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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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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 -