Then, from the perspective of operator theory, a circulant matrix is the kernel of a discrete integral transform, namely the convolution operator for the function (c_0,c_1,\dots,c_{n-1}); this is a discrete circular convolution.
The C^*-algebra of all circulant matrices with complex entries is isomorphic to the group C^*-algebra of \mathbf{Z}/n\mathbf{Z}.