C*-algebra in the context of Functional analysis

The asterisk (/ˈæstərɪsk/ *), from Late Latin asteriscus, from Ancient Greek ἀστερίσκος, asteriskos, "little star", is a typographical symbol.

An asterisk is usually five- or six-pointed in print and six- or eight-pointed when handwritten, though more complex forms exist. Its most common use is to call out a footnote. It is also often used to censor words considered offensive. Computer scientists and mathematicians often vocalize it as star (as, for example, in the A* search algorithm or C*-algebra).

In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.

Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces.