Table Of Content

ANALOGS OF CUNTZ ALGEBRAS ON Lp SPACES N.CHRISTOPHERPHILLIPS 2 1 0 Abstract. Ford=2,3,...andp∈[1,∞),wedefineaclassofrepresentations 2 ρoftheLeavittalgebraLd onspaces oftheformLp(X,µ),whichwecallthe spatial representations. We prove that for fixed d and p, the Banach algebra n a Odp = ρ(Ld) is the same for all spatial representations ρ. When p = 2, we J recovertheusualCuntzalgebraOd.Wegiveanumberofequivalentconditions forarepresentationtobespatial. Weshowthatfordistinctp1,p2∈[1,∞)and 0 arbitrary d1,d2 ∈ {2,3,...}, there is no nonzero continuous homomorphism 2 fromOp1 toOp2. d1 d2 ] A F ThealgebrasthatwecalltheLeavittalgebrasL (seeDefinition1.1)areaspecial d h. case of algebras introduced in characteristic 2 by Leavitt [20], and generalized to t arbitrary ground fields in [21]. The Cuntz algebra O , introduced in [9], can be a d defined as the norm closure of the range of a *-representation of L on a Hilbert m d space. (Cuntz did not define O this way.) The Cuntz algebras have turned out d [ to be one of the most fundamental families of examples of C*-algebras. Leavitt 1 algebras were long obscure, but they have recently attracted renewed attention. v In this paper we study the analogs of Cuntz algebras on Lp spaces. That is, we 6 consider the norm closure of the range of a representation of L on a space of the 9 d form Lp(X,µ). It turns out that there is a rich theory of such algebras, of which 1 4 we exhibit the beginning in this paper. . Our main results are as follows. There are many possible Lp analogs of Cuntz 1 0 algebras (although we mostly do not know for sure that they really are essentially 2 different), but there is a natural class of such algebras, namely those that come 1 from what we call spatial representations (Definition 7.4(2)). Spatial representa- v: tions are those for which the standard generators form an Lp analog of what is i called a row isometry in multivariable operator theory on Hilbert space. (The sur- X vey article [11] emphasizes the more general row contractions on Hilbert spaces, r but row isometries also play a significant role. See especially Section 6.2 of [11].) a We give a number of rather different equivalent conditions for a representation to be spatial—further evidence that this is a natural class. For fixed p∈[1,∞)\{2}, we prove a uniqueness theorem: the Banach algebras coming from any two spatial representations are isometrically isomorphic via an isomorphism which sends the standard generators to the standard generators. We call the Banach algebra obtained this way Op. The usual Cuntz algebra is O2. We further obtain a strong d d dependence on p. Specifically, for p 6= p and any d ,d ∈ {2,3,...}, there is no 1 2 1 2 nonzero continuous homomorphism from Op1 to Op2. d1 d2 Date:19January2012. 2000 Mathematics Subject Classification. Primary46H05,47L10; Secondary46H35. This material is based upon work supported by the US National Science Foundation under GrantsDMS-0701076andDMS-1101742. ItwasalsopartiallysupportedbytheCentredeRecerca Matema`tica(Barcelona)througharesearchvisitconducted during2011. 1 2 N.CHRISTOPHERPHILLIPS Some of our results are valid, with the same proofs, for the infinite Leavitt algebra L (which gives Lp analogs of O ) and for the Cohn algebras (which give ∞ ∞ Lp analogsofCuntz’salgebrasE ). Insuchcases,weincludethe correspondingre- d sults. However,inmanycases,theresults,oratleasttheproofs,mustbemodified. We do not go in that direction in this paper. The methods here have little in common with C*-algebra methods. Indeed, the results on spatial representations have no analog for C*-algebras, and the result onnonexistence of nonzerocontinuous homomorphisms does not make sense if one only considers C*-algebras. Uniqueness of Op is of course true when p=2, but, as d far as we can tell, our proof for p6=2 does not work when p=2. This paper is organized as follows. In Section 1, we define Leavitt algebras and Cohn algebras, and give some basic facts about them which will be needed in the rest of the paper. In Section 2, we discuss representations on Banach spaces. We define several natural conditions on representations (weaker than being spatial). We describe ways to get new representations from old ones, some of which work in general and some of which are special to Leavitt and Cohn algebras. Section 3 contains a large collection of examples of representations on Lp spaces. InSection6wedevelopthetheoryof(semi)spatialpartialisometriesonLpspaces associatedtoσ-finite measurespaces. Theresultsherearethe basictechnicaltools neededtoproveourmainresults. Roughlyspeaking,asemispatialpartialisometry from Lp(X,µ) to Lp(Y,ν) comes from a map from a subset of Y to a subset of X. Lamperti’s Theorem [18], which plays a key role, asserts that for p∈(0,∞)\{2}, every isometry from Lp(X,µ) to Lp(Y,ν) is semispatial. ItisunfortunatelynotreallytruethatsemispatialpartialisometriesfromLp(X,µ) to Lp(Y,ν) come from point maps. Instead, they come from suitable homomorphisms of σ-algebras. For our theory of spatial partial isometries, we need a much more extensive theory of these than we have been able to find in the literature. In Section4we recallsomestandardfacts aboutBooleanσ-algebras,andinSection5 we discuss the maps on functions and measures induced by a suitable homomorphism of the Boolean σ-algebras of measurable sets mod null sets. Sections 7, 8, and 9 contain our main results. In Section 7, we give equivalent conditionsforrepresentationsofLeavittalgebrasonLp spacestobe spatial. Along the way, we define spatial representationsof the algebra M of d×d matrices, and d we give a number of equivalent conditions for a representationof M to be spatial. d InSection8,weprovethatandtwospatialrepresentationsoftheLeavittalgebraL d on Lp spaces give the same norm on L , and thus lead to isometrically isomorphic d Banach algebras. In Section 9, we prove the nonexistence of nonzero continuous homomorphisms between the resulting algebras for different values of p. In [22], we will show that Op is an amenable purely infinite simple Banach d algebra, and in [23], we will show that its topological K-theory is the same as for the ordinary Cuntz algebra O . The methods in these papers are much closer to d C*-algebra methods. Scalars will always be C. Much of what we do also works for real scalars. We use complex scalars for our proof of the equivalence of several of the conditions in Theorem 7.2 for a representation of M to be spatial. We do not know whether d complex scalars are really necessary. We have tried to make this paper accessible to operator algebraists who are not familiar with operators on spaces of the form Lp(X,µ). CUNTZ ALGEBRAS ON Lp SPACES 3 We are grateful to Joe Diestel, Ilijas Farah, Coenraad Labuschagne, and Volker Runde for useful email discussions and for providing references. We are especially grateful to Bill Johnson for extensive discussions about Banach spaces, and to GuillermoCortinãsandMar´ıaEugeniaRodr´ıguez,whocarefullyreadanearlydraft and whose comments led to numerous correctionsand improvements. Some of this workwascarriedoutduringavisittothe InstitutoSuperiorTécnico,Universidade Técnicade Lisboa, andduring anextended researchvisitto the Centre de Recerca Matema`tica (Barcelona). I also thank the Research Institute for Mathematical Sciences of Kyoto University for its support through a visiting professorship. I am grateful to all these institutions for their hospitality. 1. Leavitt and Cohn algebras Inthissection,wedefineLeavittalgebrasandsomeoftheirrelatives. Wedescribe agrading,alinearinvolution,andaconjugatelinearinvolution. Wegivesomeuseful computational lemmas. Definition 1.1. Let d ∈ {2,3,4,...}. We define the Leavitt algebra L to be the d universal complex associative algebra on generators s ,s ,...,s ,t ,t ,...,t sat- 1 2 d 1 2 d isfying the relations (1.1) t s =1 for j ∈{1,2,...,d}, j j (1.2) t s =0 for j,k ∈{1,2,...,d} with j 6=k, j k and d (1.3) s t =1. j j j=1 X These algebras were introduced in Section 3 of [20] (except that the base field there is Z/2Z), andthey are simple (Theorem2 of [21], with anarbitrarychoice of the field). Definition 1.2. Let d ∈ {2,3,4,...}. We define the Cohn algebra C to be the d universal complex associative algebra on generators s ,s ,...,s ,t ,t ,...,t sat- 1 2 d 1 2 d isfying the relations (1.1) and (1.2) (but not (1.3)). These algebras are a special case of algebras introduced at the beginning of Section 5 of [8]. What we have called C is called U in [8], and also in [8] the d 1,d fieldisallowedtobe arbitrary. Ournotation,andthename“Cohnalgebra”,follow Definition 1.1 of [4], except that we specifically take the field to be C and suppress it in the notation. Definition 1.3. Let d ∈ {2,3,4,...}. We define the (infinite) Leavitt algebra L ∞ to be the universal complex associative algebra on generators s ,s ,...,t ,t ,... 1 2 1 2 satisfying the relations (1.4) t s =1 for j ∈Z j j >0 and (1.5) t s =0 for j,k ∈Z with j 6=k. j k >0 When it is necessaryto distinguish the generatorsof L from those of L and C , ∞ d d (∞) (∞) (∞) (∞) we denote them by s ,s ,...,t ,t ,.... 1 2 1 2 4 N.CHRISTOPHERPHILLIPS This algebra is simple, by Example 3.1(ii) of [3]. Remark 1.4. The algebras L , C , and L are all examples of Leavitt path d d ∞ algebras. For L see Example 1.4(iii) of [2], for C see Section 1.5 of [1], and for d d L see Example 3.1(ii) of [3]. (Warning: There are two possible conventions for ∞ thechoiceofthedirectionofthearrowsinthegraph,andbothareincommonuse.) Lemma 1.5. Let C be as in Definition 1.2 and let L be as in Definition 1.1, d d with the generators named as there (using the same names in both kinds of algebras). For d ,d ∈{2,3,4,...,∞} with d <d , there is a unique homomorphism 1 2 1 2 ι : C →L suchthatι (s )=s andι (t )=t forj ∈{1,2,...,d }. d1,d2 d1 d2 d1,d2 j j d1,d2 j j 1 Moreover, ι is injective. d1,d2 Proof. Existenceanduniqueness ofι areimmediate fromthe definitions ofthe d1,d2 algebras as universal algebras on generators and relations. We proveinjectivity. We presumethatthere isa purelyalgebraicproof,butone caneasilyseethisbycomparingwiththe C*-algebras,followingRemark2.9below. LetE be the extended Cuntz algebra,as in Remark 2.9. There is a commutative d1 diagram C −−ιd−1,−d→2 L d1 d2   Eyd1 −−−−→ Oyd2. The left vertical map is injective by Theorem 7.3 of [25] and Remark 1.4, and the bottom horizontal map is well known to be injective. Therefore ι is injective. d1,d2 (cid:3) Lemma 1.6. Let A be any of L (Definition 1.1), C (Definition 1.2), or L d d ∞ (Definition 1.3). (1) Thereexists aunique conjugatelinearantimultiplicative involutiona7→a∗ on A such that s∗ =t and t∗ =s for all j. j j j j (2) There existsa unique linear antimultiplicativeinvolutiona7→a′ onA such that s′ =t and t′ =s for all j. j j j j The properties of a 7→ a∗ are just the algebraic properties of the adjoint of a complex matrix: (a+b)∗ =a∗+b∗, (λa)∗ =λa∗, (ab)∗ =b∗a∗, and (a∗)∗ =a for all a,b∈A and λ∈C. The properties of a7→a′ are the same, except that it is linear: (λa)′ =λa′ for all a∈A and λ∈C. Proof of Lemma 1.6. See Remark3.4 of[25], where explicitformulas,validfor any graph algebra, are given, and Remark 1.4. Both parts may also be easily obtained using the universal properties of algebras on generators and relations: a 7→ a is a homomorphism from A to its opposite algebra, and a 7→ a∗ is the composition of a7→a with a homomorphism from A to its complex conjugate algebra. (cid:3) OnecangetLemma1.6(1)byusingthefact(Theorem7.3of[25]andRemark1.4) that there are injective maps from L , C , and L to C*-algebras which preserve d d ∞ the intended involution. (For L and L , injectivity is automatic because the d ∞ algebras are simple. See Remark 2.9 for definitions of *-representations on Hilbert spaces.) CUNTZ ALGEBRAS ON Lp SPACES 5 Proposition 1.7. Let A be any of L (Definition 1.1), C (Definition 1.2), or L d d ∞ (Definition 1.3). Then there is a unique Z-grading on A determined by deg(s )=1 and deg(t )=−1 j j for all j. Proof. The proof is easy. (See after Definition 3.12 in [25].) (cid:3) WewillneedsomeofthefineralgebraicstructureofL ,andassociatednotation. d We roughly follow the beginning of Section 1 of [9], starting with 1.1 of [9]. Notation 1.8. Let d ∈ {2,3,4,...,∞}, and let n ∈ Z . For d < ∞, we define ≥0 Wd = {1,2,...,d}n, and we define W∞ = (Z )n. Thus, Wd is the set of all n n >0 n sequences α = α(1),α(2),...,α(n) with α(l) ∈ {1,2,...,d} (or α(l) ∈ Z if >0 d=∞) for l=1,2,...,n. We set (cid:0) (cid:1) ∞ Wd = Wd. ∞ n n=0 a We call the elements of Wd words (on {1,2,...,d} or Z as appropriate). If ∞ >0 α ∈ Wd, the length of α, written l(α), is the unique number n ∈ Z such that ∞ ≥0 α∈Wd. Note that there is a unique word of length zero, namely the empty word, n whichwewriteas∅.Forα∈Wd andβ ∈Wd,wedenotebyαβ theconcatenation, m n a word in Wd . m+n Notation 1.9. Let A be any of L (Definition 1.1), C (Definition 1.2), or L d d ∞ (Definition 1.3). Let n ∈ Z , and let α = α(1),α(2),...,α(n) ∈ Wd. If n ≥ 1, ≥0 n we define s ,t ∈A by α α (cid:0) (cid:1) s =s s ···s s and t =t t ···t t . α α(1) α(2) α(n−1) α(n) α α(n) α(n−1) α(2) α(1) We take s =t =1. ∅ ∅ For emphasis: in the definition of t , we take the generators t in reverse α α(l) order. We do this to get convenient formulas in Lemma 1.10(3). In particular, when working with Cuntz algebras,one simply uses s∗ in place of t , and we want j j to have s∗ =t . α α Lemma 1.10. Let the notation be as in Notation 1.8 and Notation 1.9. (1) Let α,β ∈Wd. Then s =s s and t =t t . ∞ αβ α β αβ β α (2) In the Z-grading on A of Proposition 1.7, we have deg(s ) = l(α) and α deg(t )=−l(α) for all α∈Wd. α ∞ (3) Let α∈Wd. Then the involutions of Lemma 1.6 satisfy s′ =s∗ =t and ∞ α α α t′ =t∗ =s . α α α (4) Let a ,a ,...,a ∈{s ,s ,...}∪{t ,t ,...}. 1 2 n 1 2 1 2 Suppose a a ···a 6= 0. Then there exist unique α,β ∈ Wd such that 1 2 n ∞ a a ···a =s t . 1 2 n α β (5) Let α,β ∈ Wd satisfy l(α) = l(β). Then t s = 1 if α = β, and t s = 0 ∞ β α β α otherwise. Proof. Parts (1), (2), (3), and (5) are obvious. (Part (5) is also in Lemma 1.2(b) of [9].) Using Part (3), we see that Part (4) is Lemma 1.3 of [9]. (cid:3) 6 N.CHRISTOPHERPHILLIPS Lemma1.11. Letd∈{2,3,4,...},letL beasinDefinition1.1,andletm∈Z . d ≥0 Thenthecollection(s t ) isasystemofmatrixunitsforaunitalsubalgebra α β α,β∈Wd m ofL isomorphictoM .Thatis,identifyingM withthelinearmapsonavector d dm dm space with basis Wd, with matrix units e for α,β ∈ Wd, there is a unique m α,β m homomorphism ϕm: Mdm →Ld such that ϕm(eα,β)=sαtβ. Proof. We prove that s t = 1, by induction on m. The case m = 1 is α∈Wd α α m relation (1.3) in Definition 1.1. Assuming the result holds for m, use this and the P case m=1 at the last step to get d d s t = s s t t = s s t t =1. α α β j j β β j j β j=1 α∈XWmd+1 βX∈Wmd Xj=1 β∈XWmd (cid:18)X (cid:19) The statement of the lemma now follows from Lemma 1.10(5), or is Proposi- tion 1.4 of [9]. (cid:3) Lemma 1.12. Let d ∈ {2,3,4,...}, let m ∈ Z , and let a ,a ,...,a ∈ L . >0 1 2 m d Then there exist n ∈ Z , a finite set F ⊂ Wd, and numbers λ ∈ C for ≥0 ∞ k,α,β k =1,2,...,m, α∈F, and β ∈Wd, such that n (1.6) a = λ s t k k,α,β α β αX∈FβX∈Wnd for k =1,2,...,m. Proof. Since a ,a ,...,a are linear combinations of products of the s and t , it 1 2 m j j suffices to prove the statement when a ,a ,...,a are products of the s and t . 1 2 m j j By Lemma 1.10(4), we may assume a =s t with α ,β ∈Wd. Set k αk βk k k ∞ n=max l(β ), l(β ), ..., l(β ) . 1 2 m For k=1,2,...,m, set lk =n−l(cid:0)(βk). Take (cid:1) m F = α α: α∈Wd . k lk k=1 [ (cid:8) (cid:9) Lemma 1.11 and Lemma 1.10(1) imply a =s t = s s t t = s t . k αk βk αk α α βk αkα βkα α∈XWldk α∈XWldk This expression has the form in (1.6). (cid:3) Definition 1.13. Let A be any of L , C , or L . Let λ = (λ ,λ ,...,λ ) ∈ Cd. d d ∞ 1 2 d (For A = L , take d = ∞, take Cd = ∞ C, and take λ = (λ ,λ ,...).) Define ∞ j=1 1 2 s ,t ∈A by λ λ L d d s = λ s and t = λ t . λ j j λ j j j=1 j=1 X X In principle, this notation conflicts with Notation 1.9, but no confusion should arise. Lemma 1.14. Let the notation be as in Definition 1.13. Let λ,γ ∈Cd. Then t s = λ γ ·1. λ γ j j j (cid:16)X (cid:17) CUNTZ ALGEBRAS ON Lp SPACES 7 Proof. This is immediate from the relations t s = 1 for j = k and t s = 0 for j k j k j 6=k. (cid:3) 2. Representations on Banach spaces In this section, we discuss representations of Leavitt and Cohn algebras on Ba- nach spaces. Much of what we say makes sense for representations on general Banach spaces, but some only works for representations on spaces of the form Lp(X,µ).Someoftheconstructionsworkforgeneralalgebras,butsomearespecial to representations of Leavitt and Cohn algebras. Some of what we do is intended primarily to establish notation and conventions. (For example, representationsare required to be unital, and isomorphisms are required to be surjective.) All Banach spaces in this article will be over C. Notation 2.1. LetE andF beBanachspaces. WedenotebyL(E,F)theBanach space of all bounded linear operators from E to F, and by K(E,F)⊂L(E,F) the closed subspace of all compact linear operators from E to F. When E =F, we get the Banach algebra L(E) and the closed ideal K(E)⊂L(E). The followingdefinitionsummarizesterminologyfor Banachspacesthatwe use. We will need both isometries and isomorphisms of Banach spaces. Definition 2.2. If E andF areBanachspaces,we say that a∈L(E,F) is an isomorphism if a is bijective. (By the Open Mapping Theorem, a−1 is also bounded.) If an isomorphism exists, we say E and F are isomorphic. We say that a ∈ L(E,F) is an isometry if kaξk = kξk for every ξ ∈ E. (We do not require that a be surjective.) If there is a surjective isometry from E to F, we say that E and F are isometrically isomorphic. If A and B are Banach algebras, we say that a homomorphism ϕ: A→B is an isomorphism if it is continuous and bijective. (By the Open Mapping Theorem, ϕ−1 is also continuous.) If an isomorphismexists, we say A and B are isomorphic. If in addition ϕ is isometric, we call it an isometric isomorphism. If such a map exists, we say A and B are isometrically isomorphic. For emphasis (because some authors do not use this convention): isomorphisms are required to be surjective. The following notation for duals is intended to avoid conflict with the notation for adjoints. Notation 2.3. Let E be a Banach space. We denote by E′ its dual Banach space L(E,C), consisting of all bounded linear functionals on E. If F is another Banach space and a ∈ L(E,F), we denote by a′ the element of L(F′,E′) defined by a′(ω)(ξ)=ω(aξ) for ξ ∈E and ω ∈F′. We will also need some notation for specific spaces. Notation 2.4. For any set S, we give lp(S) the usual meaning (using counting measure on S), and we take (as usual) lp = lp(Z ). For d ∈ Z and p ∈ [1,∞], >0 >0 we let lp = lp {1,2,...,d} . We further let Mp = L lp with the usual operator d d d norm, and we algebraically identify Mp with the algebra M of d × d complex (cid:0) (cid:1) d (cid:0) (cid:1) d matrices in the standard way. 8 N.CHRISTOPHERPHILLIPS We warn of a notational conflict. Many articles on Banach spaces use L (X,µ) p rather than Lp(X,µ), and use ld for what we call lp. Our convention is chosen to p d avoidconflictwiththestandardnotationfortheLeavittalgebraL ofDefinition1.1. d For fixed d, the norms on the various Mp are of course equivalent, but they are d not equal. The following example illustrates this. Example 2.5. Let d ∈ Z . Let η,µ ∈ Cd, and let ω : Cd → C be the linear >0 η functional ω (ξ) = d η ξ . Let a ∈ M be the rank one operator given by η j=1 j j d aξ =ω (ξ)µ. Let p,q∈[1,∞]satisfy 1+1 =1, andregarda asanelementofMp. η P p q d Then one can calculate that kak=kµk kηk . p q The following terminology and related observation will be used many times. Definition 2.6. Let(X,B,µ)beameasurespaceandletp∈[1,∞].Forafunction ξ ∈ Lp(X,µ) (or, more generally, any measurable function on X) and a subset E ⊂X, we will say that ξ is supported in E if ξ(x)=0 for almostall x∈X\E. If I is a countable set, and (ξ ) is a family of elements of Lp(X,µ) or measurable i i∈I functionsonX,wesaythattheξ havedisjoint supports iftherearedisjointsubsets i E ⊂X such that ξ is supported in E for all i∈I. i i i Remark 2.7. Let(X,B,µ)beameasurespace,letp∈[1,∞),letI beacountable set, and let ξ ∈Lp(X,µ), for i∈I, have disjoint supports. Then i p ξ = kξ kp. i i p i∈I p (cid:13)X (cid:13) Xi∈I (cid:13) (cid:13) Definition 2.8. LetAbe any(cid:13)unitalc(cid:13)omplexalgebra. LetE beanonzeroBanach space. A representation of A on E is a unital algebra homomorphism from A to L(E). We do notsayanythingaboutcontinuity. We willmostlybe interestedinrepre- sentations of L , C , and L , for which we do not use a topology on the algebra, d d ∞ or of M , for which all representations are continuous by finite dimensionality. d Remark 2.9. The well known representations of L , C , and L are those on a d d ∞ Hilbert space H. Choose any d isometries w ,w ,...,w ∈ L(H) (or, for the case 1 2 d of L , isometries w ,w ,... ∈ L(H)) with orthogonal ranges. Then we obtain ∞ 1 2 a representation ρ: C → L(H) or ρ: L → L(H) by setting ρ(s ) = w and d ∞ j j ρ(t ) = w∗ for all j. If d < ∞ and d w w∗ = 1, we get a representation j j j=1 j j of L . These representations are even *-representations: making A a *-algebra as d P in Lemma 1.6(1), we have ρ(a∗)=ρ(a)∗ for all a∈A. The closures ρ(A) do not depend on the choice of ρ (in case A = C , provided d ρ 1− d s s∗ 6= 0), and are the usual Cuntz algebra O when A = L (in- j=1 j j d d cl(cid:16)udingPthe case d(cid:17)= ∞), and the extended Cuntz algebras Ed when A = Cd. See Theorem 1.12 of [9] for L and L , and see Lemma 3.1 of [10] for C . d ∞ d FurtherexamplesofrepresentationsofL ,C ,andL willbegiveninSection3. d d ∞ RepresentationsofL ,C ,andL haveakindofrigidityproperty. Itisstronger d d ∞ for L than for the others: a representation is determined by the images of the s d j or by the images of the t . j Lemma 2.10. Let A be any of L (Definition 1.1), C (Definition 1.2), or L d d ∞ (Definition 1.3). Let B be a unital algebra over C, and let ϕ,ψ: L →B be unital d CUNTZ ALGEBRAS ON Lp SPACES 9 homomorphisms such that for all j we have ϕ(s ) = ψ(s ) and ϕ(s t ) = ψ(s t ). j j j j j j Then ϕ = ψ. The same conclusion holds if we replace ϕ(s ) = ψ(s ) with ϕ(t ) = j j j ψ(t ). j Proof. Assume ϕ(s ) = ψ(s ) for all j. Using the relations t s t = t at the first j j j j j j step and ϕ(t )ϕ(s )=1 at the last step, we calculate: j j ϕ(t )=ϕ(t )ϕ(s t )=ϕ(t )ψ(s t )=ϕ(t )ψ(s )ψ(t )=ϕ(t )ϕ(s )ψ(t )=ψ(t ). j j j j j j j j j j j j j j The first statement follows. If instead ϕ(t ) = ψ(t ) for all j, similar reasoning j j (using s t s =s ) gives j j j j ϕ(s )=ϕ(s t )ϕ(s ) j j j j =ψ(s t )ϕ(s )=ψ(s )ψ(t )ϕ(s )=ψ(s )ϕ(t )ϕ(s )=ψ(s ). j j j j j j j j j j This completes the proof. (cid:3) Lemma 2.11. Let d ∈ {2,3,4,...}, let B be a unital algebra over C, and let ϕ,ψ: L →Bbeunitalhomomorphismssuchthatϕ(s )=ψ(s )forj ∈{1,2,...,d}. d j j Then ϕ = ψ. The same conclusion holds if we replace ϕ(s ) = ψ(s ) with ϕ(t ) = j j j ψ(t ). j Proof. For j ∈ {1,2,...,d}, define idempotents e ,f ∈ B by e = ϕ(s t ) and j j j j j f =ψ(s t ). By Lemma 2.10, it suffices to show that e =f for all j. j j j j j First assume that ϕ(s ) = ψ(s ) for all j. Using this statement at the first and j j third steps, t s = 0 for j 6= k at the second step, and d e = 1 at the last j k k=1 k step, we have P d d (2.1) f e =ψ(s t s )ϕ(t )= ψ(s t s )ϕ(t )= f e =f . j j j j j j j j k k j k j k=1 k=1 X X If now j 6=k, then f e =f e e =0. k j k k j This equation, together with d f =1, gives k=1 k d P (2.2) e = f e =f e . j k j j j k=1 X The proof is completed by combining (2.1) and (2.2). Now assume that ϕ(t )=ψ(t ) for all j. With similar justifications, we get j j d d f e =ψ(s )ϕ(t s t )= ψ(s )ϕ(t s t )= f e =e . j j j j j j k k j j k j j k=1 k=1 X X So for j 6= k we have f e = f f e = 0. Combining these results gives f = j k j k k j d f e =f e , whence e =f . (cid:3) k=1 j k j j j j PThe analog of Lemma 2.11 for L∞ and Cd is false. See Example 3.5. Thefollowingdefinitiongivesseveralnaturalconditionstoaskofarepresentation of L , C , or L on a Banach space E. The condition in (3) is motivated by the d d ∞ following property of a *-representation ρ of L or C on a Hilbert space H (as in d d Remark 2.9): for λ ,λ ,...,λ ∈C and ξ ∈H, we have 1 2 d d (2.3) ρ λ s ξ =k(λ ,λ ,...,λ )k kξk. j j 1 2 d 2 j=1 (cid:13) (cid:18) (cid:19) (cid:13) (cid:13) X (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) 10 N.CHRISTOPHERPHILLIPS InDefinition7.4andDefinition7.6,wewillseefurtherconditionsonrepresentations which are natural when E =Lp(X,µ). Definition 2.12. Let A be any of L (Definition 1.1), C (Definition 1.2) or L d d ∞ (Definition 1.3). Let E be a nonzero Banach space, and let ρ: A → L(E) be a representation. (1) Wesaythatρiscontractiveongenerators ifforeveryj,wehavekρ(s )k≤1 j and kρ(t )k≤1. j (2) We say that ρ is forward isometric if ρ(s ) is an isometry for every j. j (3) We say that ρ is strongly forward isometric if ρ is forward isometric and (following Definition 1.13) for every λ ∈ Cd, the element ρ(s ) is a scalar λ multiple of an isometry. Remark 2.13. A representation which is contractive on generators is clearly forward isometric. A representation of L which is contractive on generators need not be strongly d forward isometric. See Example 3.11 below. We will see in Example 3.5 below that a strongly forward isometric representation of L need not be contractive on ∞ generators. We do not know whether this can happen for L with d finite. d We now describe several ways to make new representations from old ones. The firsttwo(directsums andtensoringwiththe identityonsomeotherBanachspace) work for representations of general algebras. They also work for more general choices of norms on the direct sum and tensor product than we consider here. For simplicity, we restrict to specific choices which are suitable for representations on spaces of the form Lp(X,µ). Lemma 2.14. LetAbe aunitalcomplexalgebra,andletp∈[1,∞]. Letn∈Z , >0 and for l = 1,2,...,n let (X ,B,µ ) be a σ-finite measure space and let ρ : A → l l l l L(Lp(X ,µ )) be a representation. Equip E = n Lp(X ,µ ) with the norm l l l=1 l l k(ξ ,ξ ,...,ξ )k= nLkξkp 1/p. 1 2 n l p l=1 Then there is a unique representation ρ: A(cid:16)X→L(E) suc(cid:17)h that ρ(a)(ξ ,ξ ,...,ξ )= ρ (a)ξ , ρ (a)ξ , ..., ρ (a)ξ 1 2 n 1 1 2 2 n n for a∈Ld and ξl ∈Lp(Xl,µl) for l =(cid:0)1,2,...,n. If A is any of Ld,(cid:1)Cd, or L∞, and each ρ is contractive on generators or forward isometric, then so is ρ. l Proof. This is immediate. (cid:3) Remark 2.15. The norm used in Lemma 2.14 identifies E with Lp( n X ), l=1 l using the obvious measure. We write this space as ` Lp(X ,µ )⊕ Lp(X ,µ )⊕ ···⊕ Lp(X ,µ ). 1 1 p 2 2 p p n n We write the representation ρ as ρ=ρ ⊕ ρ ⊕ ···⊕ ρ , 1 p 2 p p n and call it the p-direct sum of ρ ,ρ ,...,ρ . 1 2 n Example 3.11 below shows that if ρ ,ρ ,...,ρ are strongly forward isometric, 1 2 n it does not follow that ρ is strongly forwardisometric. One can form a p-direct sum ρ over an infinite index set I provided i∈I i sup kρ (a)k<∞ for all a∈A. i∈I i L