functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...

, the Calkin algebra, named after John Williams Calkin, is the

quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

of ''B''(''H''), the

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

bounded linear operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vect ...

s on a separable infinite-dimensional Hilbert space ''H'', by the

ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...

''K''(''H'') of

compact operator In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact c ...

s. Here the addition in ''B''(''H'') is addition of operators and the multiplication in ''B''(''H'') is composition of operators; it is easy to verify that these operations make ''B''(''H'') into a ring. When scalar multiplication is also included, ''B''(''H'') becomes in fact an algebra over the same field over which ''H'' is a Hilbert space.

Properties

* Since ''K''(''H'') is a maximal norm-closed ideal in ''B''(''H''), the Calkin algebra is

simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

. In fact, ''K''(''H'') is the only closed ideal in ''B''(''H''). * As a quotient of a C*-algebra by a two-sided ideal, the Calkin algebra is a C*-algebra itself and there is a

short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...

0 \to K(H) \to B(H) \to B(H)/K(H) \to 0

:which induces a six-term cyclic exact sequence in

K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...

. Those operators in ''B''(''H'') which are mapped to an invertible element of the Calkin algebra are called

Fredholm operator In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : '' ...

s, and their index can be described both using K-theory and directly. One can conclude, for instance, that the collection of unitary operators in the Calkin algebra consists of 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 not isomorphic to an algebra of operators on a separable Hilbert space. The Gelfand-Naimark-Segal construction implies that the Calkin algebra is isomorphic to an algebra of operators on a nonseparable Hilbert space, but while for many other C*-algebras there are explicit descriptions of such Hilbert spaces, the Calkin algebra does not have an explicit representation. * The existence of an outer automorphism of the Calkin algebra is shown to be independent of ZFC, by work of Phillips and Weaver, and Farah.

Generalizations

* One can define a Calkin algebra for any infinite-dimensional complex Hilbert space, not just separable ones. * An analogous construction can be made by replacing ''H'' with a Banach space, which is also called a Calkin algebra.{{cite journal, last=Appell, first=Jürgen, title=Measures of noncompactness, condensing operators and fixed points: An application-oriented survey , journal=Fixed Point Theory, volume=6, issue=2, pages=157–229, year=2005 * The Calkin algebra is the

Corona algebra In mathematics, the multiplier algebra, denoted by ''M''(''A''), of a C*-algebra ''A'' is a unital C*-algebra that is the largest unital C*-algebra that contains ''A'' as an ideal in a "non-degenerate" way. It is the noncommutative generalization ...

of the algebra of compact operators on a Hilbert space.

References

Operator theory C*-algebras K-theory