The \(T\)-universal Korovkin closures of algebra subspaces \(H\subset A\), where \(T:A\to B\) is a spectral linear contraction between commutative Banach algebras with identity, or a positive operator between commutative involutive symmetric Banach algebras with bounded approximate identities are described by means of intersections of uniqueness sets with respect to certain linear forms. The \(T\)-universal Korovkin closures relative to operators \(T\) of type \(T=\sum_{i=1}^ p\lambda_ iT_ i\), where \(T_ i\) are algebraic homomorphisms (or algebraic *-homomorphisms in the case of involutive algebras) and \(\lambda_ i\geq 0\), are investigated in particular. There are characterized \(T\)-universal Korovkin spaces, i.e. the spaces whose \(T\)-universal Korovkin closure coincides with the initial algebra \(A\). Several previous results by the author, related with convergence sets for linear positive forms of discrete type, are extended as well.