In mathematics, 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*. A *-representation π of *A* on a Hilbert space *H* is **irreducible** if, and only if, there is no closed subspace *K* different from *H* and {0} which is invariant under all operators π(*x*) with *x* ∈ *A*. We implicitly assume that irreducible representation means *non-null* irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces. As explained below, the spectrum *Â* is also naturally a topological space; this is similar to the notion of the spectrum of a ring.

- Primitive spectrum
- Examples
- Commutative C*-algebras
- The C*-algebra of bounded operators
- Finite-dimensional C*-algebras
- Other characterizations of the spectrum
- The space Irrn(A)
- Mackey–Borel structure
- Algebraic primitive spectra
- References

One of the most important applications of this concept is to provide a notion of dual object for any locally compact group. This dual object is suitable for formulating a Fourier transform and a Plancherel theorem for unimodular separable locally compact groups of type I and a decomposition theorem for arbitrary representations of separable locally compact groups of type I. The resulting duality theory for locally compact groups is however much weaker than the Tannaka–Krein duality theory for compact topological groups or Pontryagin duality for locally compact *abelian* groups, both of which are complete invariants. That the dual is not a complete invariant is easily seen as the dual of any finite-dimensional full matrix algebra M_{n}(**C**) consists of a single point.

The topology of *Â* can be defined in several equivalent ways. We first define it in terms of the **primitive spectrum** .

The primitive spectrum of *A* is the set of primitive ideals Prim(*A*) of *A*, where a primitive ideal is the kernel of an irreducible *-representation. The set of primitive ideals is a topological space with the **hull-kernel topology** (or **Jacobson topology**). This is defined as follows: If *X* is a set of primitive ideals, its **hull-kernel closure** is

Hull-kernel closure is easily shown to be an idempotent operation, that is

and it can be shown to satisfy the Kuratowski closure axioms. As a consequence, it can be shown that there is a unique topology τ on Prim(*A*) such that the closure of a set *X* with respect to τ is identical to the hull-kernel closure of *X*.

Since unitarily equivalent representations have the same kernel, the map π ↦ ker(π) factors through a surjective map

We use the map *k* to define the topology on *Â* as follows:

**Definition**. The open sets of *Â* are inverse images *k*^{−1}(*U*) of open subsets *U* of Prim(*A*). This is indeed a topology.

The hull-kernel topology is an analogue for non-commutative rings of the Zariski topology for commutative rings.

The topology on *Â* induced from the hull-kernel topology has other characterizations in terms of states of *A*.

The spectrum of a commutative C*-algebra *A* coincides with the Gelfand dual of *A* (not to be confused with the dual *A'* of the Banach space *A*). In particular, suppose *X* is a compact Hausdorff space. Then there is a natural homeomorphism

This mapping is defined by

I(*x*) is a closed maximal ideal in C(*X*) so is in fact primitive. For details of the proof, see the Dixmier reference. For a commutative C*-algebra,

Let *H* be a separable infinite-dimensional Hilbert space. *L*(*H*) has two norm-closed *-ideals: *I*_{0} = {0} and the ideal *K* = *K*(*H*) of compact operators. Thus as a set, Prim(*L*(*H*)) = {*I*_{0}, *K*}. Now

- {
*K*} is a closed subset of Prim(*L*(*H*)). - The closure of {
*I*_{0}} is Prim(*L*(*H*)).

Thus Prim(*L*(*H*)) is a non-Hausdorff space.

The spectrum of *L*(*H*) on the other hand is much larger. There are many inequivalent irreducible representations with kernel *K*(*H*) or with kernel {0}.

Suppose *A* is a finite-dimensional C*-algebra. It is known *A* is isomorphic to a finite direct sum of full matrix algebras:

where min(*A*) are the minimal central projections of *A*. The spectrum of *A* is canonically isomorphic to min(*A*) with the discrete topology. For finite-dimensional C*-algebras, we also have the isomorphism

The hull-kernel topology is easy to describe abstractly, but in practice for C*-algebras associated to locally compact topological groups, other characterizations of the topology on the spectrum in terms of positive definite functions are desirable.

In fact, the topology on *Â* is intimately connected with the concept of weak containment of representations as is shown by the following:

**Theorem**. Let*S*be a subset of*Â*. Then the following are equivalent for an irreducible representation π;- The equivalence class of π in
*Â*is in the closure of*S* - Every state associated to π, that is one of the form

- with ||ξ|| = 1, is the weak limit of states associated to representations in
*S*.

- The equivalence class of π in

The second condition means exactly that π is weakly contained in *S*.

The GNS construction is a recipe for associating states of a C*-algebra *A* to representations of *A*. By one of the basic theorems associated to the GNS construction, a state *f* is pure if and only if the associated representation π_{f} is irreducible. Moreover, the mapping κ : PureState(*A*) → *Â* defined by *f* ↦ π_{f} is a surjective map.

From the previous theorem one can easily prove the following;

**Theorem**The mapping- given by the GNS construction is continuous and open.

There is yet another characterization of the topology on *Â* which arises by considering the space of representations as a topological space with an appropriate pointwise convergence topology. More precisely, let *n* be a cardinal number and let *H _{n}* be the canonical Hilbert space of dimension

Irr_{n}(*A*) is the space of irreducible *-representations of *A* on *H _{n}* with the point-weak topology. In terms of convergence of nets, this topology is defined by π

It turns out that this topology on Irr_{n}(*A*) is the same as the point-strong topology, i.e. π_{i} → π if and only if

**Theorem**. Let*Â*be the subset of_{n}*Â*consisting of equivalence classes of representations whose underlying Hilbert space has dimension*n*. The canonical map Irr_{n}(*A*) →*Â*is continuous and open. In particular,_{n}*Â*can be regarded as the quotient topological space of Irr_{n}_{n}(*A*) under unitary equivalence.

**Remark**. The piecing together of the various *Â _{n}* can be quite complicated.

*Â* is a topological space and thus can also be regarded as a Borel space. A famous conjecture of G. Mackey proposed that a *separable* locally compact group is of type I if and only if the Borel space is standard, i.e. is isomorphic (in the category of Borel spaces) to the underlying Borel space of a complete separable metric space. Mackey called Borel spaces with this property **smooth**. This conjecture was proved by James Glimm for separable C*-algebras in the 1961 paper listed in the references below.

**Definition**. A non-degenerate *-representation π of a separable C*-algebra *A* is a **factor representation** if and only if the center of the von Neumann algebra generated by π(*A*) is one-dimensional. A C*-algebra *A* is of type I if and only if any separable factor representation of *A* is a finite or countable multiple of an irreducible one.

Examples of separable locally compact groups *G* such that C*(*G*) is of type I are connected (real) nilpotent Lie groups and connected real semi-simple Lie groups. Thus the Heisenberg groups are all of type I. Compact and abelian groups are also of type I.

**Theorem**. If*A*is separable,*Â*is smooth if and only if*A*is of type I.

The result implies a far-reaching generalization of the structure of representations of separable type I C*-algebras and correspondingly of separable locally compact groups of type I.

Since a C*-algebra *A* is a ring, we can also consider the set of primitive ideals of *A*, where *A* is regarded algebraically. For a ring an ideal is primitive if and only if it is the annihilator of a simple module. It turns out that for a C*-algebra *A*, an ideal is algebraically primitive if and only if it is primitive in the sense defined above.

**Theorem**. Let*A*be a C*-algebra. Any algebraically irreducible representation of*A*on a complex vector space is algebraically equivalent to a topologically irreducible *-representation on a Hilbert space. Topologically irreducible *-representations on a Hilbert space are algebraically isomorphic if and only if they are unitarily equivalent.

This is the Corollary of Theorem 2.9.5 of the Dixmier reference.

If *G* is a locally compact group, the topology on dual space of the group C*-algebra C*(*G*) of *G* is called the **Fell topology**, named after J. M. G. Fell.

In mathematics, more specifically in functional analysis, a **Banach space** is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

**C ^{∗}-algebras** are subjects of research in functional analysis, a branch of mathematics. A C*-algebra is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra

In algebra and algebraic geometry, the **spectrum** of a commutative ring *R*, denoted by , is the set of all prime ideals of *R*. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an **affine scheme**.

In mathematics, a **topological group** is a group G together with a topology on G such that both the group's binary operation and the function mapping group elements to their respective inverses are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

In mathematics and theoretical physics, a **representation of a Lie group** is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras.

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.

In mathematics, 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. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group *G*. The theorem is a collection of results generalizing the significant facts about the decomposition of the regular representation of any finite group, as discovered by Ferdinand Georg Frobenius and Issai Schur.

In functional analysis, a discipline within mathematics, given a C*-algebra *A*, the **Gelfand–Naimark–Segal construction** establishes a correspondence between cyclic *-representations of *A* and certain linear functionals on *A*. The correspondence is shown by an explicit construction of the *-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal.

In mathematics, the **Gelfand–Naimark theorem** states that an arbitrary C*-algebra *A* is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 and was a significant point in the development of the theory of C*-algebras since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an operator algebra.

In mathematics, the **Gelfand representation** in functional analysis has two related meanings:

In mathematics, specifically in harmonic analysis and the theory of topological groups, **Pontryagin duality** explains the general properties of the Fourier transform on locally compact abelian groups, such as , the circle, or finite cyclic groups. The **Pontryagin duality theorem** itself states that locally compact abelian groups identify naturally with their bidual.

In functional analysis and related areas of mathematics, the **group algebra** is any of various constructions to assign to a locally compact group an operator algebra, such that representations of the algebra are related to representations of the group. As such, they are similar to the group ring associated to a discrete group.

In mathematics, a left **primitive ideal** in ring theory is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals.

In mathematics, a **compact** (**topological**) **group** is a topological group whose topology is compact. Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.

In mathematics, **Tannaka–Krein duality** theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative. The theory is named for two men, the Soviet mathematician Mark Grigorievich Krein, and the Japanese Tadao Tannaka. In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category of representations Π(*G*) with some additional structure, formed by the finite-dimensional representations of *G*.

In mathematics, the **multiplier algebra**, denoted by *M*(*A*), of a C*-algebra *A* is a unital C*-algebra which is the largest unital C*-algebra that contains *A* as an ideal in a "non-degenerate" way. It is the noncommutative generalization of Stone–Čech compactification. Multiplier algebras were introduced by Busby (1968).

This is a **glossary of algebraic geometry**.

This is a **glossary of representation theory** in mathematics.

In mathematics, the **representation theory of semisimple Lie algebras** is one of crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly by E. Cartan and H. Weyl and because of that, the theory is also known as the **Cartan–Weyl theory**. The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra ; in particular, it gives a way to parametrize irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the theorem of the highest weight.

This is a glossary for the terminology in a mathematical field of functional analysis.

- J. Dixmier,
*Les C*-algèbres et leurs représentations*, Gauthier-Villars, 1969. - J. Glimm,
*Type I C*-algebras*, Annals of Mathematics, vol 73, 1961. - G. Mackey,
*The Theory of Group Representations*, The University of Chicago Press, 1955.

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