There is a large literature on harmonic analysis on locally compact group, that
I am just beginning to discover. However I have not seen so far anything that emphasizes the *central* functions on $G$. A good reference on this subject could thus be very useful,
as I'd like to understand generalizations to as large as possible a class of locally compact groups of the following elementary results on finite groups $G$:

*Any central function* $f: G \rightarrow \mathbb C$ *can be written in a unique way* $$(1) \ \ \ \ \ f = \sum c_\pi \ \ tr\ \pi,$$ *where* $\pi$ *runs in the sum on the set of equivalence classes of complex irreducible representations of* $G$. *Moreover we have nice formulae
relating various norms of* $f$ *and the* $c_\pi$, *such as* $$(2)\ \ \ \ \ ||f||^2 = \sum |c_\pi|^2 ,$$
*and* $$(3)\ \ \ \ \ \ sup_{||h||=1, h \in L^2(G)} ||f \ast h|| = sup_{\pi} |c_\pi|,$$ *where* $||f||=\sqrt{\int |f(g)|^2 \ dg}$, $dg$ *being the Haar probability measure*, *and* $f\ast h$ *is the usual convolution product*.

Let me explain what kind of generalization I am looking for, focussing first on (1). First, how to define the $c_\pi$ for a given $f$? For $G$ a finite group, $$ (4) \ \ \ c_\pi = \int f(g) \overline{tr \ \pi(g)} dg = \overline{ tr\ \pi(\bar f)},$$ where $\pi(f) := \int f(g) \pi(g) \ dg$ as usual. Thanks to the answer and comments to my earlier question, one can generalize this definition to the following situation: $G$ is second countable, unimodular, type I (hence has a Plancherel measure on its dual $\hat{G}$), and $f$ satisfies some regularity condition (edit: specifically, I'd like to take $f$ in the Eymard's Fourier algebra $A(G)$, i.e. the convolution of two functions in $L^2(G)$). Then $\pi(f)$ is trace-class for almost every $\pi$ for the Plancherel measure, and one can define $c_\pi(f)$, almost, everywhere on the support of the Plancherel measure, that is on the reduced dual $\hat{G}_r$) by the same formula (4) as in the case of finite group. But then I cannot answer these questions, that seem natural to me:

If $f$ is moreover central, is it true that the $c_\pi(f)$ for $\pi$ in the reduced dual determine $f$? If so, is there a formula analogue to (1)? If not, what if we assume in addition that the group $G$ is amenable, so that the dual and the reduced dual are the same? What about (2) and (3)?

PS: I somehow feel that as an ex-PhD-student of an expert in the trace formula, I should know the answer to all this inside-out. Fortunately, my ex-advisor does not read MO :-)

andconstant on conjugacy classes? $\endgroup$8more comments