Geometry of the unit ball and representation theory for operator algebras
The author of this paper introduces the concept of the set \(\text{cp}(E)\) of contractive perturbations of a non-empty subset \(E\) of the unit ball \(A_1\) in a complex Banach space \(A\), and goes on to show that for a point \(a\) in \(A_1\) of norm one, the real linear span of \(a-\text{face}(a)\) coincides with the real linear span of \(\text{cp}(\{a\})\). The element \(a\) is said to be geometrically compact if \(\text{cp}(\text{cp}(\{a\}))\) is norm-compact. The author shows that if the unit ball \(A_1\) in the complex Banach space \(A\) has a non-trivial compact face \(F\), then for any element \(a\) on \(F\), the linear span of \(a- F\), which is independent of \(a\) and denoted by \(S(F)\), contains a geometrically compact element. In the case in which \(A\) is a \(C^*\)-algebra, an element \(a\) is geometrically compact if and only if there is a faithful representation \(\pi\) of \(A\) such that \(\pi(a)\) is a compact operator. The author goes on to study the situation in which \(A\) is a not necessarily selfadjoint operator algebra. He shows that \(A_1\) has a non-trivial compact face \(F\) if and only if \(S(F)\) possesses a non-scalar element \(a\), the spectrum of which has at most one limit point, and that if \(F\) is finite-dimensional, then \(S(F)\) is a finite-dimensional hereditary subalgebra of \(A\) such that, for any \(a\) in \(F\), \[ F=(a+ S(F))\cap A_1. \] The author goes on to apply his interesting results to a class of operator algebras defined to be operator simple which generalise the ranges of the reduced atomic representations of \(C^*\)-algebras, and gives a series of stimulating examples and applications.
scientific article; zbMATH DE number 2138445