Lecture Notes on C-algebras - UVic.ca

1 Lecture Notes onC -algebrasIan F. PutnamJanuary 3, 20192 Contents1 Basics ofC Definition .. Examples .. Spectrum .. CommutativeC -algebras .. Further consequences of theC -condition .. Positivity .. Finite-dimensionalC -algebras .. Non-unitalC -algebras .. Ideals and quotients .. Traces .. Representations .. The GNS construction .. von Neumann algebras .. 602 GroupC Group representations .. Group algebras .. Finite groups .. TheC - algebra of a discrete group .. Abelian groups .. The infinite dihedral group.

2 The groupF2.. Locally compact groups .. 913 GroupoidC Groupoids .. Topological groupoids .. TheC - algebra of an etale groupoid .. fundamental lemma .. -algebraCc(G) .. left regular representation .. (G) andC r(G) .. The structure of groupoidC -algebras .. expectation ontoC(G0) .. on groupoidC -algebras .. in groupoidC -algebras .. AF-algebras .. 135 PrefaceIt is a great pleasure to thank those who have made helpful suggestions andfound errors: Christian Skau, Charles Starling and Maria 1 Basics ofC DefinitionWe begin with the definition of aC -algebraAis a (non-empty) set with the followingalgebraic operations:1.

3 Addition, which is commutative and associative2. multiplication, which is associative3. multiplication by complex scalars4. an involutiona7 a (that is,(a ) =a, for allainA)Both types of multiplication distribute over addition. Fora,binA, we have(ab) =b a . The involution is conjugate linear; that is, fora,binAand inC, we have( a+b) = a +b . Fora,binAand , inC, we have (ab) = ( a)b=a( b)and( )a= ( a).In addition,Ahas a norm in which it is a Banach algebra ; that is, a =| | a , a+b a + b , ab a b ,for alla,binAand inC, andAis complete in the metricd(a,b) = a b .Finally, for allainA, we have a a = a 1. BASICS OFC -ALGEBRASVery simply,Ahas an algebraic structure and a topological structurecoming from a norm.

4 The condition thatAbe a Banach algebra expressesa compatibility between these structures. The final condition, usually re-ferred to as theC -condition, may seem slightly mysterious, but it is a verystrong link between the algebraic and topological structures, as we shall is probably also worth mentioning two items which , the algebra need not have a unit for the multiplication. If it doeshave a unit, we write it as 1 or 1 Aand say thatAis unital. Secondly, themultiplication isnotnecessarily commutative. That is, it is not generally thecase thatab=ba, for alla,b. The first of these two issues turns out to be arelatively minor one (which will be dealt with in Section ).

5 The latter isessential and, in many ways, is the heart of the might have expected an axiom stating that the involution is isomet-ric. In fact, it is a simple consequence of the ones given, particularly theC an element of aC -algebraA, then a = a . a Banach algebra a 2= a a a a and so a a .Replacingawitha then yields the , we introduce some terminology for elements in aC - algebra . For agivenainA, the elementa is usually called theadjointofa. The first termin the following definition is then rather obvious. The second is much lessso, but is used for historical reasons from operator theory. The remainingterms all have a geometric flavour.

6 If one considers the elements inB(H),operators on a Hilbert space, each of these purely algebraic terms can be givenan equivalent formulation in geometric terms of the action of the operatoron the Hilbert aC An elementaisself-adjointifa = An elementaisnormalifa a=aa .3. An elementpis aprojectionifp2=p=p ; that is,pis a EXAMPLES94. Assuming thatAis unital, an elementuis aunitaryifu u= 1 =uu ;that is,uis invertible andu 1=u .5. Assuming thatAis unital, an elementuis anisometryifu u= An elementuis apartial isometryifu uis a An elementaispositiveif it may be writtena=b b, for this case, we often writea 0for ExamplesExample , the complex numbers.

7 More than just an example, it isthe a complex Hilbert space with inner product denoted< , >. The collection of bounded linear operators onH, denoted byB(H),is aC - algebra . The linear structure is clear. The product is by compositionof operators. The operation is the adjoint; for any operatoraonH, itsadjoint is defined by the equation< a , >=< ,a >, for all and inH. Finally, the norm is given by a = sup{ a | H, 1},for anyainB(H).Example any positive integer, we letMn(C)denote the set ofn ncomplex matrices. It is aC - algebra using the usual algebraic operationsfor matrices. The operation is to take the transpose of the matrix and thentake complex conjugates of all its entries.

8 For the norm, we must resort backto the same definition as our last example a = sup{ a 2| Cn, 2 1},where 2is the usual`2-norm course, this example is a special case of the last usingH=Cn, andusing a fixed basis to represent linear transformations as a compact Hausdorff space and considerC(X) ={f:X C|fcontinuous}.10 CHAPTER 1. BASICS OFC -ALGEBRASThe algebraic operations of addition, scalar multiplication and multiplicationare all point-wise. The is point-wise complex conjugation. The norm is theusual supremum norm f = sup{|f(x)||x X}for anyfinC(X). This particular examples has the two additional featuresthatC(X)is both unital and this slightly, letXbe a locally compact Hausdorff space andconsiderC0(X) ={f:X C|fcontinuous, vanishing at infinity}.

9 Recall that a functionfis said to vanish at infinity if, for every >0, thereis a compact setKsuch that|f(x)|< , for allxinX\K. The algebraicoperations and the norm are done in exactly the same way as the case example is also commutative, but is unital if and only ifXis compact(in which case it is the same asC(X)).Example thatA BareC -algebras, we form their directsumA B={(a,b)|a A,b B}.The algebraic operations are all performed coordinate-wise and the norm isgiven by (a,b) = max{ a , b },for is an obvious extension of this notion to finite direct sums. Also,ifAn,n 1is a sequence ofC -algebras, their direct sum is defined as n=1An={(a1,a2.)}

10 |an An,for alln,limn an = 0}.Aside from noting the condition above on the norms, there is not much elseto consider it as a - algebra with coordinate-wise addition, multiplication and conjugation. (In other words, it isC({1,2}).)1. Prove that with the norm ( 1, 2) =| 1|+| 2|AisnotaC SPECTRUM112. Prove that ( 1, 2) = max{| 1|,| 2|}is theonlynorm which makesAinto aC - algebra . (Hint: Proceed asfollows. First prove that, in anyC - algebra norm,(1,0),(1,1),(0,1)all have norm one. Secondly, show that for any( 1, 2)inA, there isa unitaryusuch thatu( 1, 2) = (| 1|,| 2|). From this, deduce that ( 1, 2) = (| 1|,| 2|) . Next, show that ifa,bhave norm one and0 t 1, then ta+ (1 t)b 1.)