In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group and the representations are strongly continuous. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ...

The theory of unitary representations of groups is closely connected with harmonic analysis. In the case of an abelian group G, a fairly complete picture of the representation theory of G is given by Pontryagin duality. In general, the unitary equivalence classes of irreducible unitary representations of G makes up its unitary dual. This set can be identified to the spectrum of the C*-algebra associated to G by the group C*-algebra construction. This is a topological space. Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ... In mathematics, the term irreducible is used in several ways. ... The spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i. ... In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

The general form of the Plancherel theorem tries to describe the regular representation of G on L²(G) by means of a measure on the unitary dual. For G abelian this is given by the Pontryain duality theory. For G compact, this is done by the Peter-Weyl theorem; in that case the unitary dual is a discrete space, and the measure attaches an atom to each point of mass equal to its degree. In mathematics, the Plancherel theorem is a result in harmonic analysis, first proved by Michel Plancherel. ... Measure can mean: To perform a measurement. ... Several specialized usages of the terms compact and compactness exist. ... The Peter-Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...

One of the pioneers in constructing a general theory of unitary representations was George Mackey. George Mackey is an American mathematician, working mainly in the fields of representation theory and group actions, and related parts of functional analysis. ...

The theory has been widely applied in quantum mechanics since the 1920s. Fig. ... Sometimes referred to as the Roaring Twenties. Events and trends Technology John T. Thompson invents Thompson submachine gun, also known as Tommy Gun. ...

Formal definitions

Let G be a topological group. A strongly continuous unitary representation of G on a Hilbert space H is a group homomorphism from G into the unitary group of H,

such that g → π(g) ξ is a norm continuous function for every ξ ∈ H.

Note that if G is a Lie group, this representation is necessarily smooth (respectively real analytic) with respect to the differentiable structure (respectively real analytic structure) of the Lie group. In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

Complete reducibility

A unitary representation is completely reducible, in the sense that for any closed invariant subspace, the orthogonal complement is again a closed invariant subspace. This is at the level of an observation, but is a fundamental property. For example, it implies that finite dimensional unitary representations are always a direct sum of irreducible representations, in the algebraic sense. In mathematics, an invariant subspace of a linear mapping over some vector space V is a subspace W of V such that , literally: T(W) is contained in W. An invariant subspace of T is said to be T invariant. ... In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of an inner product space V is the set of all vectors in V that are orthogonal to every vector in W, i. ...

Since unitary representations are much easier to handle than the general case, it is natural to consider unitarizable representations, those that become unitary on the introduction of a suitable complex Hilbert space structure. This works very well for finite groups, and more generally for compact groups, by an averaging argument applied to an arbitrary hermitian structure. For example, a natural proof of Maschke's theorem is by this route. In mathematics, the general features of the representation theory of a finite group G, over the complex numbers, were discovered by Ferdinand Georg Frobenius in the years before 1900. ... In mathematics, a compact (topological, often understood) group is a topological group that is also a compact space. ... In mathematics, in particular group representation theory, Maschkes theorem is the basic result proving that linear representations of a finite group over the complex numbers break up into irreducible pieces. ...

In general, for non-compact groups, it is a more serious question which representations are unitarizable.