  Let $X=\mathbb R^2$ and let $q\in\mathbb C$, $|q|=1$. For $x=(x^1,x^2)$ and
$y=(y^1,y^2)$ from $X^2$, we define a function $Q(x,y)$ to be equal to $q$ if
$x^1<y^1$, to $\bar q$ if $x^1>y^1$, and to $\Re q$ if $x^1=y^1$. Let
$\partial_x^+$, $\partial_x^-$ ($x\in X$) be operator-valued distributions such
that $\partial_x^+$ is the adjoint of $\partial_x^-$. We say that
$\partial_x^+$, $\partial_x^-$ satisfy the anyon commutation relations (ACR) if
$\partial^+_x\partial_y^+=Q(y,x)\partial_y^+\partial_x^+$ for $x\ne y$ and
$\partial^-_x\partial_y^+=\delta(x-y)+Q(x,y)\partial_y^+\partial^-_x$ for
$(x,y)\in X^2$. In particular, for $q=1$, the ACR become the canonical
commutation relations and for $q=-1$, the ACR become the canonical
anticommutation relations. We define the ACR algebra as the algebra generated
by operator-valued integrals of $\partial_x^+$, $\partial_x^-$. We construct a
class of gauge-invariant quasi-free states on the ACR algebra. Each state from
this class is completely determined by a positive self-adjoint operator $T$ on
the real space $L^2(X,dx)$ which commutes with any operator of multiplication
by a bounded function $\psi(x^1)$. In the case $\Re q<0$, the operator $T$
additionally satisfies $0\le T\le -1/\Re q$. Further, for $T=\kappa^2\mathbf 1$
($\kappa>0$), we discuss the corresponding particle density
$\rho(x):=\partial_x^+\partial_x^-$. For $\Re q\in(0,1]$, using a
renormalization, we rigorously define a vacuum state on the commutative algebra
generated by operator-valued integrals of $\rho(x)$. This state is given by a
negative binomial point process. A scaling limit of these states as
$\kappa\to\infty$ gives the gamma random measure, depending on parameter $\Re
q$.
