Spectrum Operators

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Spectrum of Discrete Schrodinger Operators

by

@FlL'<in 'Wei

A thesis submitted to the School of Graduate Studies in partial fulfillment of the requirements for the degree of

St. John's

Master of Science

Department of Mathematics and Statistics

Memorial University of Newfoundland October 2013

Newfoundland

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Abstract

The purpose of this thesis is to explore the spectra of discrete Schroclinger operators of a special form. \Ne consider the specifi way to identify the Schrodinger operator Hx₀ in our model with an element in the crossed product C(X) ^><l0 Z which is generated by the commutative unital C* -algebra C(X) and countable discrete group Z via the action a ^wit^h^respect^t^o^theuniversal norm. We show that the crossed product C(Y) ^><l0Z, where Y = Orbr.p(x0), is i omorphic to the concrete algebra O"x₀(C(X) ^><l0 Z), where O"x₀ is an integrated representation of C(X) ^><l0 Z induced by point evaluation fLx₀^. As a orollary, we conclude that the pectra of the Schrodinger operators Hx₀ and Hx, are the ·ame 'vvhen the closures of the orbits of the two points x₀, x1 E X are the same, and apply this result to ome special kinds of systems. Aft r considering the classification of the ·pectra of eli crete chrodinger operators, we give some examples to show the calctdation of the spectrum by using K-theory.

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Acknowledgements

I am grateful to my supervisor, Dr. Zhuang Niu, and my co-supervisor, Dr. Niikhail V. Kotchetov, for their professional guidance, patience and encouragement.

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Introduction

The Schrodinger equation was proposed by physicist Erwin Schrodinger in 1926.

There are two types of Schrodinger equations, time-dependent and time-independent.

The time-independent Schrodinger equation is used when dealing with stationary states, i.e., the states that do not change over time, so the wavefunction is a function of position. In the time-dependent Schrodinger equation, the wavefunction is a function of position and time.

\!Ve have that Kinetic Energy (T)

+

Potential Energy (V)=Total Energy (E) from classical mechanics. The Schrodinger equation uses this fundamental principle in terms of its wavefunction:

where 1/Jn is the wavefunction,

fi

is the Hamiltonian operator, and En is the nth energy eigenvalue corresponding to 1/Jn (solutions exist for the time-independent Schrodinger equation only for certain values of energy). In the time-independent Schrodinger equation, the Hamiltonian operator is equivalent to the total energy operator.

In this thesis, we consider the one-dimensional discrete Schrodinger operator H = Hxo of the form

( H xo 1/J) ( n) = 1/J ( n

+

1)

+

1/J ( n - 1 )

+

V ( n) 1/J ( n) , (0.0.1)

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on €²(/£), where the potential

v :

1£ -1 JR. is given by

with the point _.1::0in a compact metric space X , <p a homeomorphism of X onto itself and

f

a continuous function from X to JR..

The operator Hi. a self-adjoint bounded operator on €²(/£), and hence its spectrum is a non-empty compact set in the real line.

In the first chapter, the basic definitions of *-algebras, normcd algebras, Banach algebras and C* -algebras are reviewed; the equivalence of a topological system (X, <p) and a dynamical system (C(X ), a) is shown (see e.g. [29]), where C(X) is the C*- algebra of all continuous complex-valued functions on X and a i an automorphism of C(X) of the form (1.2.1). Moreover, given a single automorphi m a in Aut(C(X)), we know it gives rise to an action of the group 1£ on C(X ) (sec Definition 1.3.7) by an ^:= a¹¹^. In the following sections, we also denote by a this action of 1£ induced by a single automorphism a. In thi way, hence, we form a C*-dynamical system (C(X), 1£, a) (see Definition 1.3.10) and then obtain the crossed product C(X) ><Ia 1£

(see e.g. Theorem 1.4.1, [24], [30]). Considering the map ax0 = 11-xo ^><1 A, which is the representation (see Definition 1.3.13 and Equation (1.4.1)) of C(X ) ><~rp 1£ on the Hilbert ·pace €²(/£) corresponding to the covariant repre entation (ilxo, A) (see Definition 1.3.16 and Equations (1.4.3)) induced by a representation 11-xo of C(X) on C, we will identify the Schrodinger operator Hxo of the form (0.0.1) with the image of an element in the crossed product C(X) ><Ia 1£ under the representation axo·

In the second chapter, the properties of short xact sequences of C* -algebras are pre- sented and then used to show (see Theorem 2.2.1) that ax₀(C(X) ><laZ) is isomorphic to C(X) ^><10 1£ where Y = Orb'P(x_{0 ),}and 0Tbrp(x_{0 )} is the orbit of the point x₀in X

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under rp. As a consequence, the spectrum of the Schrodinger operator Hxo of the form (0.0.1) is determined by Y (see Corollary 2.2.6). Moreover, this result is applied to three kinds of dynamical systems: minimal systems, almost minimal systems and essentially minimal systems.

It is important to label gaps in the spectrum of the Schrodinger operator Hx₀ of the form (0.0.1), where a gap means a connected component in the s t lR \sp(Hx₀). In the last chapter, we review K₀-groups, K₁-groups and the Pimsner- Voiculescu sequence of a C*-algebra, which is the main tool to label gaps in the spectrum (see e.g. [3], [4], [6],[9], [10], [11], [13]. [17], [20], [21]). In the end, some special Schrodinger operators whose spectra are Cantor sets will be given as examples for the calculation of the spectrum (sec e.g. [5], [6], [7], [25]).

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Chapter 1 Realization of the Schrodinger operators in crossed products

1.1 Pre li m inaries to C * -alge bras

In this part, we will review the definition of different kinds of algebras and the spectrum of an clement in a C* -algebra.

Definition 1.1.1. An algebra (over C) is a vector space A endowed with a pmduct A x A ~ A, (a, b) Nab such that

(i) a(bc) = (ab)c for all a, b, c E A (associativity),

(ii) a(b +c)

=

ob ac and (b + c)a

=

ba + ca for all a. b, c E A (distributivity),

(iii) (oa)(j3b) = (oj3)(ab) for all a, (3 E C and a, bE A

(compatibility with scalar multiplication).

Definition 1.1.2. (a) A *-algebra is an algebra A provided with a map* : A~ A, aN a* such that, for all a. bE A and a E C,

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(i) (a+b)*=a*+b*,

(ii) (cw)* = aa*,

(iii) (ab)* = b*a*, (iv) (a*)* =a.

The mapping a t--t a* is called the involution.

(b) A normed algebra is an algebra A with a norm 11·11 : A--+ ~+ (with the convention that 0 belongs to ~+ ), a t--t llall such that llabll ::; llallllbll for all a, bE A.

(c) A Banach algebra is a normed algebr-a which is complete in its norm.

(d) A C*-algebra A is a Banach algebr-a which is at the same time a *-algebra such that the norm satisfies

(1.1.1)

for all a EA.

An algebra A is unital if it has a multiplicative identity, which will be denoted by e or eA. It follows from the condition (1.1.1) that llell = 1 for any nontrivial unital C* -algebra. A C* -algebra is said to be separ-able if it contains a countable dense subset.

A sub-C* -algebr-a of a C* -algebra A is a non-empty closed subset of A which is a

*-algebra vvith respect to the operations given on A.

Let A be a C* -algebra, and let F be a subset of A. The sub-C* -algebra of A gener-ated by F, denoted by C*(F), is the intersection of all sub-C*-algebras of A that contain F. We write C*(a1 , a2 ... , an) instead of C*( {a_{1 ,}a2 ... , a".}), when a₁, a2 , ... , an EA.

Theorem 1.1.3. Let A be a C*-algebra. Then the involution is isometric, i.e., llall =

II

a*

II·

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Proof. By the second and third axioms for C*-algebra A, we know that lla*all =

llall²^::::; II a* IIIIa II for any element a E A. If II all = 0, then a = a* = 0, so II all = II a* ¹^1.

If II all > 0, then we obtain II all ::::; II a* II· On the other hand, since (a*)* = a, in the similar way, we obtain that II a* II ::::; II all· Thus, II all = II a* II for all a E A. D

An element a in a C* -algebra A is called

(i) selj~adjoint if a* = a;

(ii) a projection if a = a* = a²;

(iii) normal if a*a = aa*;

(iv) (if A is unital) unitary if a*a = eA = aa*,

( v) (if A is unital) invertible if there is an element b in A such that ab = ba = eA.

If A is a C* -algebra, then the unitalization of A is the unique (up to canonical isometric

*-isomorphism) C* -algebra

A

with multiplicative unit which contains A as a closed, two-sided ideal of linear codimension one. The algebra A can be constructed as follows (see e.g. [22, Page 5]): If A is contained in a unital C* -algebra B whose unit e8 does not belong to A, then

A

is equal (or isomorphic) to the sub-C*-algebra A+ C · ea of B. If A has a unit eA, and if ex is the unit in

A,

then f =ex- eA is a projection in A, and

A =

{a+ af: a E A, a E C}.

Definition 1.1.4. If A is any unital C* -algebra, the spectrum of an element a E A

is the set

sp(a) := {A E C I AeA- a is not invertible in A}.

If A is not unital, then sp( a) is defined to be the spectrum of a in the unitalization

A.

(It follows that if A is non-unital, then 0 E sp(a) for every a E A).

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The spectral radius of a is

r(a) = sup{[A[: A E sp(a)}.

The spectrum sp(a) is a nonempty compact subset of C, and the spectral radius satisfies r(a) ~ [[a[[ (see e.g. [16, Lemma 1.2.4]).

The following theorem can be found in [23, Theorem 18.9].

Theorem 1.1.5 (Spectral Radius Formula). For- ever-y x E A,

(1.1.2)

For a normal element a E A, Equation (1.1.2) reduces to T(a) = [[a[[. By (1.1.1), we have that [[a[[ = ~ = Jr-(a*a), or, equivalently,

[[a[[²= sup{ A E C [A A -a*a is not invertible in A}.

This implies that the norm in a C* -algebra is uniquely determined by product and involution.

Definition 1.1.6. An element a in a C*-algebm A is positive if it is nor-mal and sp(a) ^<;;:;JR+. We wW write a 2:: 0 to indicate that a is positive in this sense.

The set of positive elements in A is denoted by A+. An element a in A is positive if and only if a= x*x for some x E A (see e.g. [22, page 6]).

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1.2 The equivalence of a dynamical system and a topological system

A classical dynamical system consists of a compact Hausdorff space X and a homeomorphism cp of X onto itself.

Note that the Schrodinger operator Hxo of the form (0.0.1) is a bounded self-adjoint operator on £²(/Z) and is determined by the topological system (X, cp, x0 ).

Given a topological system (X, cp), consider the C* -algebra C(X) of all the continuous complex-valued functions on a compact Hausdorff space X. Define

(1.2.1)

Obviously, a E Aut(C(X)). Let us show, conversely, that a dynamical system (C(X), a), a ^EAut(C(X)), induces a corresponding topological system (X, cp) such that ( 1. 2.1) holds.

Definition 1.2.1. Let A be an algebra over C. A multiplicative linear functional is a nonzero linear- functional cp : A ---+ C such that

cp(xy) = cp(x)cp(y), 't!x, yEA.

Multiplicative linear-functionals ar-e also called characters of A.

Definition 1.2.2. Let A be a Banach algebra. A left ideal (right ideal) of A is a closed linear- subalgebra I s;;; A for which a E I implies that ba E I (ab E I) for all bE A.

An ideal in A is a subspace that is both a left and a right ideal (i.e., a two-sided ideal). If I =1- A, I is a proper-ideal. Maximal ideals ar-e proper ideals which are not contained

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in any larger proper ideals.

Definition 1.2.3. Let <p : A ---+ B be a map between C* -algebras A and B. The map

<p : A ---+ B is called a *-homomorphism if it is a linear and multiplicative map which satisfies <p(a*) = <p(a)* for all a E A; the map <p : A---+ B is called a *-isomorphism if it is a *-homomorphism which is bijective; the map <p : A ---+ B is called an isometric

*-isomorphism if it is a *-isomoTphism which is isometric.

Theorem 1.2.4 ([15, Theorem 1.5.15]). EveTy *-homomorphism <p : A ---+ B of C* -algebras is noTm-decreasing, and cp( A) is always a sub-C* -algebra of B. If <p is injective. then it is an isometry.

By Theorem 1.2.4. we know that the kernel of a *-homomorphi m <p A ---+ B of C* -algebras is closed in A and the image i · closed in B.

The kernel of a *-homomorphism of C* -algebras 1r : A ---+ B is an ideal in A. Con- versely, for any ideal I ~ A, A/ I i a C* -algebra and I i the kernel of the quotient map from A to A/ I.

Let H be a Hilbert space. Denote by B(H) the C* -algebra of all bounded linear operators on H and by U(H) the group of all unitary operators on H.

The following theorem is known as Gelfand-Naimark-Segal Th orem that can be found

in e.g. [22, Theorem 1.1.3], in which the GNS-construction is u ·ed to construct a Hilbert space H for a given C* -algebra A such that A can be i ometrically embedded into B(H) as a sub-C*-algebra.

Theorem 1.2.5 (Gelfand-Naimark-Segal Theorem). For each C*-algebra A there ex- ist a HilbeTt space H and an isometric *-homomorphism <p from A into B(H). If A is separable, then H can be chosen to be a separable Hilbert space.

The following Gelfand-Naimark Theorem (see e.g. [22, Theorem 1.2.3]) gives a uni- versal model for any commutative C* -algebra.

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Theorem 1.2.6 (Gelfand-Naimark Theorem). Every commutative C*-algebra is iso- metr-ically *-isomor-phic to the C*-algebra C0(X) for- some locally compact Hausdorff space X.

Recall that C₀(X) is the C* -algebra of all continuous functions

f :

X --t CC that vanish at infinity: for each E > 0 there is a compact subset ]( of X such that

IJ(x)l :::;

E for all x E X \ K. The norm on C₀(X) is the supremum norm. If X is compact, then Co(X) = C(X).

In addition to Gelfand-Naimark Theorem, we have the following properties (see [22, Page 7]):

(i) The C* -algebra C₀(X) is unital if and only if X is compact.

(ii) X and Y are homeomorphic if and only if C₀(X) and C₀(Y) are isometrically

*-isomorphic.

(iii) There is a bijective correspondence between open sets of X and ideals in C₀(X), which is set up as follows: the ideal corresponding to the open subset U of X is the set {f E C₀(X)

I

f vanishes on uc} ~ C0(U).

In the following, denote by 6 (A) the set of all nonzero characters of a commutative Banach algebra A. It is a compact Hausdorff space with respect to the following topology: A neighborhood basis at rp₀E 6 (A) is given by the collection of sets

where ^E> 0, n E .N, and x1, · · · , Xn are arbitrary elements of A. This topology on 6(A) is called the Gelfand topology.

If A has an identity, then 6 (A) is also called the maximal ideal space of A.

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Definition 1.2.7 (Gelfand transform). Suppose that A is a commutative Banach algebra with 6.(A) nonempty. Then the Gelfand transform of a E A is the function

a :

6.(A) -t

cc

given by

a

(h) = h (a) .

The space X corresponding to the commutative C* -algebra A in Gelfand- aimark Theorem can be chosen as the space 6.(A) with Gelfand topology, and then the Gelfand transform provides an isometric *-isomorphism A-t C0(X).

Let X be a compact Hausdorff space. By Gelfand-Naimark Theorem, we have that C(X) is isomorphic to C(6.(C(X))), hence X is homeomorphic to 6.(C(X)). Explic-

itly, for any x E X,

fJx : C(X) -t CC, !Jx(f) = f(x)

is a character of C(X), and the mapping x f-t fJx is a homeomorphism X -t 6.(C(X)).

Definition 1.2.8. The function fJ1: is called a point evaluation of x E X.

Any a E Aut(C(X)) defines a permutation

a:

^6.^(C(^X)⁾-t 6.(C(X)) as follows:

~ ~ ~ o a for any~ E 6.(C(X)).

Since the set 6.(C(X)) coincides with the set of all point evaluations of C(X), i.e.,

6.(C(X)) = ^{fJx ^:^X E X}, the permutation

a

induces a permutation <p of X by

~Lx ~ ^fJx^o^a=^:^fJ'P-^l(^x)^· ^Thus,^we^have^thata(!) =

f

o <p- ¹for all

f

E C(X), and

then one can show that <p is a homeomorphism and uniquely determined by a.

As discussed above, a topological system (X, <p) corresponds to a dynamical system (C(X), a) uniquely, which implies that studying a topological system (X, cp) is equivalent to investigating the corresponding dynamical system ( C (X), a).

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In the following sections, we always denote by a the automorphism of C(X) corresponding to the homeomorphism <p : X ~ X.

1.3 C * -dynamical systems

Now we review some basic definitions and properties related to the concept of a C*- dynamical system.

Definition 1.3.1. A topological group is a group ( G, ·) together with a topology ^T such that

(i) points are closed in (G, T), and

(ii) the map G x G ~ G, (s, r) ~----+ sr-¹is continuous.

Example 1.3.2. Any group G equipped with the discrete topology is a topological

group, joT example, ^zn. The groups ^]Rn and ^']fn= {z = (zl, ... , ^Zn⁾ E e n I lzil =

1 for all i}, are topological groups in their usual topologies.

Definition 1.3.3. A (locally) compact group is a topological group for which the under-lying topology is (locally) compact.

Example 1.3.4. Any discrete group is locally compact; JR.n is locally compact; 'Ifn is compact.

Definition 1.3.5 (Group actions on sets). A group G acts on the left on a set X if there is a map G x X ~ X,

(s, x) ~----+ s · x (1.3.1)

such that

e · X = X and S · ( T · X) = ST · X

for all s, T E G and all x E X, where e is the identity element of G.

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If G is a topological group and X is a topological space, then we say the action is continuous if the mapping (1.3.1) is continuous. In this case, X is called a left G-space and the pair ( G, X ) is called a transformation group. If both G and X are locally compact, then (G, X) is called a locally compact transformation group.

Example 1.3.6. Let <p be a homeomorphism of a space X onto itself. Then Z acts on X by n · x := <pn(x), and (Z, X ) is a transformation group.

Definition 1.3.7 (Group actions on C*-algebras). Let G be a topological group and let A be a C* -algebra. A map a : G -+ Aut( A), t 1---7 at is called an action of G on A if

(1) for any t, s E G, we have at o a5 = at5 ,

(2) for any a ^EA, the map G-+ A, t 1---7 at(a) is continuous.

Let ( G, X ) be a locally compact transformation group. Then for each s E G, the map

x 1---7 s ·x is a homeomorphism of X. Therefore we obtain a homomorphism

a : G --t Aut(C₀(X)) (1.3.2)

defined by

as(J)(x) ^:= f(s-¹^·x).

Indeed,

(1.3.3)

since

The set Aut(A) of automorphisms of a C*-algebra A is a group under composition.

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Definition 1.3.8. The point-norm topology on Aut( A) is the topology of point-wise convergence of functions on A; thus ai -t a in the point-norm topology if and only if llai(a)- a(a)ll -t 0 for all a EA.

Lemma 1.3.9 ([30, Lemma 2.5]). Suppose that (G, X) is a locally compact trans- formation group and that Aut(C₀(X)) is given the point-norm topology. Then the

associated homomorphism (1.3.2) of G into Aut(C₀(X)) is continuous.

Definition 1.3.10. A C*-dynamical system is a triple (A,G,a) consisting of a C*- algebra A, a locally compact gr·oup G and an action a of G on A.

Example 1.3.11. For any a E Aut(A), we can define an action of Z on A by n · a = an (a). Such a dynamical system (A, Z, a), with A = C (X), already appeared in Section 1. 2.

Equation (1.3.3) and Lemma 1.3.9 tell us that the homomorphism (1.3.2) is an action of the topological group G on the C*-algebra C₀(X), which means that there is a C*-dynamical system (C₀(X), G, a) induced by a transformation group (G, X). The following proposition tells us that, conversely, a transformation group ( G, X) can be induced by a C*-dynamical system (C0(X), G, a).

Proposition 1.3.12 ([30, Lemma 2.7]). Suppose that (C₀(X), G, a) is a C* -dynamical system (with X locally compact). Then there is a transformation group (G, X) such that

as(f)(x) = f(s-¹· x).

Definition 1.3.13 (A representation of a C* -algebra). A representation 1r of a C*- algebra A on a Hilbert space 7-i. is a *-homomorphism of A into B(H). A representa- tion ^1ris nondegenerate if the set

{1r(a)~

I

a E A, ~ E H}

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is dense in H. We say that a r-epr-esentation 1r is faithful if it is an injective map.

Definition 1.3.14 (A unitary representation of a group). A unitary representation of a gr-oup G on a Hilber-t space H is a homomor-phism U : G -+ U(H) such that the map t f--7 Ut ( x) is continuous for- ev r-y vector- x E H.

For a locally compact group G with a left Haar measure p, (see [30, S ction 1.3]), denot by L²(G) the Hill crt space of equivalence classes of Borel measurable functions

f

^on

G with complex values such that

fc

lf (s)l²dp,(s) < oo; denote by L (G) the Banach space of all ssentially bounded functions G -+ C with re pect to the Haar measure.

Iore generally. let H be a complex Hilbert space. We define L²(G, H) to be the space of equival ucc classes of Borel measurable functions

f

on G with values in H. If G is a countable cliscr tc group, then the Haar measure is just counting measure, so we can think of elements of L²(G) and L²(G, H) as sequences indexed by G. The traditional notation for L²(G) and L²(G, H) in this case is

e

²^(G)^and^€²^(G,^H^),^resp^ectively.

Example 1.3.15. Let G be a locally compact gr-oup and consider- L²(G). Then, for-

T E G, the translation >.(r) is defined by

>.(r-).f(s) := .f(r--¹s),

f

^EL²(G).

Since translation >.(T) is a unitar-y operator- on L²(G), it follow that >. : G -+

U(L²(G)) is a Tepr-esentation of G. It is called the left regular representation. MoTe generally. we can define a left Tegular- r-epr-esentation ofG on L²(G, H) for- any Hilber-t space H.

Definition 1.3.16 (Covariant repre ·entation). Suppose that (A, G, a:) is a C* -dynamical system and that H is a HilbeTt space. Then a covariant representation of (A, G, a:) into B(H) is a paiT (1r, U) consisting of a r-epr-esentation 1r: A -+ B(H) and a unitar-y

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representation U : G----+ U(H) on the same Hilbert space such that

U51r(a)U; = 1r(a5(a)) for all a E A, s E G.

Example 1.3.17. Covariant representations of a dynamical system (C, G, triv) coT- respond to unitaTy TepTesentations of G.

1.4 Construction of crossed product s

In the following, we will concentrate on a C*-dynamical system (A, G, a), where A is

a unital separable C* -algebra and G is a countabl discrete group.

Consider the space Cc( G, A) of continuous functions from G to A with compact sup- port, i.e.,

Cc(G, A)= {f =

L

âtV^,^tÎâtEA for all t E G, and at -!- 0 for only finitely many t E G},

LEG

where {ut: t E G} is the basis of the space Cc(G, A) given by

{

eA, if t = s,

Ut(s) =

0, ift-f-s.

Instead of the usual point-wise multiplication, we define the multiplication of Cc( G, A) by the formal rules

Thu we obtain the twisted convolution product:

fg = L (Latat(bt-ls))us E Cc(G,A).

sEC LEG

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The involution is determined by the rule u~ = 'Lit-^I^,so

j* = (Lat'Lit)* ⁼ L at(a;-t)'Lit E Cc(G,A).

LEG tEG

One can check that the space Cc(G, A) endowed with the multiplication and involution as above is a *-algebra.

Note that a covariant representation (1r, U) of the C*-dynamical system (A, G, a) induces a *-representation of Cc(G, A) by

CJ(f) = CJ ( L at 'Lit) = L 1r( at) Ut.

tEC tEC

(1.4.1)

Indeed,

tEC tEG sEC

and

CJ(f)CJ(g) = L L 1f(at)Ut1f(bu)Uu = L ( L 1f(atat(bt-ts))) Us= CJ(fg).

tEC uEG sEC tEC

This representation CJ is denoted by 7f Xl U and called the integrated representation induced by (1r,U).

Conversely, when A is unital, a *-representation CJ of Cc(G, A) yields a covariant representation ( 7f, U) of (A, G, a) as follows: