  In this paper, we propose a new class of positive definite kernels based on
the spectral truncation, which has been discussed in the fields of
noncommutative geometry and $C^*$-algebra. We focus on kernels whose inputs and
outputs are functions and generalize existing kernels, such as polynomial,
product, and separable kernels, by introducing a truncation parameter $n$ that
describes the noncommutativity of the products appearing in the kernels. When
$n$ goes to infinity, the proposed kernels tend to the existing commutative
kernels. If $n$ is finite, they exhibit different behavior, and the
noncommutativity induces interactions along the data function domain. We show
that the truncation parameter $n$ is a governing factor leading to performance
enhancement: by setting an appropriate $n$, we can balance the representation
power and the complexity of the representation space. The flexibility of the
proposed class of kernels allows us to go beyond previous commutative kernels.
