In mathematics, a **universal C*-algebra** is a C*-algebra described in terms of generators and relations. In contrast to rings or algebras, where one can consider quotients by free rings to construct universal objects, C*-algebras must be realizable as algebras of bounded operators on a Hilbert space by the Gelfand-Naimark-Segal construction and the relations must prescribe a uniform bound on the norm of each generator. This means that depending on the generators and relations, a universal C*-algebra may not exist. In particular, free C*-algebras do not exist.

There are several problems with defining relations for C*-algebras. One is, as previously mentioned, due to the non-existence of free C*-algebras, not every set of relations defines a C*-algebra. Another problem is that one would often want to include order relations, formulas involving continuous functional calculus, and spectral data as relations. For that reason, we use a relatively roundabout way of defining C*-algebra relations. The basic motivation behind the following definitions is that we will define relations as the category of their representations.

Given a set *X*, the *null C*-relation* on *X* is the category with objects consisting of pairs (*j*, *A*), where *A* is a C*-algebra and *j* is a function from *X* to *A* and with morphisms from (*j*, *A*) to (*k*, *B*) consisting of *-homomorphisms φ from *A* to *B* satisfying φ ∘ *j* = *k*. A *C*-relation* on *X* is a full subcategory of satisfying:

- the unique function
*X*to {0} is an object; - given an injective *-homomorphism φ from
*A*to*B*and a function*f*from*X*to*A*, if φ ∘*f*is an object, then*f*is an object; - given a *-homomorphism φ from
*A*to*B*and a function*f*from*X*to*A*, if*f*is an object, then φ ∘*f*is an object; - if
*f*_{i}is an object for*i*=1,2,...,n, then is also an object. Furthermore, if*f*_{i}is an object for*i*in an nonempty index set*I*implies the product is also an object, then the C*-relation is*compact*.

Given a C*-relation *R* on a set *X*. then a function ι from *X* to a C*-algebra *U* is called a *universal representation* for *R* if

- given a C*-algebra
*A*and a *-homomorphism φ from*U*to*A*, φ ∘ ι is an object of*R*; - given a C*-algebra
*A*and an object (*f*,*A*) in*R*, there exists a unique *-homomorphism φ from*U*to*A*such that*f*= φ ∘ ι. Notice that ι and*U*are unique up to isomorphism and*U*is called the*universal C*-algebra for R*.

A C*-relation *R* has a universal representation if and only if *R* is compact.

Given a *-polynomial *p* on a set *X*, we can define a full subcategory of with objects (*j*, *A*) such that *p* ∘ *j* = 0. For convenience, we can call *p* a relation, and we can recover the classical concept of relations. Unfortunately, not every *-polynomial will define a compact C*-relation. ^{ [1] }

Alternatively, one can use a more concrete characterization of universal C*-algebras that more closely resembles the construction in abstract algebra. Unfortunately, this restricts the types of relations that are possible. Given a set *G*, a *relation* on *G* is a set *R* consisting of pairs (*p*, η) where *p* is a *-polynomial on *X* and η is a non-negative real number. A *representation* of (*G*, *R*) on a Hilbert space *H* is a function ρ from *X* to the algebra of bounded operators on *H* such that for all (*p*, η) in *R*. The pair (*G*, *R*) is called *admissible* if a representation exists and the direct sum of representations is also a representation. Then

is finite and defines a seminorm satisfying the C*-norm condition on the free algebra on *X*. The completion of the quotient of the free algebra by the ideal is called the *universal C*-algebra* of (*G*,*R*). ^{ [2] }

- The noncommutative torus can be defined as a universal C*-algebra generated by two unitaries with a commutation relation.
- The Cuntz algebras, graph C*-algebras and k-graph C*-algebras are universal C*-algebras generated by partial isometries.
- The universal C*-algebra generated by a unitary element
*u*has presentation . By continuous functional calculus, this C*-algebra is the algebra of continuous functions on the unit circle in the complex plane. Any C*-algebra generated by a unitary element is isomorphic to a quotient of this universal C*-algebra.^{ [2] }

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The representation theory of groups is a part of mathematics which examines how groups act on given structures.

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In mathematics, the **Grothendieck group** construction constructs an abelian group from a commutative monoid *M* in the most universal way, in the sense that any abelian group containing a homomorphic image of *M* will also contain a homomorphic image of the Grothendieck group of *M*. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.

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In mathematics, specifically in symplectic geometry, the **momentum map** is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including **symplectic** (**Marsden–Weinstein**) **quotients**, discussed below, and symplectic cuts and sums.

In mathematics, a **vector-valued differential form** on a manifold *M* is a differential form on *M* with values in a vector space *V*. More generally, it is a differential form with values in some vector bundle *E* over *M*. Ordinary differential forms can be viewed as **R**-valued differential forms.

In universal algebra and in model theory, a **structure** consists of a set along with a collection of finitary operations and relations that are defined on it.

In algebraic geometry, a **morphism of schemes** generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes.

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

- ↑ Loring, Terry A. (1 September 2010). "C*-Algebra Relations".
*Mathematica Scandinavica*.**107**(1): 43–72. ISSN 1903-1807 . Retrieved 27 March 2017. - 1 2 Blackadar, Bruce (1 December 1985). "Shape theory for $C^*$-algebras".
*Mathematica Scandinavica*.**56**(0): 249–275. ISSN 1903-1807 . Retrieved 27 March 2017.

- Loring, T. (1997),
*Lifting Solutions to Perturbing Problems in C*-Algebras*, Fields Institute Monographs,**8**, American Mathematical Society, ISBN 0-8218-0602-5

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