###### Abstract

The Gelfand–Naimark–Sigal representation construction is considered in a general case of topological involutive algebras of quantum systems, including quantum fields, and inequivalent state spaces of these systems are characterized. We aim to show that, from the physical viewpoint, they can be treated as classical fields by analogy with a Higgs vacuum field.

Inequivalent Vacuum States in Algebraic Quantum Theory

G. Sardanashvily

Department of Theoretical Physics, Moscow State University, Moscow, Russia

###### Contents

- 1 Introduction
- 2 GNS construction. Bounded operators
- 3 Inequivalent vacua
- 4 Example. Infinite qubit systems
- 5 Example. Locally compact groups
- 6 GNS construction. Unbounded operators
- 7 Example. Commutative nuclear groups
- 8 Infinite canonical commutation relations
- 9 Free quantum fields
- 10 Euclidean QFT
- 11 Higgs vacuum
- 12 Automorphisms of quantum systems
- 13 Spontaneous symmetry breaking
- 14 Appendixes

## 1 Introduction

No long ago, one thought of vacuum in quantum field theory (QFT) as possessing no physical characteristics, and thus being invariant under any symmetry transformation. It is exemplified by a particleless Fock vacuum (Section 8). Contemporary gauge models of fundamental interactions however have arrived at a concept of the Higgs vacuum (HV). By contrast to the Fock one, HV is equipped with nonzero characteristics, and consequently is non-invariant under transformations.

For instance, HV in Standard Model of particle physics is represented by a constant background classical Higgs field, in fact, inserted by hand into a field Lagrangian, whereas its true physical nature still remains unclear. In particular, somebody treats it as a sui generis condensate by analogy with the Cooper one, and its appearance is regarded as a phase transition characterized by some symmetry breakdown [3, 5, 31].

Thus, we come to a concept of different inequivalent and, in particular, non-invariant vacua [53]. Here, we consider some models of these vacua in the framework of algebraic quantum theory (AQT). We aim to show that, from the physical viewpoint, their characteristics are classical just as we observe in a case of the above-mentioned Higgs vacuum.

In AQT, a quantum system is characterized by a topological involutive algebra and a family of continuous positive forms on . Elements of are treated as quantum objects, and we call the quantum algebra. In this framework, values of positive forms on are regarded as numerical averages of elements of . In the spirit of Copenhagen interpretation of quantum theory, one can think of positive forms on as being classical objects.

A corner stone of AQT is the following Gelfand–Naimark–Segal (GNS) representation theorem [24, 27, 57].

Theorem 1.1. Let be a unital topological involutive algebra and a positive continuous form on such that (i.e., is a state). There exists a strongly cyclic Hermitian representation of in a Hilbert space with a cyclic vector such that

(1.1) |

It should be emphasized that a Hilbert space in Theorem 1 is a completion of the quotient of an algebra with respect to an ideal generated by elements such that , and the cyclic vector is the image of the identity in this quotient. Thus, a carrier space of a representation of and its cyclic vector in Theorem 1 comes from a quantum algebra , and they also can be treated as quantum objects.

Since (1.1) is a cyclic vector, we can think of it as being a vacuum vector and, accordingly, a state as being a vacuum of an algebra . Let us note that a vacuum vector is a quantum object, whereas a vacuum of is the classical one. In particular, a quantum algebra acts on quantum vectors, but not on its vacua (states).

Since vacua are classical objects, they are parameterized by classical characteristics. A problem is that different vacua of define inequivalent cyclic representations of a quantum algebra in general. In this case, they are called inequivalent.

We say that a quantum algebra performs a transition between its vacua and if there exist elements such that and for all . In this case cyclic representations and and, accordingly, vacua and are equivalent (Theorem 6). A problem thus is to characterize inequivalent vacua of a quantum system.

One can say something in the following three variants.

(i) If a quantum algebra is a unital -algebra, Theorem 1 comes to well-known GNS (Theorem 2), and we have a cyclic representation of by bounded operators in a Hilbert space (Section 2). This is a case of quantum mechanics.

(ii) A quantum algebra is a nuclear involutive algebra (Theorem 6). In particular, this is just the case of quantum field theory (Sections ?9 and 10).

(iii) Given a group of automorphisms of a quantum algebra , its vacuum is invariant only under a proper subgroup of . This is the case of spontaneous symmetry breaking in a quantum system (Sections 12 and 13).

If a quantum algebra is a unital -algebra, one can show that a set of states of is a weakly-closed convex hull of a set of pure states of , and it is weakly* compact (Theorem 3). A set of pure states of , in turn, is a topological bundle over the spectrum of whose fibres are projective Hilbert space. The spectrum of is a set of its nonequivalent irreducible representations provided with the inverse image of the Jacobson topology. It is quasi-compact.

In accordance with Theorem 3 a unital -algebra of a quantum system performs invertible transitions between different vacua iff they are equivalent. At the same time, one can enlarge an algebra to some algebra so that all states of become equivalent states of (Theorem 3). Moreover, this algebra contains the superselection operator (3.17) which belongs to the commutant of and whose distinct eigenvalues characterize different vacua of .

In Section 4, an infinite qubit system modelled on an arbitrary set is studied. Its quantum -algebra possesses pure states whose set is a set of maps (4.18) of a set to the unit sphere in . They are equivalent iff the relation (4.20) is satisfied and, in particular, if maps and differ from each other on a finite subset of . By analogy with a Higgs vacuum, one can treat the maps (4.18) as classical vacuum fields.

In Section 5, we consider an example of a locally compact group and its group algebra of equivalence classes of complex integrable functions on . This is a Banach involutive algebra with an approximate identity. There is one-to-one correspondence between the representations of this algebra and the strongly continuous unitary representations of a group (Theorem 5). Continuous positive forms on and, accordingly, its cyclic representations are parameterized by continuous positive-definite functions on as classical vacuum fields (Theorem 5). If is square-integrable, the corresponding cyclic representation of is contained in the regular representation (5.28). In this case, distinct square integrable continuous positive-definite functions and on define inequivalent irreducible representations if they obey the relations (5.29).

However, this is not the case of unnormed topological -algebras. In order to say something, we restrict our consideration to nuclear algebras (Section 6 and 7).

This technique is applied to the analysis of inequivalent representations of infinite canonical commutative relations (Section 8) and, in particular, free quantum fields, whose states characterized by different masses are inequivalent.

Section 10 addresses the true functional integral formulation of Euclidean quantum theory (Section 10). These integrals fail to be translationally invariant that enables one to model a Higgs vacuum a translationall inequivalent state (Section 11).

Sections 12 and 13 are devoted to the phenomenon of spontaneous symmetry breaking when a state of a quantum algebra fails to be stationary only with respect to some some proper subgroup of a group of automorphisms of . Then a set of inequivalent states of these algebra generated by these automorphisms is a subset of the quotient .

## 2 GNS construction. Bounded operators

We start with a GNS representation of a topological involutive algebra by bounded operators in a Hilbert space. This is the case of Banach involutive algebras with an approximate identity (Theorem 2). Without a loss of generality, we however restrict our consideration to GNS representations of -algebras because any involutive Banach algebra with an approximate identity defines the enveloping -algebra such that there is one-to-one correspondence between the representations of and those of (Remark 2).

Let us recall the standard terminology [18, 24]. A complex associative algebra is called involutive (a -algebra) if it is provided with an involution such that

An element is normal if , and it is Hermitian or self-adjoint if . If is a unital algebra, a normal element such that is called the unitary one.

A -algebra is called the normed algebra (resp. the Banach algebra) if it is a normed (resp. complete normed) vector space whose norm obeys the multiplicative conditions

A Banach -algebra is said to be a -algebra if for all . If is a unital -algebra, then . A -algebra is provided with a normed topology, i.e., it is a topological -algebra.

Remark 2.1. It should be emphasized that by a morphism of normed algebras is meant a morphism of the underlying -algebras, without any condition on the norms and continuity. At the same time, an isomorphism of normed algebras means always an isometric morphism. Any morphism of -algebras is automatically continuous due to the property

(2.2) |

Any -algebra can be extended to a unital algebra by the adjunction of the identity to . The unital extension of also is a -algebra with respect to the operation

If is a -algebra, a norm on is uniquely prolonged to the norm

on which makes a -algebra.

One says that a Banach algebra admits an approximate identity if there is a family of elements of , indexed by a directed set , which possesses the following properties:

for all ,

and for every .

It should be noted that the existence of an approximate identity is an essential condition for many results (see, e.g., Theorems 2 and 2).

For instance, a -algebra has an approximate identity. Conversely, any Banach -algebra with an approximate identity admits the enveloping -algebra (Remark 2) [18, 24].

An important example of -algebras is an algebra of bounded (and, equivalently, continuous) operators in a Hilbert space (Section 14.2). Every closed -subalgebra of is a -algebra and, conversely, every -algebra is isomorphic to a -algebra of this type (Theorem 2).

An algebra is endowed with the operator norm

(2.3) |

This norm brings the -algebra of bounded operators in a Hilbert space into a -algebra. The corresponding topology on is called the normed operator topology.

One also provides with the strong and weak operator topologies, defined by the families of seminorms

respectively. The normed operator topology is finer than the strong one which, in turn, is finer than the weak operator topology. The strong and weak operator topologies on a subgroup of unitary operators coincide with each other.

It should be emphasized that fails to be a topological algebra with respect to strong and weak operator topologies. Nevertheless, the involution in also is continuous with respect to the weak operator topology, while the operations

where is a fixed element of , are continuous with respect to all the above mentioned operator topologies.

Remark 2.2. Let be a subset of . The commutant of is a set of elements of which commute with all elements of . It is a subalgebra of . Let denote the bicommutant. Clearly, . A -subalgebra of is called the von Neumann algebra if . This property holds iff is strongly (or, equivalently, weakly) closed in [18]. For instance, is a von Neumann algebra. Since a strongly (weakly) closed subalgebra of also is closed with respect to the normed operator topology on , any von Neumann algebra is a -algebra.

Remark 2.3. A bounded operator in a Hilbert space is called completely continuous if it is compact, i.e., it sends any bounded set into a set whose closure is compact. An operator is completely continuous iff it can be represented by the series

(2.4) |

where are elements of a basis for and are positive numbers which tend to zero as . For instance, every degenerate operator (i.e., an operator of finite rank which sends onto its finite-dimensional subspace) is completely continuous. A completely continuous operator is called the Hilbert–Schmidt operator if the series

converges. Hilbert–Schmidt operators make up an involutive Banach algebra with respect to this norm, and it is a two-sided ideal of an algebra . A completely continuous operator in a Hilbert space is called a nuclear operator if the series

converges. Nuclear operators make up an involutive Banach algebra with respect to this norm, and it is a two-sided ideal of an algebra . Any nuclear operator is the Hilbert–Schmidt one. Moreover, the product of arbitrary two Hilbert–Schmidt operators is a nuclear operator, and every nuclear operator is of this type.

Let us consider representations of -algebras by bounded operators in Hilbert spaces [18, 39]. It is a morphism of a -algebra to an algebra of bounded operators in a Hilbert space , called the carrier space of . Representations throughout are assumed to be non-degenerate, i.e., there is no element of such that or, equivalently, is dense in .

Theorem 2.1. If is a -algebra, there exists its exact (isomorphic) representation.

Theorem 2.2. A representation of a -algebra is uniquely prolonged to a representation of the unital extension of .

Let , , be a family of representations of a -algebra in Hilbert spaces . If the set of numbers is bounded for each , one can construct a bounded operator in a Hilbert sum whose restriction to each is .

Theorem 2.3. This is the case of a -algebra due to the property (2.2). Then is a representation of in , called the Hilbert sum

(2.5) |

of representations .

Given a representation of a -algebra in a Hilbert space , an element is said to be a cyclic vector for if the closure of is equal to . Accordingly, (or a more strictly a pair ) is called the cyclic representation.

Theorem 2.4. Every representation of a -algebra is a Hilbert sum of cyclic representations.

Remark 2.4. It should be emphasized, that given a cyclic representation of a -algebra in a Hilbert space , a different element of is a cyclic for iff there exist some elements such that and .

Let be a -algebra, its representation in a Hilbert space , and an element of . Then a map

(2.6) |

is a positive form on . It is called the vector form defined by and .

Therefore, let us consider positive forms on a -algebra . Given a positive form , a Hermitian form

(2.7) |

makes a pre-Hilbert space. If is a normed -algebra, continuous positive forms on are provided with a norm

(2.8) |

Theorem 2.5. Let be a unital Banach -algebra such that . Then any positive form on is continuous.

In particular, positive forms on a -algebra always are continuous. Conversely, a continuous form on an unital -algebra is positive iff . It follows from this equality that positive forms on a unital -algebra obey a relation

(2.9) |

Let us note that a continuous positive form on a topological -algebra admits different prolongations onto the unital extension of . Such a prolongation is unique in the following case [18].

Theorem 2.6. Let be a positive form on a Banach -algebra with an approximate identity. It is extended to a unique positive form on the unital extension of such that .

A key point is that any positive form on a -algebra equals a vector form defined by some cyclic representation of in accordance with the following GNS representation construction [18, 24].

Theorem 2.7. Let be a positive form on a Banach -algebra with an approximate identity and its continuous positive prolongation onto the unital extension (Theorems 2 and 2). Let be a left ideal of consisting of those elements such that . The quotient is a Hausdorff pre-Hilbert space with respect to the Hermitian form obtained from (2.7) by passage to the quotient. We abbreviate with the completion of and with the canonical image of in . For each , let be an operator in obtained from the left multiplication by in by passage to the quotient. Then the following holds.

(i) Each has a unique extension to a bounded operator in a Hilbert space .

(ii) A map is a representation of in .

(iii) A representation admits a cyclic vector .

(iv) for each .

The representation and the cyclic vector in Theorem 2 are said to be defined by a form , and a form equals the vector form defined by and .

As was mentioned above, we further restrict our consideration of the GNS construction in Theorem 2 to unital -algebras in view of the following [18, 24].

Remark 2.5. Let be an involutive Banach algebra with an approximate identity, and let be the set of pure states of (Remark 3). For each , we put

(2.10) |

It is a seminorm on such that . If is a -algebra, due to the relation (2.2) and the existence of an isomorphic representation of . Let denote the kernel of . It consists of such that . Then the completion of the factor algebra with respect to the quotient of the seminorm (2.10), is a -algebra, called the enveloping -algebra of . There is the canonical morphism . Clearly, if is a -algebra. The enveloping -algebra possesses the following important properties.

If is a representation of , there is exactly one representation of such that . Moreover, the map is a bijection of a set of representations of onto a set of representations of .

If is a continuous positive form on , there exists exactly one positive form on such that . Moreover, . The map is a bijection of a set of continuous positive forms on onto a set of positive forms on .

## 3 Inequivalent vacua

Let be a unital -algebra of a quantum systems. As was mentioned above, positive forms on a algebra are said to be equivalent if they define its equivalent cyclic representations.

Remark 3.1. Let us recall that two representations and of a -algebra in Hilbert spaces and are equivalent if there is an isomorphism such that

(3.11) |

In particular, if representations are equivalent, their kernels coincide with each other.

Given two positive forms and on a unital -algebra , we meet the following three variants.

(i) If , there is an isomorphism of the corresponding Hilbert spaces such that the relation (3.11) holds, and moreover

(3.12) |

(ii) Let positive forms and be equivalent, but different. Then their equivalence morphism fails to obey the relation (3.12).

(iii) Positive forms and on are inequivalent.

In particular, let be a representation of in a Hilbert space , and let be an element of which defines the vector form (2.6) on . Then a representation contains a summand which is equivalent to the cyclic representation of defined by a vector form .

There are the following criteria of equivalence of positive forms.

Theorem 3.1. Positive forms on a unital -algebra are equivalent only if their kernels contain a common largest closed two-sided ideal.

Proof. The result follows from the fact that the kernel of a cyclic representation defined by a positive form on a unital -algebra is a largest closed two-sided ideal of the kernel of this form [18]

Theorem 3.2. Positive forms and on a unital -algebra are equivalent iff there exist elements such that

Proof. Let a positive form define a cyclic representation of in . Let us consider an element . In accordance with Remark 2, this element is a cyclic element for a representation . It provides a positive form on such that . Then a positive form defines a cyclic representation of in a Hilbert space which is isomorphic as to a cyclic representation in such that the relation (3.12) holds. Conversely, let positive forms and be equivalent. Then a positive form defines an isomorphic cyclic representation in , but with a different cyclic vector . Then the result follows from Remark 2.

In particular, it follows from Theorem 3 that given a positive form on , the state of is equivalent to . Speaking on equivalent positive forms on , we therefore can restrict our consideration to states.

For instance, any cyclic representation of a -algebra is a summand of the Hilbert sum (2.5):

(3.13) |

of cyclic representations of where runs through a set of all states of . Since for any element there exists a state such that , the representation is injective and, consequently, isometric and isomorphic.

A space of continuous forms on a -algebra is the (topological) dual of a Banach space . It can be provided both with a normed topology defined by the norm (2.8) and a weak topology (Section 14.1). It follows from the relation (2.9), that a subset of states is convex and its extreme points are pure states.

Remark 3.2. Let us recall that a positive form on a -algebra is said to be dominated by a positive form if is a positive form [24, 18]. A non-zero positive form on a -algebra is called pure if every positive form on which is dominated by reads , .

A key point is the following [18]

Theorem 3.3. The cyclic representation of of a -algebra defined by a positive form on is irreducible iff is a pure form [18]

In particular, any vector form defined by a vector of a carrier Hilbert space of an irreducible representation is a pure form.

Remark 3.3. Let us note that a representation of a -algebra in a Hilbert space is called topologically irreducible if the following equivalent conditions hold:

the only closed subspaces of invariant under are 0 and ;

the commutant of in is a set of scalar operators;

every non-zero element of is a cyclic vector for .

At the same time, irreducibility of in the algebraic sense means that the only subspaces of invariant under are 0 and . If is a -algebra, the notions of topologically and algebraically irreducible representations are equivalent. It should be emphasized that a representation of a -algebra need not be a Hilbert sum of the irreducible ones.

An algebraically irreducible representation of a -algebra is characterized by its kernel . This is a two-sided ideal, called primitive. Certainly, algebraically irreducible representations with different kernels are inequivalent, whereas equivalent irreducible representations possesses the same kernel. Thus, we have a surjection

(3.14) |

of a set of equivalence classes of algebraically irreducible representations of a -algebra onto a set Prim of primitive ideals of .

A set Prim is equipped with the so called Jacobson topology [18]. This topology is not Hausdorff, but obeys the Fréchet axiom, i.e., for any two distinct points of Prim, there is a neighborhood of one of them which does not contain the other. Then a set is endowed with the coarsest topology such that the surjection (3.14) is continuous. Provided with this topology, is called the spectrum of a -algebra . In particular, one can show the following.

Theorem 3.4. If a -algebra is unital, its spectrum is quasi-compact, i.e., it satisfies the Borel–Lebesgue axiom, but need not be Hausdorff.

Theorem 3.5. The spectrum of a -algebra is a locally quasi-compact space.

Example 3.4. A -algebra is said to be elementary if it is isomorphic to an algebra of compact operators in some Hilbert space (Example 2). Every non-trivial irreducible representation of an elementary algebra is equivalent to its isomorphic representation by compact operators in [18]. Hence, the spectrum of an elementary algebra is a singleton set.

By analogy with Theorem 3, one can state the following relations between equivalent pure states of a -algebra.

Theorem 3.6. Pure states and of a unital -algebra are equivalent iff there exists a unitary element such that the relation

(3.15) |

holds.

Proof. A key point is that, if is a pure state of a unital -algebra, a pseudo-Hilbert space in Theorem 2 is complete, i.e., .

Corollary 3.7. Let be an irreducible representation of a unital -algebra in a Hilbert space . Given two distinct elements and of (they are cyclic for ), the vector forms on defined by and are equal iff there exists , , such that .

Corollary 3.8. There is one-to-one correspondence between the pure states of a unital -algebra associated to the same irreducible representation of in a Hilbert space and the one-dimensional complex subspaces of , i.e, these pure states constitute a projective Hilbert space .

There is an additional important criterion of equivalence of pure states of a unital -algebra [23].

Theorem 3.9. Pure states and of a unital -algebra are equivalent if .

Let denote a set of pure states of a unital -algebra . Theorem 3 implies a surjection . One can show that, if is provided with a relative weak topology, this surjection is continuous and open, i.e., it is a topological fibre bundle whose fibres are projective Hilbert spaces [18].

Turning to a set of states of a unital algebra -algebra , we have the following.

Theorem 3.10. A set is a weakly-closed convex hull of a set of pure states of . It is weakly* compact [18].

Herewith, by virtue of Theorem 3, any set of mutually inequivalent pure states of a unital -algebra is totally disconnected in a normed topology, i.e., its connected components are points only.

By virtue of Theorem 3, elements of a quantum algebra can not perform invertible transitions between its inequivalent states. At the same time, one can show the following.

Theorem 3.11. There exists a wider unital -algebra such that inequivalent states of become its equivalent ones.

Proof. Let us consider the Hilbert sum (3.13) of cyclic representations of whose carrier space is a Hilbert sum

(3.16) |

Let be a unital algebra of bounded operators in (3.16). Since the representation of is exact, an algebra is isomorphic to a subalgebra of . Any state of is equivalent to a vector state of which also is that of . Since all vector states of are equivalent, all states of are equivalent as those of .

## 4 Example. Infinite qubit systems

Let be a two-dimensional complex space equipped with the standard positive non-degenerate Hermitian form . Let be an algebra of complex -matrices seen as a -algebra. A system of qubits is usually described by a Hilbert space and a -algebra , which coincides with the algebra of bounded operators in [32]. One can straightforwardly generalize this description to an infinite set of qubits by analogy with a spin lattice [20, 24, 50]. Its algebra admits inequivalent irreducible representations.

We follow the construction of infinite tensor products of Hilbert spaces and -algebras in [20]. Let be a set of two-dimensional Hilbert spaces . Let be a complex vector space whose elements are finite linear combinations of elements of the Cartesian product of the sets . A tensor product of complex vector spaces is the quotient of with respect to a vector subspace generated by elements of the form:

, where for some element and for all the others,

, , where for some element and for all the others.

Given a map

(4.18) |

let us consider an element

(4.19) |

Let us denote the subspace of spanned by vectors where only for a finite number of elements . It is called the -tensor product of vector spaces , . Then is a pre-Hilbert space with respect to a positive non-degenerate Hermitian form

Its completion is a Hilbert space whose orthonormal basis consists of the elements , , , such that and , where is an orthonormal basis for .

Let now be a set of unital -algebras . These algebras are provided with the operator norm

where are the Pauli matrices. Given the family , let us construct the -tensor product of vector spaces . One can regard its elements as tensor products of elements of , , for finite subsets of and of the identities , . It is easily justified that is a normed -algebra with respect to the operations

and a norm

Its completion is a -algebra treated as a quantum algebra of a qubit system modelled over a set . Then the following holds [20].

Theorem 4.1. Given the element (4.19), the natural representation of a -algebra in the pre-Hilbert space is extended to an irreducible representation of a -algebra in the Hilbert space such that is an algebra of all bounded operators in . Conversely, all irreducible representations of are of this type.

An element in Theorem 4 defines a pure state of an algebra . Consequently, a set of pure states of this algebra is a set of maps (4.18).

Theorem 4.2. Pure states and of an algebra are equivalent iff

(4.20) |

In particular, the relation (4.20) holds if maps and differ from each other on a finite subset of .

By analogy with a Higgs vacuum, one can treat the maps (4.18) as classical vacuum fields.

## 5 Example. Locally compact groups

Let be a locally compact group provided with a Haar measure (Section 14.4). A space of equivalence classes of complex integrable functions (or, simply, complex integrable functions) on is an involutive Banach algebra (Section 14.3) with an approximate identity. As was mentioned above, there is one-to-one correspondence between the representations of this algebra and the strongly continuous unitary representations of a group (Theorem 5). Thus, one can employ the GNS construction in order to describe these representations of [18, 24].

Let a left Haar measure on hold fixed, and by an integrability condition throughout is meant the -integrability.

A uniformly (resp. strongly) continuous unitary representation of a locally compact group in a Hilbert space is a continuous homeomorphism of to a subgroup of unitary operators in provided with the normed (resp. strong) operator topology. A uniformly continuous representation is strongly continuous. However, the uniform continuity of a representation is rather rigorous condition. For instance, a uniformly continuous irreducible unitary representation of a connected locally compact real Lie group is necessarily finite-dimensional. Therefore, one usually studies strongly continuous representations of locally compact groups.

In this case, any element of a carrier Hilbert space yields the continuous map . Since strong and weak operator topologies on a unitary group coincide, we have a bounded continuous complex function

(5.21) |

on for any fixed elements . It is called the coefficient of a representation . There is an obvious equality

The Banach space of integrable complex functions on is provided with the structure of an involutive Banach algebra with respect to the contraction (14.129) and the involution

where is the modular function of . It is called the group algebra of . A map defines an isometric monomorphism of to a Banach algebra of bounded complex measures on provided with the involution . Unless otherwise stated, will be identified with its image in . In particular, a group algebra admits an approximate identity which converges to the Dirac measure .

Remark 5.1. The group algebra is not a -algebra. Its enveloping -algebra is called the -algebra of a locally compact group .

Unitary representations of a locally compact group and representations of a group algebra are related as follows [18].

Theorem 5.1. There is one-to-one correspondence between the (strongly continuous) unitary representations of a locally compact group and the representations (5.23) of its group algebra .

Proof. Let be a (strongly continuous) unitary representation of in a Hilbert space . Given a bounded positive measure on , let us consider the integrals

of the coefficient functions (5.21) for all . There exists a bounded operator in such that

It is called the operator-valued integral of with respect to the measure , and is denoted by

(5.22) |

The assignment provides a representation of a Banach -algebra in . Its restriction

(5.23) |

to is non-degenerate. One says that the representations (5.22) of and (5.23) of are determined by a unitary representation of . Conversely, let be a representation of a Banach -algebra in a Hilbert space . There is a monomorphism of a group onto a subgroup of Dirac measures , , of an algebra . Let be an approximate identity in . Then converges to an element of which can be seen as a representation of the unit element of . Accordingly, converges to . Thereby, we obtain the (strongly continuous) unitary representation of a group in a Hilbert space . Moreover, the representation (5.23) of determined by this representation of coincides with the original representation of .

Moreover, and have the same cyclic vectors and closed invariant subspaces. In particular, a representation of is topologically irreducible iff the associated representation of is so. It should be emphasized that, since is not a -algebra, its topologically irreducible representations need not be algebraically irreducible. By irreducible representations of a group , we will mean only its topologically irreducible representations.

Theorem 5 enables us to apply the GNS construction (Theorem 2) in order to characterize unitary representations of by means of positive continuous forms on .

In accordance with Remark 14.3, a continuous form on a group algebra is defined as

(5.24) |

by an element of a space of infinite integrable complex functions on (Remark 14.3). However, a function should satisfy the following additional condition in order that the form (5.24) to be positive.

A continuous complex function on is called positive-definite if

for any finite set of elements of and any complex numbers . In particular, if and , we obtain

i.e., is bounded.

Lemma 5.2. The continuous form (5.24) on is positive iff locally almost everywhere equals a continuous positive-definite function.

Then cyclic representations of a group algebra and the unitary cyclic representations of a locally compact group are defined by continuous positive-definite functions on in accordance with the following theorem.

Theorem 5.3. Let be a representation of in a Hilbert space and a cyclic vector for which are determined by the form (5.24). Then the associated unitary representation of in is characterized by a relation

(5.25) |

Conversely, a complex function on is continuous positive-definite iff there exists a unitary representation of and a cyclic vector for such that the equality (5.25) holds.

By analogy with a Higgs vacuum, one can think of functions in Theorem 5 as being classical vacuum fields.

Example 5.2. Let a group acts on a Hausdorff topological space on the left. Let be a quasi-invariant measure on under a transformation group , i.e., where is the Radon–Nikodym derivative in Theorem 14.3. Then there is a representation

(5.26) |

of in a Hilbert space of square -integrable complex functions on [17]. It is a unitary representation due to the equality

A group can be equipped with the coarsest topology such that the representation (5.26) is strongly continuous. For instance, let be a locally compact group, and let be a left Haar measure. Then the representation (5.26) comes to the left-regular representation

(5.27) |

of in a Hilbert space of square integrable complex functions on . Note that the above mentioned coarsest topology on is coarser then the original one, i.e., the representation (5.27) is strongly continuous.

Let us consider unitary representations of a locally compact group which are contained in its left-regular representation (5.27). In accordance with the expression (5.22), the corresponding representation of a group algebra in reads

(5.28) |

Let be a unimodular group. There is the following criterion that its unitary representation is contained in the left-regular one.

Theorem 5.4. If a continuous positive-definite function on a unimodular locally compact group is square integrable, then the representation of determined by is contained in the left-regular representation (5.27) of . Conversely, let be a cyclic unitary representation of which is contained in , and let be a cyclic vector for . Then a continuous positive-definite function on is square integrable.

The representation in Theorem 5 is constructed as follows. Given a square integrable continuous positive-definite function on , there exists a positive-definite function