  In this article, we study a commutative Banach algebra structure on the space
$L^1(\mathbb{R}^{2n})\oplus \mathcal{T}^1$, where the $\mathcal{T}^1$ denotes
the trace class operators on $L^2(\mathbb{R}^{n})$. The product of this space
is given by the convolutions in quantum harmonic analysis. Towards this goal,
we study the closed ideals of this space, and in particular its Gelfand theory.
We additionally develop the concept of quantum Segal algebras as an analogue of
Segal algebras. We prove that many of the properties of Segal algebras have
transfers to quantum Segal algebras. However, it should be noted that in
contrast to Segal algebras, quantum Segal algebras are not ideals of the
ambient space. We also give examples of different constructions that yield
quantum Segal algebras.
