Functional Continuity of Unital B0-algebras with Orthogonal Bases
نویسنده
چکیده
A topological algebra is a complex associative algebra which is also a Hausdorff topological vector space such that the multiplication is separately continuous. A locally convex algebra is a topological algebra whose topology is determined by a family of seminorms. A complete metrizable locally convex algebra is called a B0-algebra. The topology of a B0-algebra A may be given by a countable family (‖.‖i)i≥1 of seminorms such that ‖x‖i ≤ ‖x‖i+1 and ‖xy‖i ≤ ‖x‖i+1‖y‖i+1 for all i ≥ 1 and x,y ∈ A. A multiplicative linear functional on a complex algebra A is an algebra homomorphism from A to the complex field. Let A be a topological algebra. M∗(A) denotes the set of all nonzero multiplicative linear functionals on A. M(A) denotes the set of all nonzero continuous multiplicative linear functionals on A. A seminorm p on A is lower semicontinuous if the set {x ∈ A : p(x)≤ 1} is closed in A. Let A be a topological algebra. A sequence (en)n≥1 in A is a basis if for each x ∈ A there is a unique sequence (αn)n≥1 of complex numbers such that x =
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