Operator Algebras and Noncommutative Geometry

An operator algebra in functional analysis is an algebra of continuous linear operators on a topological vector space, with multiplication determined by the composition of mappings.

The results of the study of operator algebras are expressed in algebraic terms, with highly analytic methodologies. Although operator algebras are typically thought of as a subdivision of functional analysis, they have direct applications in representation theory, differential geometry, quantum statistical physics, quantum information, and quantum field theory.

Operator algebras can be used to simultaneously investigate arbitrary sets of operators with little algebraic relation. Operator algebras can be thought of as a generalisation of the spectral theory of a single operator from this perspective. Operator algebras are non-commutative rings in general.

Within the algebra of continuous linear operators, an operator algebra is usually required to be closed in a specific operator topology. It’s a group of operators that have both algebraic and topological closure features. Such qualities are axiomized in various disciplines, and algebras with a specific topological structure are studied.

Noncommutative geometry

Noncommutative geometry (NCG) is a branch of mathematics concerned with the building of spaces that are locally presented by noncommutative algebras of functions and with a geometric approach to noncommutative algebras (possibly in some generalised sense). A noncommutative algebra is an associative algebra in which the multiplication is not commutative, i.e., xy does not always equal yx; or, more broadly, an algebraic structure in which one of the principal binary operations is not commutative; it also allows additional structures, such as topology or norm, to be carried by the noncommutative algebra of functions.

Gelfand Theorem

The Gelfand–Naimark theorem says that a C*-subalgebra of limited operators on a Hilbert space is isometrically *-isomorphic to an arbitrary C*-algebra A. Israel Gelfand and Mark Naimark proved this result in 1943, and it was a watershed moment in the development of C*-algebra theory because it showed the possibility of treating a C*-algebra as an abstract algebraic entity without reference to particular realisations as an operator algebra.

Orbifold theorem

An orbifold (for “orbit-manifold”) is a generalisation of a manifold in the mathematical fields of topology and geometry. An orbifold is a topological space that is locally a finite group quotient of Euclidean space, roughly speaking.

It is the theory linked to vertex algebra’s fixed point subalgebra under the action of a finite group of automorphisms in two-dimensional conformal field theory.

A manifold’s quotient space under the suitably discontinuous action of a potentially infinite group of diffeomorphisms with finite isotropy subgroups is the most common example of underlying space.

This holds true for every action of a finite group; for example, a manifold with a boundary has a natural orbifold structure since it is the quotient of its double by the action of Z₂. 

Orbifold fundamental group

The orbifold fundamental group can be defined in a variety of ways. More advanced methods employ orbifold covering spaces or groupoid classification spaces. The most basic technique (used by Haefliger and also known to Thurston) extends the traditional description of the fundamental group’s loop concept.

A path in the underlying space that has an explicit piecewise lift of path segments to orbifold charts and explicit group elements indicating paths in overlapping charts is called an orbifold path; if the underlying path is a loop, it is called an orbifold loop. If two orbifold paths in orbifold charts are related through multiplication by group elements, they are recognised. The orbifold basic group is made up of orbifold loop homotopy classes.

Conclusion

An operator algebra is a Hilbert space algebra of continuous linear operators. These algebras can be linked to a wide range of mathematical and mathematical physics topics. The results of the study of operator algebras are expressed in algebraic terms, with highly analytic methodologies. Although operator algebras are typically thought of as a subdivision of functional analysis, they have direct applications in representation theory, differential geometry,  etc. 

Operator algebras can be used to simultaneously investigate arbitrary sets of operators with little algebraic relation. 

Noncommutative geometry (NCG) is a branch of mathematics concerned with the building of spaces that are locally presented by noncommutative algebras of functions and with a geometric approach to noncommutative algebras (possibly in some generalised sense).

The Gelfand–Naimark theorem says that a C*-subalgebra of limited operators on a Hilbert space is isometrically *-isomorphic to an arbitrary C*-algebra A.

An orbifold (for “orbit-manifold”) is a generalisation of a manifold in the mathematical fields of topology and geometry.

It is the theory linked to vertex algebra’s fixed point subalgebra under the action of a finite group of automorphisms in two-dimensional conformal field theory.