Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra
| Autor: | Choi, Yemon, Samei, Ebrahim, Stokke, Ross |
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| Rok vydání: | 2013 |
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| Zdroj: | Math. Scand. 117 (2015) no. 2, 258--303 |
| Druh dokumentu: | Working Paper |
| Popis: | 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 virtual diagonal in the usual sense. Our approach does not involve invariant means for $G$. Comment: v2: AMS-LaTeX, 11pt, 40 pages. Various minor improvements and corrections, including changes to notation and additional references; also new material in Sections 5 and 6. Incorporates referee's revisions. To appear in Mathematica Scandinavica |
| Databáze: | arXiv |
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