Bochner's theorem for measures of positive type Is there a version of Bochner's theorem characterizing measures of positive type on a locally compact group? By a measure of positive type on the group $\Gamma$, I mean a measure $\mu$ satisfying $\int f^* * f~d\mu \geq 0$ for all $f \in C_c(\Gamma)$. If so, what is a reference?
I am mainly interested in the case when $\Gamma$ is a compact group.
 A: On $\Gamma = \mathbf R^n$, there is a Bochner theorem for not only measures but distributions of positive type, in Schwartz, Théorie des distributions, Chap. VII, §9:

Théorème XVIII (Bochner) Pour qu'une distribution $T$ soit $>>0$, il faut et il suffit qu'elle soit transformée de Fourier d'une mesure $\mu\geqslant0$ à croissance lente.

For general locally compact groups, there is a theory of distributions by Bruhat, but I don't know if anyone proved a Bochner theorem in that context.
If you only want measures, there is a Bochner theorem on locally compact abelian groups, by Argabright and Gil de Lamadrid:

THEOREM 4.1. Every positive definite measure on $\Gamma$ is transformable. A measure $\mu\in\mathfrak M_T(\Gamma)$ is positive definite if and only if $\hat\mu\geqslant0$.

(Here they call a mesure $\mu$ on $\Gamma$  transformable, $\mu\in\mathfrak M_T(\Gamma)$, if there is a measure $\hat\mu$ on the Pontryagin dual group $G$ such that $\mu(f^**f) = \hat\mu(|\check f|^2)$ for all compactly supported continuous functions on $\Gamma$.)
Finally if you insist on non-abelian $\Gamma$, Argabright and Gil de Lamadrid point to a generalization by Godement (1957)(p.101) where it is called "Théorème de Plancherel". (I haven't read it.) 
A: This result seems to be attributed by George Mackey to Gelfand-Raikov 1943, see
http://books.google.com/books?id=TvHNj3U8nroC&pg=PA81&lpg=PA81&dq=bochner+theorem+locally+compact+noncommutative+topological+group&source=bl&ots=OVmCAAspep&sig=HkV7vbEd_m6kORl2s7SWD05XaxY&hl=en&sa=X&ei=2tLWU77vDZSvyASt8YLADg&ved=0CDwQ6AEwBA#v=onepage&q=bochner%20theorem%20locally%20compact%20noncommutative%20topological%20group&f=false
I can't seem to find an electronic version of Gelfand-Raikov.
