Summary: Given a sequence of independent standard Gaussian variables \((Z_n)\), the classical Pisier algebra \(\mathcal{P}\) is the class of all continuous functions \(f\) on the unit circle \(\mathbb{T}\) such that for each \(t\in\mathbb{T}\), the random Fourier series \(\sum_{n\in\mathbb{Z}}{Z}_n\hat{f}(n)\times\exp(2\pi \mathrm{i}nt)\) converges in \(L^2\) and the corresponding sums constitute a Gaussian process that admits a continuous version. It was constructed by G. Pisier in [Isr. J. Math. 34, 38–44 (1979; Zbl 0428.46035)] to answer a long-standing question raised by Katznelson. In this paper, we consider the general random Fourier series \(\sum_{n\in\mathbb{Z}}{\xi}_n\hat{f}(n)\times\exp(2\pi\mathrm{i}nt)\) where \(\xi=(\xi_n)\) is a discrete Gaussian process of standard Gaussian random variables but with the restriction of independence relaxed and study the corresponding class \(\mathcal{P}(\xi)\) of continuous functions \(f\) on \(\mathbb{T}\). We obtain sufficient conditions (based on some spectral properties of the covariance matrix of \((\xi_n))\) for each of the relations \(\mathcal{P}\subset \mathcal{P} (\xi)\), \(\mathcal{P}(\xi)\subset \mathcal{P}\), and \(\mathcal{P}=\mathcal{P}(\xi)\). We illustrate these results by the classical fractional Gaussian noise. Whether in general \(\mathcal P(\xi)\) is also a Banach algebra is an open problem.