  $U(n\otimes m)\ast$ 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.
