Ideals and structure of operator algebras (Q2891153)

An operator algebra is a norm closed algebra of operators on a Hilbert space. The paper under review continues the program of the authors of studying the structure of operator algebras and operator spaces using one-sided ideals of special types. In particular, \(r\)-ideals are right ideals of an operator algebra possessing a left contractive approximate identity. Swapping ``left'' and ``right'' we get the definition of \(l\)-ideal.NEWLINENEWLINE After recording many general results about one-sided ideals in Section 2, authors dedicate Section 3 to the existence of non-trivial \(r\)-ideals. Section 4 introduces the concept of a ``matricial'' algebra, that is, having a full set of matrix units whose span is dense in the algebra. In particular, there is a Wedderburn type structure theorems for operator algebras, using \(1\)-matricial algebras as the building blocks. Section 5 continues the development and presents some new characterizations of \(C^*\)-algebras consisting of compact operators. The final section considers the ideal structure of the Haagerup tensor product of operator algebras.