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

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Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra. / Choi, Yemon; Samei, Ebrahim; Stokke, Ross.

In: Mathematica Scandinavica, Vol. 117, No. 2, 14.12.2015, p. 258-303.

Research output: Contribution to journalJournal articlepeer-review

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Choi, Y, Samei, E & Stokke, R 2015, 'Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra', Mathematica Scandinavica, vol. 117, no. 2, pp. 258-303. <http://www.mscand.dk/article/view/22870>

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Choi Y, Samei E, Stokke R. Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra. Mathematica Scandinavica. 2015 Dec 14;117(2):258-303.

Author

Choi, Yemon ; Samei, Ebrahim ; Stokke, Ross. / Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra. In: Mathematica Scandinavica. 2015 ; Vol. 117, No. 2. pp. 258-303.

Bibtex

@article{47e8a6ba622f4db8a7931c8d97fdbbe3,
title = "Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra",
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. 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$G$.",
author = "Yemon Choi and Ebrahim Samei and Ross Stokke",
year = "2015",
month = dec,
day = "14",
language = "English",
volume = "117",
pages = "258--303",
journal = "Mathematica Scandinavica",
issn = "0025-5521",
publisher = "Mathematica Scandinavica",
number = "2",

}

RIS

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 -