Let $G$ be a compact abelian group (finite dimensional, but not finite) and $A$ be a $C^*$-algebra. Consider an action $\alpha: G\to Aut(A)$. In analogy with the case of finite abelian group, I believe that it is true that $A=\oplus_{\chi\in \hat{G}} A_\chi$, with $A_\chi$ being the subspace made of elements $x\in A$ such that $\alpha_g(x)=\chi(g)x$. Moreover, I think that there exist a family of linear maps (indexed by the characters of the group) $E_\chi: A\to A_\chi$ $$ E_\chi(x)=\int_G \overline{\chi(g)}\alpha_g(x) $$ Can someone tell me if it is true and give me reference to a book or paper? Thanks for the help.
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1$\begingroup$ I don't have a copy on hand, but surely this is covered in Pedersen, C*-Algebras and Their Automorphism Groups. $\endgroup$– Nik WeaverNov 15, 2016 at 23:34
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1$\begingroup$ I have worked out the details of this for the case of the circle group in "Circle actions on C*-algebras, partial automorphisms and a generalized Pimsner-Voiculescu exact sequence", J. Funct. Analysis, 122 (1994), 361-401. The computations easily generalize to compact abelian groups. Moreover I have worked a bit on this problem in the case of actions of non-compact abelian groups in "Morita-Rieffel equivalence and spectral theory for integrable automorphism groups of C*-algebras", J. Funct. Analysis, 172 (2000), 404-465. $\endgroup$– RuyNov 15, 2016 at 23:46
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1$\begingroup$ You need to put $\overline{\chi}$ instead of $\chi$ in the integral (so that one gets the identity on $A_{\chi}$ !). $\endgroup$– Simon HenryNov 16, 2016 at 12:51
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$\begingroup$ @ Ruy: Thank you much for the references. The papers contain the answer to the particular case of the circle and I appreciate it. Thanks again $\endgroup$– John N.Nov 16, 2016 at 21:05
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1$\begingroup$ @John, this is just a small point regarding the workings of Mathoverflow. If you mention someone by name in a comment, preceeding the person's name by the character "@", that person is alerted about it. But this only works if there is no space between the "@" and the name, which you inadvertently inserted twice above. $\endgroup$– RuyNov 19, 2016 at 13:05
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The facts that $A$ is a $C^*$-algebra and that $G$ is abelian are irrelevant: such a decomposition holds more generally for any Banach space $A$ with a continuous linear action of a compact group. The reference I know for this is Representations of a compact group on a Banach space by Shiga, available here. There are probably earlier references for abelian groups, but I do not know them.
answered Nov 15, 2016 at 23:35
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$\begingroup$ @JohnN. : to be more precise, I am not sure whether the formula for the projection $E_\chi$ appears explicitely in Shiga's paper. Anyway, I needed the result and I reproved it in Theorem 2.5 in arxiv.org/abs/1307.2475 $\endgroup$ Nov 16, 2016 at 21:10