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pages

In the present article, we provide several constructions of C*-dynamical systems (F,G,\beta) with a compact group G in terms of Cuntz–Pimsner algebras. These systems have a minimal relative commutant of the fixed-point algebra A := F\sp G in F, i.e. A' \cap F = Z, where Z is the center of A, which is assumed to be non-trivial. In addition, we show in our models that the group action β: G -> AutF has full spectrum, i.e. any unitary irreducible representation of G is carried by a β_G-invariant Hilbert space within F.

First, we give several constructions of minimal C*-dynamical systems in terms of a single Cuntz–Pimsner algebra F = O_ℌ associated to a suitable Z-bimodule ℌ. These examples are labelled by the action of a discrete Abelian group ℭ (which we call the chain group) on Z and by the choice of a suitable class of finite dimensional representations of G. Second, we present a more elaborate contruction, where now the C*-algebra F is generated by a family of Cuntz–Pimsner algebras. Here, the product of the elements in different algebras is twisted by the chain group action. We specify the various constructions of C*-dynamical systems for the group G = SU(N), N ≥ 2.

40 pages, no figures.-- MSC2000 codes: 46L08, 47L80, 22D25.-- ArXiv pre-print available at: http://arxiv.org/abs/math/0702775

Dedicated to Klaus Fredenhagen on his 60th birthday.

MR#: MR2541934

Zbl#: Zbl pre05589445

World Scientific Publishing

International Journal of Mathematics (IJM), 2009, vol. 20, n. 6, p. 751-790

We are grateful to the DFG-Graduiertenkolleg "Hierarchie und Symmetrie in mathematischen Modellen" for supporting a visit of E.V. to the RWTH-Aachen.

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Realization of minimal C*-dynamical systems in

terms of Cuntz-Pimsner algebras

∗Fernando Lledo´ Ezio Vasselli

Department of Mathematics, Dipartimento di Matematica,

University Carlos III Madrid University of Rome ”La Sapienza”

Avda. de la Universidad 30, E-28911 Legan´es (Madrid), Spain. P.le Aldo Moro 2, I-00185 Roma, Italy

ﬂledo@math.uc3m.es vasselli@mat.uniroma2.it

September 16, 2008

Dedicated to Klaus Fredenhagen on his 60th birthday

Abstract

In the present article we provide several constructions of C*-dynamical systems (F,G,β)

with a compact groupG in terms of Cuntz-Pimsner algebras. These systems have a minimal

G ′relative commutant of the ﬁxed-point algebraA :=F inF, i.e.A∩F =Z, whereZ is the

center ofA, which is assumed to be nontrivial. In addition, we show in our models that the

group action β:G→ AutF has full spectrum, i.e. any unitary irreducible representation of

G is carried by a β -invariant Hilbert space withinF.G

First, we give several constructions of minimal C*-dynamical systems in terms of a single

Cuntz-Pimsner algebraF =O associated to a suitableZ-bimoduleH. These examples areH

labeled by the action of a discrete Abelian group C (which we call the chain group) onZ

and by the choice of a suitable class of ﬁnite dimensional representations ofG. Second, we

present a more elaborate construction, where now the C*-algebraF is generated by a family

of Cuntz-Pimsner algebras. Here the product of the elements in diﬀerent algebras is twisted

by the chain group action. We specify the various constructions of C*-dynamical systems for

the groupG = SU(N), N≥ 2.

Keywords: C*-dynamical systems, minimal relative commutant, Cuntz-Pimsner algebra, Hilbert

bimodule, duals of compact groups, tensor categories, non-simple unit

MSC-classiﬁcation: 46L08, 47L80, 22D25

Contents

1 Introduction 2

1.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Hilbert C*-systems and the chain group 6

2.1 Hilbert C*-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The chain group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

∗Institute for Pure and Applied Mathematics, RWTH-Aachen, Templergraben 55, D-52062 Aachen, Germany

(on leave). e-mail: lledo@iram.rwth-aachen.de

1

arXiv:math/0702775v4 [math.OA] 16 Sep 20083 Cuntz-Pimsner algebras 11

3.1 Basic deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Endomorphisms in Cuntz-Pimsner algebras . . . . . . . . . . . . . . . . . . . . . 13

3.3 Amplimorphisms and their associated Cuntz-Pimsner algebras. . . . . . . . . . . 14

4 Examples of minimal C*-dynamical systems 16

5 Construction of Hilbert C*-systems 21

5.1 The C*-algebra of a chain group action . . . . . . . . . . . . . . . . . . . . . . . 21

5.1.1 Crossed products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.2 Minimal and regular C*-dynamical systems . . . . . . . . . . . . . . . . . . . . . 27

6 Appendix: Tensor categories of Hilbert bimodules 31

1 Introduction

Duality of groups plays a central role in abstract harmonic analysis. Its aim is to reconstruct

ba groupG from its dualG, i.e. from the set (of equivalence classes) of continuous, unitary and

irreduciblerepresentations, endowedwithaproperalgebraic andtopological structure. Themost

famous duality result is Pontryagin’s duality theorem for locally compact Abelian groups. For

compact, not necessarily Abelian, groups there exist also classical results due to Tannaka and

Krein(see[18,19]). Motivatedbyalongstandingprobleminquantumﬁeldtheory,Doplicherand

Roberts came up with a new duality for compact groups (see [13] as well as Mu¨ger’s appendix in

[17]). In the proof of the existence of a compact gauge group of internal gauge symmetries using

only a natural set of axioms for the algebra of observables, they placed the duality of compact

groups in the framework of C*-algebras. In this situation, the C*-algebra of local observables

A speciﬁes a categorical structure generalizing the representation category of a compact group.

The objects of this category are no longer ﬁnite-dimensional Hilbert spaces (as in the classical

results by Tannaka and Krein), but only a certain semigroupT of unital endomorphisms of

the C*-algebraA. In this setting,A has a trivial center, i.e. Z :=Z(A) =C . The arrows

of the category are the intertwining operators between these endomorphisms: for any pair of

1endomorphismsσ,ρ∈T one deﬁnes

(ρ, σ):={X∈A|Xρ(A) =σ(A)X , A∈A}. (1)

This category is a natural example of a tensor C*-category, where the norm of the arrows is

the C*-norm inA. The tensor product of objects is deﬁned as composition of endomorphisms

ρ,σ →ρ◦σ and for arrows X ∈(ρ,σ ), i =1,2, one deﬁnes the tensor product byi i i

X ×X :=X ρ (X ).1 2 1 1 2

The unit object ι is the identity endomorphism, which is simple iﬀA has a trivial center (since

(ι,ι) =Z). IfA has a trivial center, then the representation category ofG embeds as a full

subcategory into the tensor C*-category of endomorphisms ofA. The concrete group dual can

bedescribedintermsof anessentially uniqueC*-dynamical system(F,G,β), whereF isa unital

C*-algebra containing theoriginal algebraA, andtheaction ofthecompact groupβ:G→AutF

bhas full spectrum. This means that for any element in the dual D∈G there is a β -invariantG

Hilbert spaceH inF such that β H ∈ D. (Recall that the scalar product of any pair ofD G D

′ ′ ∗ ′elements ψ,ψ ∈H is deﬁned ashψ,ψi := ψ ψ ∈C and any orthonormal basis inH is aD D

set of orthogonal isometries{ψ} . The support ofH the projection given by the sum of thei i D

1In this article we will write the set arrows Hom(ρ,σ) simply by (ρ,σ) for each pair ρ,σ of objects.

2

1

1P

∗end projections, i.e. suppH = ) Moreover,A is the ﬁxed point algebra of the C*-ψψ .D ii i

Gdynamical system, i.e.A =F and one has that the relative commutant ofA inF is minimal,

′i.e.A∩F =C . This clearly impliesZ =C . The C*-algebraF can also be seen as a crossed

product ofA by the semigroupT of endomorphisms ofA (cf. [12]): the endomorphisms ρ∈T

(which are inner inA) may be implemented in terms of an orthonormal basis{ψ} ⊂H inF.i i

The endomorphism is unital iﬀ the corresponding implementing Hilbert space inF has support

.

In a series of articles by Baumga¨rtel and the ﬁrst author (cf. [3, 4, 5]) the duality of compact

groups has been generalized to the case whereA has a nontrivial center, i.e.Z)C , and the

relative commutant ofA inF remains minimal, i.e.

′A∩F =Z. (2)

′(We always have the inclusionZ⊆A∩F.) We deﬁnea Hilbert C*-system to bea C*-dynamical

system (F,G,β) with a group action that has full spectrum and for which the Hilbert spaces

inF carrying the irreducible representations ofG have support (see Section 2.1 for a precise

deﬁnition). These particular C*-dynamical systems have a rich structured and many relevant

properties hold, for instance, a Parseval like identity (cf. [5, Section 2]). Moreover, there is

an abstract characterization by means of a suitable non full inclusion of C*-categoriesT ⊂T,C

whereT isasymmetrictensorcategory withsimpleunit,conjugates, subobjectsanddirectsumsC

(cf. [5]). A similar construction appeared in by Mu¨ger in [24], using crossed products of braided

tensor *-categories with simple units w.r.t. a full symmetric subcategory.

The C*-dynamical systems (F,G,β) in this more general context provide natural examples

of tensor C*-categories with a nonsimple unit, since (ι,ι) =Z. The analysis of these kind of

categories demands the extension of basic notions. For example, a new deﬁnition of irreducible

object is needed (cf. [4, 5]). In this case the intertwiner space (ι,ι))C is a unital Abelian

C*-algebra and an object ρ∈T is said to be irreducible if the following condition holds:

(ρ,ρ) =1 ×(ι,ι), (3)ρ

where 1 is the unit of the C*-algebra (ρ,ρ). In other words, (ρ,ρ) is generated by 1 as a (ι,ι)-ρ ρ

module. Another new property that appears in the context of non-simple units is the action of a

discrete Abelian group on (ι,ι). To any irreducible object ρ one can associate an automorphism

α ∈AutZ by means ofρ

1 ⊗Z =α (Z)⊗1 , Z∈Z . (4)ρ ρ ρ

bUsing this family of automorphisms{α} we deﬁne an equivalence relation onG, the dual of theρ ρ

compact groupG, and the corresponding equivalence classes become the elements of a discrete

Abelian group C(G), which we call the chain group ofG. The chain group is isomorphic to the

character group of the center ofG and the map ρ →α induces an action of the chain group onρ

Z,

α:C(G)→AutZ, (5)

(seeSection2.2). TheobstructiontohaveT symmetricisencodedintheactionα:T issymmetric

if and only if α is trivial (cf. [5, Section 7]).

Thesestructuresaresoinvolved thatitisadiﬃculttasktoproduceexplicitexamplesofHilbert

C*-systemswithnon-simpleunit. Indeed,uptonowithasbeendoneonlyfor Abeliangroupsand

in the setting of the C*-algebras of the canonical commutation resp. anticommutation relations

in [1, 2]. Some indirect examples based on the abstract characterization in terms of the inclusion

of C*-categoriesT ⊂T, can be found [5, Section 6].C

The aim of the present article is to provide a large class of minimal C*-dynamical systems

and Hilbert C*-systems for compact non-Abelian groups. These examples are labeled by the

3

1

1

1

1

1

1action of the chain group on the unital Abelian C*-algebraZ given in (5). A crucial part of

our examples are the Cuntz-Pimsner algebras introduced by Pimsner in his seminal article [27].

This is a new family of C*-algebrasO that are naturally generated by a Hilbert bimoduleM

M over a C*-algebraA. These algebras generalize Cuntz-Krieger algebras as well as crossed-

products by the groupZ. In Pimsner’s constructionO is given as a quotient of a ToeplitzM

like algebra acting on a concrete Fock space associated toM. An alternative abstract approach

to Cuntz-Pimsner algebras in terms of C*-categories is given in [10, 20, 28]. In our models we

Cuntz-Pimsner algebrasO associated to a certain freeZ-bimodules H =H⊗Z. The factorH

H denotes a generating ﬁnite dimensional Hilbert space with an orthonormal basis speciﬁed by

isometries{ψ} . The leftZ-action of the bimodule is deﬁned in terms of the chain group actioni i

(5).

1.1 Main results

To state our ﬁrst main result we need to introduce the familyG of all ﬁnite-dimensional repre-0

sentations V of the compact groupG that satisfy the following two properties: ﬁrst, V admits

an irreducible subrepresentation of dimension or multiplicity≥2 and, second, there is a natural

n n

numbern∈N such that⊗V contains the trivial representation ι, i.e. ι≺⊗V. Then we show:

Main Theorem 1 (Theorem 4.9) LetG be a compact group,Z a unital Abelian C*-algebra and

consider a ﬁxed chain group action α: C(G)→ Aut(Z). Then for any V ∈ G there exists a0

Z-bimodule H =H ⊗Z with leftZ-action given in terms of α and a C*-dynamical systemV V

(O ,G,β ), satisfying the following properties:H VV

′ G(i) (O ,G,β ) is minimal, i.e.A ∩O = Z, where A := O is the correspondingH V H VV V V HV

ﬁxed-point algebra.

′(ii) The Abelian C*-algebraZ coincides with the center of the ﬁxed-point algebraA , i.e. A ∩V V

A =Z.V

Moreover, ifG is a compact Lie group, then the Hilbert spectrum of (O ,G,β ) is full, i.e. forH VV

beach irreducible class D∈G there is an invariant Hilbert spaceH ⊂O (in this case notD HV

necessarily of support ) such that β H speciﬁes an irreducible representation of class D.V D

An important step in the proof is to show that the corresponding bimodulesH are nonsingular.V

This notion was introduced in [10] and is important for analyzing the relative commutants in the

corresponding Cuntz-Pimsner algebras (see Section 3 for further details). We give a characteri-

zation of the class of nonsingular bimodulesthat will appear in this article (cf. Proposition 3.12).

The preceding theorem may be applied to the group SU(N) in order to deﬁne a corresponding

minimal C*-dynamical system with full spectrum (cf. Example 4.10).

To present examples of minimal C*-dynamical systems with full spectrum, where the Hilbert

spaces inF that carry the irreducible representations of the group have support , we need a

more elaborate construction: to begin with, we introduce a C*-algebra generated by a family of

Cuntz-Pimsner algebras that are labeled by any family G of unitary, ﬁnite-dimensional repre-

sentations ofG (see Subsection 5.1 for precise presentation of this algebra). This construction

is interesting in itself and can be performed for coeﬃcient algebras R which are not necessarily

Abelian. Concretely we show:

Main Theorem 2 (Theorem 5.6) Let G be a compact group, R a unital C*-algebra and

α: C(G) → AutR a ﬁxed action of the chain group C(G). Then, for every set G of ﬁnite-

αdimensional representations ofG, there exists a universal C*-algebra R⋊ G generated by R

and the Cuntz-Pimsner algebras{O } , where the product of the elements in the diﬀerentHV V ∈G

4

1

1algebras is twisted by the chain group action α.

αThe C*-algebra R⋊ G (which we will also denote simply byF) generalizes some well-known

constructions, obtained for particular choices of the family of representationsG, such as Cuntz-

Pimsneralgebras, crossedproductsbysingleendomorphisms(a` laStacey) orcrossedproductsby

αAbelian groups. Hilbert space representations ofR⋊ G are labeled by covariant representations

of the C*-dynamical system (R,C(G),α).

Now, we restrict the result of the Main Theorem 2 to the caseG=G with Abelian coeﬃcient0

αalgebra R =Z. The C*-algebraF =Z⋊ G speciﬁes prototypes of Hilbert C*-systems for0

non-Abelian groups in the context of non-simple units satisfying all the required properties:

Main Theorem 3 (Theorem 5.14) LetG be a compact group,Z a unital Abelian C*-algebra and

α:C(G)→AutZ a ﬁxed chain group action. Given the set of ﬁnite-dimensional representations

αG introduced above and the C*-algebraF :=Z⋊ G of the preceding theorem, there exists a0 0

′minimal C*-dynamical system (F,G,β), i.e.A ∩F =Z, whereA is the corresponding ﬁxed

′point algebra. Moreover,Z coincides with the center ofA, i.e.Z =A∩A, and for any V ∈G0

the Hilbert spaceH ⊂O ⊂F has support .V HV

We may apply the preceding theorem to the groupG :=SU(2). Here we choose as the family

of ﬁnite-dimensional representationsG all irreducible representations ofG with dimension≥ 2.0

This gives an explicit example of a Hilbert C*-system for SU(2) (cf. Example 5.15).

Thestructureofthearticleisasfollows: InSection2wepresentthemaindeﬁnitionsandresults

concerning Hilbert C*-systems and the chain group. In Section 3 we recall the main features of

Cuntz-Pimsner algebras that will be needed later. In the following section we present a family

of minimal C*-dynamical systems for a compact groupG and a single Cuntz-Pimsner algebra.

This family of examples is labeled by the chain group action (5) and the elements of a suitable

classG of ﬁnite-dimensional representations ofG. In Section 5 we construct ﬁrst a C*-algebra0

F generated by the Cuntz-Pimsner algebras{O } as described above. Then we show thatH V∈GV 0

withF we can construct a Hilbert C*-system in a natural way. We conclude this article with

a short appendix restating some of the previous concrete results in terms of tensor categories of

Hilbert bimodules.

1.2 Outlook

Doplicher and Roberts show in the setting of the new duality of compact groups that essentially

every concrete dual of a compact groupG may be realized in a natural way within a C*-algebra

F, which is the C*-tensor product of Cuntz algebras (cf. [11]). Under additional assumptions it

is shown that the corresponding ﬁxed point algebra is simple and therefore must have a trivial

center. The results in this paper generalize this situation. In fact, one may also realize concrete

αgroup dualswithin the C*-algebraF :=Z⋊ G constructed in the Main Theorem 3, wherenow0

αthe corresponding ﬁxed point algebra has a nontrivial centerZ. IfZ =C , thenZ⋊ G reduces

to the tensor product of Cuntz algebras labeled by the ﬁnite dimensional representations of the

compact group contained inG.

As mentioned above our models provide natural examples of tensor C*-categories with a non-

simple unit. These structureshave been studiedrecently in several problems inmathematics and

mathematical physics: inthegeneral context of 2-categories (see [33] anreferences cited therein),

in the study of group duality and vector bundles [30, 31], and in the context of superselection

theory in the presence of quantum constraints [2]. Finally, algebras of quantum observables with

nontrivialcenterZ alsoappearinlowerdimensionalquantumﬁeldtheorieswithbraidingsymme-

try (see e.g. [15], [23,§2]). In particular, in the latter reference the vacuum representation of the

global observable algebra is not faithful and maps central elements to scalars. In the mathemat-

5

1

1ical setting of this article, the analogue of the observable algebra is analyzed without making use

of Hilbert space representations that trivialize the center. Moreover, the representation theory

of a compact group is described by endomorphisms (i.e. the analogue of superselection sectors)

that preserve the center. It is clear that our models do not ﬁt completely in the frame given by

lower dimensional quantum ﬁeld theories, since, for example, we do not use any braiding symme-

try. Nevertheless, we hope that some pieces of the analysis considered here can also be applied.

E.g. the generalization of the notion of irreducible objects and the analysis of their restriction to

the centerZ that in our context led to the deﬁnition of the chain group or the importance of

Cuntz-Pimsner algebras associated toZ-bimodules.

2 Hilbert C*-systems and the chain group

For convenience of the reader we recall the main deﬁnitions and results concerning Hilbert C*-

systems that will be used later in the construction of the examples. We will also introduce the

notion of the chain group associated to a compact group which will be crucial in the speciﬁcation

of the examples. For a more detailed analysis of Hilbert C*-systems we refer to [5, Sections 2

and 3] and [6, Chapter 10]).

2.1 Hilbert C*-systems

Roughly speaking, a Hilbert C*-system is a special type of C*-dynamical system{F,G,β} that,

in addition, contains the information of the representation category of the compact groupG.F

denotes a unital C*-algebra and β:G∋ g →β ∈ AutF is a pointwise norm-continuous mor-g

phism. Moreover, therepresentations ofG are carriedby the algebraic Hilbertspaces, i.e. Hilbert

∗spacesH⊂F, where the scalar producth·,·i ofH is given byhA,Bi := A B for A, B∈H.

(Algebraic Hilbert spaces are also called in the literature Hilbert spaces in C*-algebras.) Hence-

forth, we consider only ﬁnite-dimensional algebraic Hilbert spaces. The support suppH ofH isPd ∗deﬁned by suppH := ψ ψ , where{ψ |j =1,..., d} is any orthonormal basis ofH.j jj=1 j

TogiveaprecisedeﬁnitionofaHilbertC*-systemweneedtointroducethespectralprojections:

bfor D∈G (the dual ofG) its spectral projection Π ∈L(F) is deﬁned byD

Z

Π (F) := χ (g)β (F)dg for all F∈F, (6)D D g

G

where χ (g) := dimD·TrU (g), U ∈D,D D D

is the so-called modiﬁed character of the class D and dg is the normalized Haar measure of the

bcompact groupG. For the trivial representation ι∈G, we put

n o

A:= ΠF = F∈F|g(F) =F, g∈G ,ι

Gi.e.A =F is the ﬁxed-point algebra inF w.r.t.G. We denote byZ =Z(A) the center ofA,

which we assume to be nontrivial.

Deﬁnition 2.1 The C*-dynamical system{F,G,β} with compact groupG is called a Hilbert

bC*-system if it has full Hilbert spectrum, i.e. for each D∈G there is a β-stable Hilbert space

H ⊂ Π F, with support and the unitary representation β H is in the equivalence classD D G D

bD∈G. A Hilbert C*-system is called minimal if

′A∩F =Z,

GwhereZ is the center of the ﬁxed-point algebraA:=F .

6

1

1Since we can identifyG withβ ⊆AutF we will often denote the Hilbert C*-system simply byG

{F,G}.

Remark 2.2 Some families of examples of minimal Hilbert C*-systems with ﬁxed point algebra

A ⊗Z, whereA has trivial center, were constructed indirectly in [5, Section 6]. Some explicitC C

examples in the context of the CAR/CCR-algebra with an Abelian group are given in [1] and [2,

Section V].

To eachG-invariant algebraic Hilbert spaceH⊂F there is assigned a corresponding inner

endomorphismρ ∈EndF given byH

d(H)X

∗ρ (F) := ψ Fψ ,H j j

j=1

where{ψ j = 1,..., d(H)} is any orthonormal basis ofH. It is easy to see thatA is stablej

under the inner endomorphism ρ. We call canonical endomorphism the restriction of ρ toH

A, i.e. ρ A∈ EndA. By abuse of notation we will also denote it simply by ρ . LetZ denoteH H

the center ofA; we say that an endomorphismρ is irreducible if

(ρ,ρ) =ρ(Z).

In the nontrivial center situation canonical endomorphisms do not characterize the algebraic

Hilbert spaces anymore. In fact, the natural generalization in this context is the following notion

offreeHilbertZ-bimodule: letHbeaG-invariant algebraicHilbertspaceinF ofﬁnitedimension

d. Then we deﬁne ﬁrst the free rightZ-moduleH by extension

d X

H :=HZ = ψ Z |Z ∈Z , (7)i i i

i=1

dwhere Ψ :={ψ} is an orthonormal basis inH. In other words, the set Ψ becomes a modulei i=1

basis ofH and dim H =d. For H ,H ∈H putZ 1 2

∗H ,H :=H H ∈Z.1 2 21H

Then, {H, ·,· } is a Hilbert right Z-module or a Hilbert Z-module, for short. Now the

H

canonical endomorphism can be also written as

dX

∗ρ (A) := ϕ Aϕ , A∈A,H j j

j=1

dwhere{ϕ} is any orthonormal basis of theZ-moduleH. Hence we actually haveρ =ρ andi H Hi=1

it is easy to show that

H∈H iﬀ HA=ρ (A)H.H

In other words ρ characterizes uniquely the HilbertZ-module H. Moreover, since for anyH

canonical endomorphism ρ = ρ we have thatZ ⊂ (ρ,ρ), it is easy to see that there is aH

canonical left action ofZ on H. Concretely, there is a natural *-homomorphismZ →L(H),

whereL(H) is the set ofZ-module morphisms (see [3, Sections 3 and 4] for more details). Hence

H becomes aZ-bimodule.

We conclude stating the isomorphism between the category of canonical endomorphisms and

the corresponding category of freeZ-bimodules (cf. [5, Proposition 4.4] and [3, Section 4]).

7Proposition 2.3 Let{F,G} be a given minimal Hilbert C*-system, where the ﬁxed point algebra

A has centerZ. Then the categoryT of all canonical endomorphisms of{F,G} is isomorphic to

the subcategoryM of the category of free HilbertZ-bimodules with objectsH =HZ, whereH isG

aG-invariant algebraic Hilbert space with suppH = , and the arrows given by the correspondingσ

G-invariant module morphismsL(H ,H ;G).1 2

The bijection of objects is given by ρ ↔H =HZ which satisﬁes the conditionsH

ρ =(AdV)◦ρ +(AdW)◦ρ ←→ H =VH +WHH 1 2 1 2

ρ ◦ρ ←→ H ·H ,1 2 1 2

∗ ∗where V,W∈A are isometries with VV +WW = and the latter product is the inner tensor

product of the HilbertZ-modules w.r.t. the *-homomorphism Z → L(H ). The bijection on2

arrows is deﬁned by

X

∗J : L(H ,H ;G)→(ρ ,ρ ) with J(T) := ψ Z ϕ .1 2 1 2 j j,k k

j,k

Here{ψ} , {ϕ} are orthonormal basis of H ,H , respectively, and (Z ) is the matrix ofj j k k 2 1 j,k j,k

the rightZ-linear operator T fromH toH which intertwines theG-actions.1 2

The preceding proposition shows that the canonical endomorphisms uniquely determine the

correspondingZ-bimodules, but not the choice of the generating algebraic Hilbert spaces. The

assumption of the minimality condition in Deﬁnition 2.1 is crucial here. From the point of view

of theZ-bimodules it is natural to consider next the following property of Hilbert C*-systems:

the existence of a special choice of algebraic Hilbert spaces within the modules that deﬁne the

canonical endomorphisms and which is compatible with products.

Deﬁnition 2.4 A Hilbert C*-system{F,G} is called regular if there is an assignmentT ∋σ→

H , whereH is aG-invariant algebraic Hilbert space with suppH = and σ =ρ (i.e. σ isσ σ σ Hσ

the canonical endomorphism of the algebraic Hilbert spaceH ), which is compatible with products:σ

σ◦τ →H·H .σ τ

Remark 2.5 InaminimalHilbertC*-systemregularitymeansthatthereisa“generating”Hilbert

spaceH ⊂ H for each τ (with H =HZ) such that the compatibility relation for productsτ τ τ τ

statedinDeﬁnition2.4holds. IfaHilbertC*-systemisminimalandZ =C thenitisnecessarily

regular.

2.2 The chain group

In the present section we recall the main motivations and deﬁnitions concerning the chain group

associated with a compact groupG. For proofs and more details see [5, Section 5] (see also [25]).

One of the fundamental new aspects of superselection theory with a nontrivial centerZ is the

fact that irreducible canonical endomorphisms act as (nontrivial) automorphisms onZ. In fact,

blet D∈G (the dual ofG) and denote by ρ := ρ the corresponding irreducible canonicalD HD

endomorphism. Then, to any class D we can associate the following automorphism onZ:

bG∋D →α :=ρ Z∈AutZ. (8)D D

bThis observation allows one to introduce a natural equivalence relation in the dualG which,

′ broughly speaking, relates elements D,D ∈G if there is a “chain of tensor products” of elements

′binG containing D andD (see Theorem 2.10 and Remark 2.11 below).

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