[Return to main list of publications]## Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra

#### Authors

Yemon Choi^{*}, Ebrahim Samei, Ross Stokke

* *Corresponding author*

#### Metadata

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
$G$.

#### Updates/comments

Yemon Choi