The notion of weights on (topological) *-algebras is defined and studied. The primary purpose is to define the notions of admissibility and approximate admissibility of weights, and to investigate when a weight is admissible or approximately admissible. The results obtained are applied to vector weights and tracial weight on unbounded operator algebras, as well as to weights on smooth subalgebras of a C *-algebra.

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Let A be a Banach algebra. Then frequently each maximal left ideal in A is closed, but there are easy examples that show that a maximal left ideal can be dense and of codimension 1 in A. It has been conjectured that these are the only two possibilities: each maximal left ideal in a Banach algebra A is either closed or of codimension 1 (or both). We shall show that this is the case for many Banach algebras that satisfy some extra condition, but we shall also show that the conjecture is not always true by constructing, for each n∈N, examples of Banach algebras that have a dense maximal left ideal of codimension n. In particular, we shall exhibit a semi‐simple Banach algebra with this property. We shall show that the questions concerning maximal left ideals in a Banach algebra A that we are considering are related to automatic continuity questions: When are A‐module homomorphisms from A into simple Banach left A‐modules automatically continuous?