TY - JOUR

T1 - Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra

AU - Choi, Yemon

AU - Samei, Ebrahim

AU - Stokke, Ross

PY - 2015/12/14

Y1 - 2015/12/14

N2 - 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 $\F(A)$, 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 virtualdiagonal in the usual sense. Our approach does not involve invariant means for$G$.

AB - 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 $\F(A)$, 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 virtualdiagonal in the usual sense. Our approach does not involve invariant means for$G$.

M3 - Journal article

VL - 117

SP - 258

EP - 303

JO - Mathematica Scandinavica

JF - Mathematica Scandinavica

SN - 0025-5521

IS - 2

ER -