## Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra

#### Authors

Yemon Choi*, Ebrahim Samei, Ross Stokke

* Corresponding author

MSC 2010: 46H20 (primary), 43A20 43A60 46H25 (secondary).

#### Status

Appeared as Math. Scand. 117 (2015) no. 2, 258–303

Preprint version available at arXiv 1307.6285 (final accepted version, incorporating referee's recommendations)

#### Reviews

[ Math Review | Zentralblatt (summary) ]

#### Abstract

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. $\newcommand{\F}{{\sf F}}$ We prove an analogous extension result, where $A^{**}$ is replaced by $\F(A)$, the \emph{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 $\F(A)$. In particular we show that $\F(A)$ is Connes-amenable if and only if $A$ admits a so-called WAP-virtual diagonal. We show that when $A=L^1(G)$, existence of a WAP-virtual diagonal is equivalent to the existence of a virtual diagonal in the usual sense. Our approach does not involve invariant means for $G$.