Noncommutative gauge theory with the symmetry $U(n\otimes m)*$ and standard-like model with fractional charges

$U(n\otimes m)\ast$ gauge field theory on noncommutative spacetime is formulated and the standard-like model with the symmetry ${\text{U}(3_c\otimes 2\otimes 1_{\text{\scriptsize$Y$}})\ast}$ is reconstructed based on it. $\text{U}(n+m)\ast$ gauge group reduces to $\text{U}(n+m)$ on the commutative spacetime which is not ${\text{U}}(N), (N=n+m)$ but isomorphic to $\text{SU}(n)\times\text{SU}(m)\times \text{U}(1)$ in this article. On the noncommutative spacetime, the representation that fields belong to is fundamental, adjoint or bi-fundamental. For this reason, one had to construct the standard model by use of bi-fundamental representations. However, we can reconstruct the standard-like model with only fundamental and adjoint representation and without using bi-fundamental representations. It is well known that the charge of fermion is 0 or $\pm1$ in the U(1) gauge theory on noncommutative spacetime. Thus, there may be no room to incorporate the noncommutative U(1) gauge theory into the standard model because the quarks have fractional charges. However, it is shown in this article that there is the noncommutative gauge theory with arbitrary charges which symmetry is for example $\text{U}(3\otimes 1)\ast$. This type of gauge theory emerges from the spontaneous breakdown of the noncommutative U(4)$\ast$ gauge theory in which the gauge field contains the 0 component $A_\mu^0(x,\theta)$. The standard-like model in this paper also has fermion fields with fractional charges. Thus, the noncommutative gauge theory with fractional U(1) charges can not exist alone, but it must coexist with noncommutative nonabelian gauge theory.