  The Generalized Locally Toeplitz (GLT) sequences of matrices have been
originated from the study of certain partial differential equations. To be more
precise, such matrix sequences arise when we numerically approximate some
partial differential equations by discretization. The study of the asymptotic
spectral behaviour of GLT sequence is very important in analysing the solution
of corresponding partial differential equations. The approximating classes of
sequences (a.c.s) and the spectral symbols are important notions in this
connection. Recently, G. Barbarino obtained some additional results regarding
the theoretical aspects of such notions. He obtained the completeness of the
space of matrix sequences with respect to pseudo metric a.c.s. Also, he
identified the space of GLT sequences with the space of measurable functions.
In this article, we follow the same research line and obtain various results
connecting the sub-algebras of matrix sequence spaces and sub-algebras of
function spaces. In some cases, these are identifications as Banach spaces and
some of them are Banach algebra identifications. In the process, we also prove
that the convergence notions in the sense of eigenvalue/singular value
clustering are equivalent to the convergence with respect to the metrics
introduced here. These convergence notions are related to the study of
preconditioners in the case of matrix/operator sequences. Finally, as an
application of our main results, we establish a Korovkin-type result in the
setting of GLT sequences.
