Let K be one of the fields R or C, A a topological associative algebra over K with separately continuous multiplication, B a subalgebra of A and M a closed maximal regular left (right or two-sided) ideal of A such that the trace M \cap B of M is a proper subset of B. In cases, when B is a Gelfand-Mazur subalgebra of A or a subalgebra of the centre Z(A) of A, such classes of topological algebras in which M \cap B (in the subset topology) is a closed maximal regular left (respectively, right or two-sided) ideal of B, are described.
ABEL, MART and ABEL, MATI (2007) "Traces of Maximal Ideals of Topological Algebras in Their Subalgebras," Turkish Journal of Mathematics: Vol. 31: No. 1, Article 6. Available at: https://journals.tubitak.gov.tr/math/vol31/iss1/6