Let $G$ be a locally compact group.
Let $\mu$ be a positive definite complex measure ([D, p295]): we have $\mu(f*f^*)\geq 0$ for any compact support continuons function $f \in C_c(G)$.
In [D, p 297], the author constructs an associated unitary representation of $G$ from scratch.
I would like construct this representation with another method which seems to me more natural: First to obtain a representation of the involutive algebra $C_c(G)$ by using the GNS construction ([T, p 134]). Then I must check the operators $\pi(f)$ are bounded. After this task, I would like find an extension of $\pi$ to a non-degenerate representation of the involutive algebra $L^1(G)$ (and deduce a unitary representation of $G$ by [D, p 284]).
1) Is it possible?
My problems are the are bounded and find an extension.
The inequality $$ \int_G f^**g^**g*f d\mu \leq K_g\int_G f^**f d\mu,\qquad f,g \in C_c(G) $$ seems to be the key in order to show that the operators of the GNS representation are bounded. But I do not see how I can prove it.
2) Is this inequality true or false?
3) What is the relationship between *-representations of $C_c(G)$ and *-representations of $L^1(G)$?
[D] Dixmier-C*-algebras
[T] Thill-Introduction to Normed *-Algebras and their Representations. http://arxiv.org/pdf/1011.1558.pdf