule. For general $\mathcal A$, all that is known is that (B) implies (A) whereas, for von Neumann algebras, (A), (B), and (C) are equivalent. We show that (C) always implies (B) whereas the converse is false for $\mathcal A = M(G)$ where $G$ is an infinite, locally compact group. Furthermore, we present partial solutions towards a characterization of (A) and (B) for $\mathcal A = B(G)$ in terms of $G$: For amenable, discrete $G$ as well as for certain compact $G$, they are equivalent to $G$ having an abelian subgroup of finite index. The question of whether or not (A) and (B) are always equivalent remains open. However, we introduce a modified definition of a normal, virtual diagonal and, using this modified definition, characterize the Connes-amenable, dual Banach algebras through the existence of an appropriate notion of virtual diagonal.