Noncommutative topology

In mathematics, noncommutative topology is a term used for the relationship between topological and C*-algebraic concepts. The term has its origins in the Gelfand–Naimark theorem, which implies the duality of the category of locally compact Hausdorff spaces and the category of commutative C*-algebras. Noncommutative topology is related to analytic noncommutative geometry.

## Examples

The premise behind noncommutative topology is that a noncommutative C*-algebra can be treated like the algebra of complex-valued continuous functions on a 'noncommutative space' which does not exist classically. Several topological properties can be formulated as properties for the C*-algebras without making reference to commutativity or the underlying space, and so have an immediate generalization. Among these are:

* compactness (unital)
* σ-compactness (σ-unital)
* dimension (real or stable rank)
* connectedness (projectionless)
* extremally disconnected spaces (AW*-algebras)

Individual elements of a commutative C*-algebra correspond with continuous functions. And so certain types of functions can correspond to certain properties of a C*-algebra. For example, self-adjoint elements of a commutative C*-algebra correspond to real-valued continuous functions. Also, projections (i.e. self-adjoint idempotents) correspond to indicator functions of clopen sets.

Categorical constructions lead to some examples. For example, the coproduct of spaces is the disjoint union and thus corresponds to the direct sum of algebras, which is the product of C*-algebras. Similarly, product topology corresponds to the coproduct of C*-algebras, the tensor product of algebras. In a more specialized setting, compactifications of topologies correspond to unitizations of algebras. So the one-point compactification corresponds to the minimal unitization of C*-algebras, the Stone–Čech compactification corresponds to the multiplier algebra, and corona sets correspond with corona algebras.

There are certain examples of properties where multiple generalizations are possible and it is not clear which is preferable. For example, probability measures can correspond either to states or tracial states. Since all states are vacuously tracial states in the commutative case, it is not clear whether the tracial condition is necessary to be a useful generalization.

## K-theory

One of the major examples of this idea is the generalization of topological K-theory to noncommutative C*-algebras in the form of operator K-theory.

A further development in this is a bivariant version of K-theory called KK-theory, which has a composition product

KK(A,B)×KK(B,C)->KK(A,C)

of which the ring structure in ordinary K-theory is a special case. The product gives the structure of a category to KK. It has been related to correspondences of algebraic varieties.[1]

## References

1. ↑ "Noncommutative geometry and motives: the thermodynamics of endomotives", Advances in Mathematics 214 (2): 761–831, 2007, doi:10.1016/j.aim.2007.03.006

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Functional analysis (topics)

Topological vector spaces|

* Asplund
* Banach (list)
* Banach lattice
* Barrelled
* Bornological
* Brauner
* F-space
* Fréchet (tame)
* Hilbert (Inner product space
* Polarization identity)
* LF-space
* Locally convex (Seminorms/Minkowski functionals)
* Mackey
* Montel
* Nuclear
* Normed (norm)
* Quasinormed
* Reflexive
* Riesz
* Smith
* Stereotype
* Strictly convex
* Webbed
* Topological tensor product (of Hilbert spaces)

Topologies of function spaces|

* Dual
* Dual space (Dual norm)
* Operator
* Ultraweak
* Weak (polar
* operator)
* Mackey
* Strong (polar
* operator)
* Ultrastrong
* Uniform convergence

Linear operators|

* Adjoint
* Bilinear (form
* operator
* sesquilinear)
* (Un)Bounded
* Closed
* Compact (on Hilbert spaces)
* (Dis)Continuous
* Densely defined
* Fredholm
* Hilbert–Schmidt
* Functionals (positive)
* Normal
* Nuclear
* Self-adjoint
* Strictly singular
* Trace class
* Transpose
* Unitary

Operator theory|

* Banach algebras
* C*-algebras
* Spectrum (C*-algebra
* radius)
* Spectral theory (of ODEs
* Spectral theorem)
* Polar decomposition
* Singular value decomposition

Theorems|

* Banach–Alaoglu
* Banach–Mazur
* Banach–Saks
* Banach–Schauder (open mapping)
* Banach–Steinhaus (Uniform boundedness)
* Bessel's inequality
* Cauchy–Schwarz inequality
* Closed graph
* Closed range
* Eberlein–Šmulian
* Freudenthal spectral
* Gelfand–Mazur
* Gelfand–Naimark
* Goldstine
* Hahn–Banach (hyperplane separation)
* Kakutani fixed-point
* Krein–Milman
* Lomonosov's invariant subspace
* Mackey–Arens
* Mazur's lemma
* M. Riesz extension
* Riesz representation
* Parseval's identity
* Schauder fixed-point

Analysis|

* Abstract Wiener space
* Bochner space
* Differentiation in Fréchet spaces
* Derivatives (Fréchet
* Gateaux
* functional
* holomorphic)
* Integrals (Bochner
* Dunford
* Gelfand–Pettis
* regulated
* Paley–Wiener
* weak)
* Functional calculus (Borel
* continuous
* holomorphic)
* Inverse function theorem (Nash–Moser theorem)
* Measures (Lebesgue
* Projection-valued
* Vector)
* Weakly measurable function

Types of sets|

* Absolutely convex
* Absorbing
* Balanced
* Bounded
* Convex
* Convex cone (subset)
* Linear cone (subset)
* Radial
* Star-shaped
* Symmetric
* Zonotope

Subsets / set operations|

* Algebraic interior (core)
* Bounding points
* Convex hull
* Extreme point
* Interior
* Minkowski addition
* Polar

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Spectral theory and *-algebras

Basic concepts|

* Involution/*-algebra
* Banach algebra
* B*-algebra
* C*-algebra
* Noncommutative topology
* Projection-valued measure
* Spectrum
* Spectrum of a C*-algebra
* Spectral radius
* Operator space

Main results|

* Gelfand–Mazur theorem
* Gelfand–Naimark theorem
* Gelfand representation
* Polar decomposition
* Singular value decomposition
* Spectral theorem
* Spectral theory of normal C*-algebras

Special Elements/Operators|

* Isospectral
* Normal operator
* Hermitian/Self-adjoint operator
* Unitary operator
* Unit

Spectrum|

* Krein–Rutman theorem
* Normal eigenvalue
* Spectrum of a C*-algebra
* Spectral radius
* Spectral asymmetry
* Spectral gap

Decomposition of a spectrum|

* (Continuous
* Point
* Residual)
* Approximate point
* Compression
* Discrete
* Spectral abscissa

Spectral Theorem|

* Borel functional calculus
* Min-max theorem
* Projection-valued measure
* Riesz projector
* Rigged Hilbert space
* Spectral theorem
* Spectral theory of compact operators
* Spectral theory of normal C*-algebras

Special algebras|

* Amenable Banach algebra
* With an Approximate identity
* Banach function algebra
* Disk algebra
* Uniform algebra

Finite-Dimensional|

* Alon–Boppana bound
* Bauer–Fike theorem
* Numerical range
* Schur–Horn theorem

Generalizations|

* Dirac spectrum
* Essential spectrum
* Pseudospectrum
* Structure space (Shilov boundary)

Miscellaneous|

* Abstract index group
* Banach algebra cohomology
* Cohen–Hewitt factorization theorem
* Extensions of symmetric operators
* Limiting absorption principle
* Unbounded operator

Examples|

* Wiener algebra

Applications|

* Almost Mathieu operator
* Corona theorem
* Hearing the shape of a drum (Dirichlet eigenvalue)
* Heat kernel
* Kuznetsov trace formula
* Lax pair
* Proto-value function
* Ramanujan graph
* Rayleigh–Faber–Krahn inequality
* Spectral geometry
* Spectral method
* Spectral theory of ordinary differential equations
* Sturm–Liouville theory
* Superstrong approximation
* Transfer operator
* Transform theory
* Weyl law

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