We shall introduce the definition of a k-mode Gaussian state and a chain
of such states which determine a C* probability space. We present examples of
such states exhibiting properties like exchangeability and stationarity. Stationary
chains are determined by block Toeplitz matrices. Using the Kac-Murdoch-Szego
theorems on asymptotic spectral distributions of Toeplitz matrices we compute the
entropy rates of some of these chains. This leaves many natural problems open.