In functional analysis, the Calkin algebra is the quotient of B(H), the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K(H) of compact operators. For functional analysis as used in psychology see the Functional analysis (psychology article In Linear algebra, the quotient of a Vector space V by a subspace N is a vector space obtained by "collapsing" N In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces In Mathematics a Topological space is called separable if it contains a countable dense subset that is there exists a sequence \{ x_n This article assumes some familiarity with Analytic geometry and the concept of a limit. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In Functional analysis, Compact operators on Hilbert spaces are a direct extension of matrices in the Hilbert spaces they are precisely the closure of Finite
Since the compact operators is a (in fact, the only) maximal norm-closed ideal in B(H), the Calkin algebra is simple. In Mathematics, specifically in Ring theory, an algebra is simple if it contains no non-trivial two-sided ideals and the set {
As a quotient of two C* algebras, the Calkin algebra is a C* algebra itself. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. There is a short exact sequence
which induces an exact sequence in K-theory. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Mathematics, K-theory is a tool used in several disciplines Those operators in B(H) which are mapped to an invertible element of the Calkin algebra are called Fredholm operators, and their index can be described both using K-theory and directly. In Mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of Integral equations It is named in honour of Erik Ivar Fredholm One can conclude, for instance, that the collection of unitary operators in the Calkin algebra are homotopy classes indexed by the integers Z. This is in contrast to B(H), where the unitary operators are path connected.
As a C* algebra, the Calkin algebra is remarkable because it is not isomorphic to an algebra of operators on a separable Hilbert space; instead, a larger Hilbert space has to be chosen (the GNS theorem says that every C* algebra is isomorphic to an algebra of operators on a Hilbert space; for many other simple C* algebras, there are explicit descriptions of such Hilbert spaces, but for the Calkin algebra, this is not the case). In Functional analysis, given a C*-algebra A, the Gelfand-Naimark-Segal construction establishes a correspondence between cyclic *-representations of
The same name is now used for the analogous construction for a Banach space. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis
References
- Calkin, J. W. (1941). Two-sided ideals and congruences in the ring of bounded operators in Hilbert space. Annals of Mathematics, 42, 839-873. The Annals of Mathematics (ISSN 0003-486X abbreviated as Ann of Math
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