The transformation (covariance) algebra $C_c(G\times X)$

Assume AC (The Axiom of Choice). Let $G$ be a locally compact Hausdorff topological group (Topological group: multiplication and inversion are continuous, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space) acting continuously on a locally compact Hausdorff space $X$ (Left group actions, transitive actions, and faithful actions, Continuity of a map of topological spaces at a point and globally, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space), with a fixed left Haar measure $dg$ (Left Haar integral and left Haar measure) and modular function ${\mathrm{\Delta }}_G$ (Modular function of a locally compact group). The transformation algebra of the action is the complex vector space $C_c(G\times X)$ of continuous complex functions with compact support (Compact support, $C_c(X)$, and $C_0(X)$), equipped with the twisted convolution

$(f_1\ast f_2)(g,x)={\int }_G{f}_1(h,x)f_2(h^{-1}g,h^{-1}x)dh$

and the involution

$f^{\ast }(g,x)={\mathrm{\Delta }}_G(g{)}^{-1}\overline{f(g^{-1},g^{-1}x)}\mathrm{.}$

Well-definedness: support and continuity. Let $K_i\subseteq G$, $L_i\subseteq X$ be compact sets with $\mathrm{supp}{f}_i\subseteq K_i\times L_i$ ($i=1,2$). If $f_2(h^{-1}g,h^{-1}x)\mathrm{\ne }0$ then $h\in g{K}_2^{-1}$ and $h^{-1}x\in L_2$, while $f_1(h,x)\mathrm{\ne }0$ forces $h\in K_1$; hence the integrand of $(f_1\ast f_2)(g,x)$ is supported in the compact set $K_1\cap g{K}_2^{-1}$, which is nonempty only for $g\in K_1{K}_2$, and the integral is a finite number by finiteness of Haar measure on compacta. For $g$ in a compact neighbourhood $V$ of a fixed $g_0$ the $h$-support lies in the fixed compact set $K=K_1\cap V{K}_2^{-1}$, and $x$ in a compact neighbourhood of a fixed $x_0$; the map $(h,g,x)\mapsto f_1(h,x)f_2(h^{-1}g,h^{-1}x)$ is continuous on $G\times G\times X$ as a composition of the continuous group operations and the continuous action (Topological group: multiplication and inversion are continuous, Left group actions, transitive actions, and faithful actions), so it is uniformly continuous on the compact set $K\times \bar{V}\times L$. Given $\epsilon >0$ there is a neighbourhood $U$ of $(g_0,x_0)$ with $\mathrm{\mid }{f}_1(h,x)f_2(h^{-1}g,h^{-1}x)-f_1(h,x_0)f_2(h^{-1}{g}_0,h^{-1}{x}_0)\mathrm{\mid }\le \epsilon$ for all $h\in K$ and $(g,x)\in U$; both integrands vanish off $K$, so $\mathrm{\mid }(f_1\ast f_2)(g,x)-(f_1\ast f_2)(g_0,x_0)\mathrm{\mid }\le \epsilon \mathrm{\mid }K\mathrm{\mid }$, where $\mathrm{\mid }K\mathrm{\mid }$ is the finite Haar measure of $K$. This proves continuity of $f_1\ast f_2$; its support is contained in $(K_1{K}_2)\times L_1$, a compact set, so $f_1\ast f_2\in C_c(G\times X)$. The product is bilinear in $(f_1,f_2)$ by linearity of the Haar integral.

Well-definedness: associativity. Fix $(g,x)$ and put $F(h,r)=f_1(h,x)f_2(h^{-1}r,h^{-1}x)f_3(r^{-1}g,r^{-1}x)$; it is continuous and compactly supported in $(h,r)$, with support in the compact set $K_1\times (K_1{K}_2\cap g{K}_3^{-1})$ by the support computation above. Writing each convolution as its defining integral, the left-hand side of $(f_1\ast f_2)\ast f_3=f_1\ast (f_2\ast f_3)$ at $(g,x)$ is the iterated integral ${\int }_G{\int }_{G}F(h,r)dhdr$, while the right-hand side is ${\int }_G{\int }_{G}F(h,hk)dkdh$; by Compactly supported kernels admit commuting radon integrals applied to the continuous compactly supported kernel $F$ the order of the first iterated integral may be interchanged, and the inner substitution $r=hk$, which preserves the left Haar integral by Left Haar integral and left Haar measure and changes $h^{-1}r\mapsto k$, $r^{-1}g\mapsto k^{-1}{h}^{-1}g$, $r^{-1}x\mapsto k^{-1}{h}^{-1}x$, identifies them. Hence $\ast$ is associative.

Well-definedness: involution. The function $f^{\ast }$ is continuous, since $(g,x)\mapsto (g^{-1},g^{-1}x)$ and ${\mathrm{\Delta }}_G$ are continuous (Modular function of a locally compact group, The modular function is a continuous homomorphism), and its support is the image of the compact set $\mathrm{supp}f$ under that homeomorphism, hence compact; so $f^{\ast }\in C_c(G\times X)$. Applying $\ast$ twice and using that ${\mathrm{\Delta }}_G$ is a continuous homomorphism into the positive reals, so that ${\mathrm{\Delta }}_G(g^{-1})={\mathrm{\Delta }}_G(g{)}^{-1}$, gives $(f^{\ast }{)}^{\ast }(g,x)={\mathrm{\Delta }}_G(g{)}^{-1}\overline{{\mathrm{\Delta }}_G(g^{-1}{)}^{-1}\stackrel{‾}{f(g,x)}}=f(g,x),$ that is, $(f^{\ast }{)}^{\ast }=f$. In the same way, for the products one computes $(f_1\ast f_2{)}^{\ast }(g,x)={\mathrm{\Delta }}_G(g{)}^{-1}{\int }_G\overline{f_1(h,g^{-1}x)}\overline{f_2(h^{-1}{g}^{-1},h^{-1}{g}^{-1}x)}dh,$ and substituting $h=g\mathrm{\ell }$ in the defining integral of $(f_2^{\ast }\ast f_1^{\ast })(g,x)$ turns its modular factor into ${\mathrm{\Delta }}_G(g{)}^{-1}$ because ${\mathrm{\Delta }}_G(g\mathrm{\ell }{)}^{-1}{\mathrm{\Delta }}_G({\mathrm{\ell }}^{-1}{)}^{-1}={\mathrm{\Delta }}_G(g{)}^{-1}$, so $(f_1\ast f_2{)}^{\ast }=f_2^{\ast }\ast f_1^{\ast }$. Together with the conjugate-linearity of $\ast$ this says that $C_c(G\times X)$ with $\ast$ and ${}^{\ast }$ is a complex associative algebra with involution; the involution of the group convolution is the special case of the published definition (Compactly supported convolution on a group).

Trivial action. If $gx=x$ for all $g,x$, then the twisted product becomes $(f_1\ast f_2)(g,x)={\int }_G{f}_1(h,x)f_2(h^{-1}g,x)dh$, which is convolution in the group variable with pointwise multiplication in the base variable, with the factor order of Compactly supported convolution on a group.

AC is inherited through Compactly supported kernels admit commuting radon integrals, which commutes the two Radon integrals in the associativity computation; no independent choice step is used, and the support, continuity and involution computations themselves make no choice.