Compact quantum group structures on type-I C ∗ -algebras
aa r X i v : . [ m a t h . OA ] A ug COMPACT QUANTUM GROUP STRUCTURES ON TYPE-I C ∗ -ALGEBRAS ALEXANDRU CHIRVASITU, JACEK KRAJCZOK, AND PIOTR M. SOŁTAN
Abstract.
We prove a number of results having to do with equipping type-I C ∗ -algebras withcompact quantum group structures, the two main ones being that such a compact quantumgroup is necessarily co-amenable, and that if the C ∗ -algebra in question is an extension of anon-zero finite direct sum of elementary C ∗ -algebras by a commutative unital C ∗ -algebra thenit must be finite-dimensional. Introduction
The theory of locally compact quantum groups is inextricably connected to the theory of oper-ator algebras. In fact, paraphrasing S.L. Woronowicz [Wor80, Section 0], any theorem on locallycompact quantum groups is one on C ∗ -algebras. In the present paper we will focus on some of theinterplay between the theory of compact quantum groups and operator algebras. Examples of suchan interplay are motivated by results such as the well known equivalence between amenability of adiscrete group Γ and nuclearity of the C ∗ -algebra C ∗ r (Γ) ([Lan73, Theorem 4.2]). This particularfact has been generalized to compact quantum groups (i.e. duals of discrete quantum groups, see[PW90, Section 3]) of Kac type by Tomatsu in [Tom06, Corollary 1.2] and in a weakened formto all compact quantum groups in [Tom06, Theorem 3.9] (see also [BT03, Theorem 3.3]). Someof these topics were pursued further e.g. in [Boc95, DLRZ02] in the language of quantum groupactions as well as in [Boc95, SV14, CN16] in the locally compact case.The moral of the above-mentioned research activity is that one can learn about certain “group-theoretical” properties of a compact quantum group G by studying purely operator theoreticproperties of the C ∗ -algebra C( G ) . Furthermore one can often show that certain C ∗ -algebras donot admit a compact quantum group structure solely on the basis of some of their properties as C ∗ -algebras. Examples of such results are given in [Soł10a, Soł10b] and also [Soł14] and most recently[KS20a]. In this last paper the second and third author show that the C ∗ -algebra known as the Toeplitz algebra (the C ∗ -algebra generated by an isometry) does not admit a structure of a compactquantum group. The main tools are built out of certain direct integral decompositions available forso called type-I quantum groups, i.e. locally compact quantum groups whose universal quantumgroup C ∗ -algebra is of type I with particular emphasis on C ∗ -algebras of type I with discrete CCRideal (see Section 1).In the present paper the techniques of [KS20a] are vastly generalized and applied to a numberof problems. Moreover the direct integral decompositions of representations (and other objects)are avoided. In the preliminary Section 1 we introduce our basic tools, recall certain objectssuch as the CCR ideal of a C ∗ -algebra and prove a number of lemmas concerning implementationof automorphisms on C ∗ -algebras with discrete CCR ideals. The main result of Section 2 isTheorem 2.7 which says that a compact quantum group G with C( G ) of type I must be co-amenable([BMT01, Section 1]). Along the way we prove a number of results about the scaling group ofa compact quantum group which allow to reprove the result of Daws ([Daw13]) about automaticadmissibility of finite dimensional representations of any discrete quantum group (cf. [Soł05a,Section 2.2]).In the final Section 3 we discuss compact quantum group structures on C ∗ -algebras which areextensions of a finite direct sum of algebras of compact operators by a commutative C ∗ -algebra.Examples of such C ∗ -algebras occur quite frequently in non-commutative geometry and include Mathematics Subject Classification.
Key words and phrases. compact quantum group, quantum space, non-commutative topology, type-I C ∗ -algebra. the Podleś spheres ([Pod87, Dą03]), the quantum real projective plane ([HS02, Section 3.2] andsome weighted quantum projective space ([BS18, Section 3]) and many others (see Section 3). Weshow that such C ∗ -algebras do not admit any compact quantum group structure which answersseveral questions left open in [Soł10a, Soł10b] and provides a number of fresh examples of naturallyoccurring quantum spaces with this property.Our exposition is based on a number of standard references. Thus we refer to classic textssuch as [Dix77, Arv76] for all necessary background on C ∗ -algebras and to [Wor87a, NT13] forthe theory of compact quantum groups. We have tried to keep the terminology and notationconsistent with recent trends and as self-explanatory as possible. In particular, for a compactquantum group G we denote by C( G ) the (usually non-commutative) C ∗ -algebra playing the roleof the algebra of continuous functions on G . The symbol Irr( G ) will denote the set of equivalenceclasses of irreducible representations of G and for a class α ∈ Irr( G ) the dimension of α will bedenoted by n α . Since in the theory of compact quantum groups we allow the C ∗ -algebras C( G ) tobe sitting strictly between the reduced and universal versions (see [BMT01]), we will write C r ( G ) and C u ( G ) for these two distinguished completions of the canonical Hopf ∗ -algebra Pol( G ) inside C( G ) . Acknowledgments.
AC is grateful for funding through NSF grants DMS-1801011 and DMS-2001128.The second and third authors were partially supported by the Polish National Agency for theAcademic Exchange, Polonium grant PPN/BIL/2018/1/00197 as well as by the FWO–PAS projectVS02619N: von Neumann algebras arising from quantum symmetries.1.
Preliminaries
All C ∗ -algebras are unital except when we specify otherwise, or with obvious exceptions suchas the algebra of compact operators on an infinite-dimensional Hilbert space.We denote by b A the spectrum of the C ∗ -algebra A , i.e. the set of equivalence classes of irreduciblenon-zero representations ([Dix77, §2.2.1 & §2.3.2]).For a Hilbert space H we denote by K ( H ) the algebra of compact operators on H . Furthermorefor a family H = { H λ } λ ∈ Λ of Hilbert spaces we set K ( H ) := c - M λ ∈ Λ K ( H λ ) (1-1)the algebra of compact operators in H = M λ ∈ Λ H λ preserving that direct sum decomposition. In general, for non-unital C ∗ -algebras A , we write A + for the minimal unitization of A .1.1. Type-I C ∗ -algebras. We will work with type-I C ∗ -algebras in the sense of [Gli61], whichprovides numerous equivalent characterizations. Textbook sources are [Dix77, Chapter 9] and[Arv76, §1.5 and Chapters 2 and 4].Recall e.g. from [Arv76, discussion preceding Definition 1.5.3] the following definition: Definition 1.1.
For a C ∗ -algebra A , the CCR ideal
CCR( A ) is the intersection, over all irreduciblerepresentations π : A −→ B ( H ) , of the pre-images π − ( K ( H )) of the ideal K ( H ) of compact operators on H .In other words, CCR( A ) consists of those elements that are compact in every irreducible rep-resentation. Definition 1.2.
A (typically type-I) C ∗ -algebra A is said to have discrete CCR ideal if its CCRideal CCR( A ) is of the form K ( H ) as in (1-1) for some family H = { H λ } λ ∈ Λ of Hilbert spaces.We occasionally also say A is discrete-CCR or CCR-discrete for brevity, though note that thisdoes not mean it is CCR!
OMPACT QUANTUM GROUP STRUCTURES ON TYPE-I C ∗ -ALGEBRAS 3 Now let A be a type-I discrete-CCR C ∗ -algebra, with CCR( A ) = K ( H ) , H = { H λ } λ ∈ Λ and set H := M λ ∈ Λ H λ . (1-2)The ideal K ( H ) ⊂ A is represented in the obvious fashion on H with each component K ( H λ ) acting naturally on H λ . This representation ρ : K ( H ) → B ( H ) extends uniquely to a representa-tion ρ : A −→ B ( H ) by e.g. [Arv76, Theorem 1.3.4]. Lemma 1.3.
Let A be a type-I C ∗ -algebra with CCR( A ) of the form (1-1) . Representation ρ isfaithful and every automorphism of A is given by conjugation by some unitary U ∈ U( H ) .Furthermore, if the family H is a singleton then that unitary is unique up to scaling by T .Proof. Recall from [Arv76, pp. 14–15] that the representation ρ is constructed from ρ : CCR( A ) = K ( H ) → B ( H ) as follows: an element a ∈ A is mapped to the unique element ρ ( a ) of B ( H ) suchthat ρ ( a ) ρ ( x ) = ρ ( ax ) for all x ∈ CCR( A ) . By construction ρ = L λ ∈ Λ ρ λ , where ρ λ is constructedanalogously from ρ λ : K ( H λ ) → B ( H λ ) . Since each ρ λ is irreducible, so is each ρ λ ([Arv76,Theorem 1.3.4]).We now note that the CCR-ideal CCR( A ) is essential. Indeed, CCR( A ) is the largest CCRideal in A (cf. [Arv76, p. 24]). But every ideal in A is type I and every type-I C ∗ -algebra containsa non-zero CCR ideal, so any non-zero ideal of A must have a non-zero intersection with CCR( A ) .It follows that ρ is faithful.Now for any α ∈ Aut( A ) the representation ρ λ ◦ α is equivalent to ρ λ , so by [Arv76, Thm. 1.3.4] ρ λ is equivalent to ρ λ ◦ α (because for a ∈ A and x ∈ CCR( A ) we have ( ρ λ ◦ α )( a )( ρ λ ◦ α )( x ) =( ρ λ ◦ α )( ax ) ). For each λ let U λ be a unitary implementing the equivalence. Then U = L λ ∈ Λ U λ implements equivalence between ρ ◦ α and ρ .As for uniqueness, it follows from the fact that when H = { H λ } λ ∈ Λ is a singleton the represen-tation ρ is irreducible, and hence the only self-intertwiners of ρ are the scalars. (cid:3) Of more interest to us, however, will be one-parameter automorphism groups (where we canalso recover some measure of uniqueness):
Lemma 1.4.
Let A be a type-I C ∗ -algebra with CCR( A ) of the form (1-1) . A one-parameterautomorphism group ( α s ) s ∈ R of A is given by conjugation by a one-parameter unitary group R ∋ s U s ∈ Y λ ∈ Λ U( H λ ) ⊂ U( H ) , preserving the decomposition (1-2) , unique up to scaling by an individual character χ λ : R → T oneach H λ .Proof. Every automorphism of A will permute the summands K ( H λ ) of K ( H ) , so a one-parametergroup will preserve each summand by continuity. But this means that on each H λ the automor-phisms ( α s ) s ∈ R are given by conjugation by a projective unitary representation ([Var85, ChapterVII, Section 2]) of R on H λ . Since projective representations of R lift to plain unitary representa-tions, for each s we have α s (cid:12)(cid:12) B ( H λ ) = conjugation by b is for a possibly-unbounded positive self-adjoint non-singular operator b on H λ . This lift is moreoverunique up to multiplication by a character R → U( H λ ) because K ( H λ ) acts irreducibly on H λ . (cid:3) ALEXANDRU CHIRVASITU, JACEK KRAJCZOK, AND PIOTR M. SOŁTAN
Scaling groups.
We recall the following well-known observation.
Lemma 1.5.
Let G and H be compact quantum groups. Any Hopf ∗ -homomorphism φ : C( G ) −→ C( H ) (i.e. a unital ∗ -homomorphism satisfying ∆ H ◦ φ = ( φ ⊗ φ ) ◦ ∆ G ) intertwines scaling groups, inthe sense that φ ◦ τ G s ( a ) = τ H s ◦ φ ( a ) , ∀ s ∈ R , a ∈ Pol( G ) . Proof.
Assume first that C( G ) = C u ( G ) , C( H ) = C u ( H ) are universal versions of the algebras ofcontinuous functions. In this case our lemma is simply a reformulation of [MRW12, Proposition3.10, equation (20)].Consider now the general case. Observe that φ restricts to a map Pol( G ) → Pol( H ) , hence bythe universal property of C u ( G ) we can extend φ | Pol( G ) to a ∗ -homomorphism ˜ φ : C u ( G ) → C u ( H ) .Clearly ˜ φ is a Hopf ∗ -homomorphism, hence by the above argument ˜ φ intertwines scaling groups.As φ and ˜ φ are equal on Pol( G ) and the canonical morphisms C u ( G ) → C( G ) , C u ( H ) → C( H ) intertwine scaling groups, we arrive at the claim. (cid:3) Admissibility and co-amenability
Throughout the discussion we denote by G a compact quantum group and by Γ = b G its discretequantum dual. The following observation will be put to use repeatedly; it is [CS19, lemma 2.3],and it follows from Lemma 1.5 upon noting that finite-dimensional representations factor throughKac quotients. Proposition 2.1.
Every finite-dimensional representation ρ : A → M n of the CQG algebra A = C( G ) is invariant under the scaling group ( τ s ) s ∈ R of G , in the sense that ρ ◦ τ s ( a ) = ρ ( a ) , ∀ s ∈ R , a ∈ Pol( G ) . Proposition 2.1 has a number of consequences. First, note the following generalization.
Corollary 2.2.
Let B be a C ∗ -algebra all of whose irreducible representations are finite-dimen-sional and A = C( G ) for a compact quantum group G . Then, every morphism ρ : A → B isinvariant under the scaling group ( τ s ) s ∈ R of G .Proof. Indeed, it follows from Proposition 2.1 that for every irreducible representation π : B → M n the composition π ◦ ρ is invariant under τ . The conclusion follows from the fact that the directsum of all π is faithful on B (i.e. every C ∗ -algebra embeds into the direct sum of its irreduciblerepresentations). (cid:3) Secondly, we obtain the following alternative proof of [Daw13, Corollary 6.6] or [Vis17, Propo-sition 3.3] which concerns admissibility of finite representations of discrete quantum. The relevantterminology is explained in [Soł05a, Daw13, DDS19].
Theorem 2.3.
Every finite-dimensional unitary representation of a discrete quantum group isadmissible.Proof.
Let G be a compact quantum group and denote by Γ the dual of G . Furthermore put A := C u ( G ) . As explained in [PW90, Theorem 3.4] (cf. [Kus01, Proposition 5.3], [SW07, Section5]), a unitary representation of Γ on C n is defined by a morphism ρ : A → M n .Proposition 2.1 ensures that ρ is invariant under the scaling group of G . But then, by [DDS19,Proposition 3.2 and Remark 3.4], the representation of Γ associated to ρ will be admissible. (cid:3) Next, we have the following sufficient criterion for the co-amenability of a compact quantumgroup G . It appears as [CS19, Proposition 2.5], and we include a slightly different proof here. Theorem 2.4.
A compact quantum group G is co-amenable if and only if the reduced algebra C r ( G ) admits a morphism ρ : C r ( G ) → M n to a finite-dimensional C ∗ -algebra. OMPACT QUANTUM GROUP STRUCTURES ON TYPE-I C ∗ -ALGEBRAS 5 Proof.
Co-amenability means the counit is bounded on C r ( G ) , so only the backwards implication‘ ⇐ ’ is interesting. We know from Proposition 2.1 that ρ is invariant under the scaling group τ s , s ∈ R , so by analytic continuation its restriction to the dense Hopf ∗ -subalgebra Pol( G ) ⊂ C r ( G ) is invariant under the squared antipode S = τ − i ([NT13, p.32]). Once we have S -invariance, co-amenability follows from [BMT02, Theorem 4.4]. (cid:3) Remark 2.5.
Theorem 2.4 generalizes [BMT01, Theorem 2.8], which requires the existence ofa bounded character , and strengthens [BMT02, Theorem 4.4] by removing the S -invariance hy-pothesis (which is automatic).For future reference, we also record the following description of the Kac quotient of a CQGalgebra. Proposition 2.6.
Let A = C( G ) and ( τ s ) s ∈ R the corresponding scaling group.The Kac quotient A kac is precisely the largest quotient of A on which τ s acts trivially, i.e. thequotient by the ideal generated by the elements τ s ( a ) − a, s ∈ R , a ∈ Pol( G ) . (2-1) Proof.
On the one hand, since ( τ s ) s ∈ R is a one-parameter group of CQG automorphisms (i.e. each τ s preserves both the multiplication and the comultiplication), the quotient A −→ B (2-2)by the ideal generated by (2-1) is indeed a CQG algebra. Since furthermore (2-2) intertwinesscaling groups (Lemma 1.5) the scaling group of B is trivial by construction and hence B is Kac;this means that (2-2) factors as A / / A kac / / B . On the other hand, the morphism
A → A kac also intertwines scaling groups. Since its codomainhas trivial scaling group, it must vanish on all elements of the form (2-1) and hence factor through B . In short, the kernels of A → A kac and (2-2) coincide. (cid:3)
Next, the goal will be to prove
Theorem 2.7.
Let G be a compact quantum group such that A = C( G ) is type-I. Then G isco-amenable. Let A = C( G ) , as in the statement, i.e. we assume that A is type-I. By [Sak66, Main Theorem] A is GCR , or postliminal in the sense of [Dix77, Definition 4.3.1]. Let I ⊂ I ⊂ · · · ⊂ I α = A (2-3)the canonical transfinite composition sequence of ideals with postliminal subquotients I β +1 / I β provided by [Dix77, Proposition 4.3.3], where α is some ordinal number. We need Lemma 2.8.
A unital type-I C ∗ -algebra A has at least one non-zero finite-dimensional irreduciblerepresentation.Proof. The ordinal α in (2-3) cannot be a limit ordinal: if it were, then by the very definition ofa composition sequence ([Dix77, Definition 4.3.2]) I α = A would be the closure of the ascendingchain of proper ideals I β , β < α , contradicting the fact that these are proper ideals in a unital C ∗ -algebra and hence are all at distance from the unit ∈ A .It follows that α = β + 1 for some ordinal β , and hence the top liminal quotient A / I β must bea (non-zero!) unital liminal C ∗ -algebra. It thus follows that all of its irreducible representationsare finite-dimensional [Dix77, 4.7.14]. (cid:3) ALEXANDRU CHIRVASITU, JACEK KRAJCZOK, AND PIOTR M. SOŁTAN
Remark 2.9.
Alternatively one could choose a proper maximal ideal I in A and note that then A / I is a type-I simple unital C ∗ -algebra. Thus any representation of A / I is faithful, and thereexists an irreducible one, say φ : A / I → B ( H ) . The range of φ contains K ( H ) , and hence it mustbe equal to K ( H ) (otherwise φ − ( K ( H )) would be a proper ideal in A / I ), but A / I is unital andand φ is faithful, so H must be finite dimensional. Proof of Theorem 2.7.
Let h be the Haar measure of G and J the ideal (cid:8) x ∈ A (cid:12)(cid:12) h ( x ∗ x ) = 0 (cid:9) ⊂ A . The quotient A / J will then be the reduced version C r ( G ) and again of type I. Since it has afinite-dimensional representation by Lemma 2.8, co-amenability follows from Theorem 2.4. (cid:3) Extensions of K ( H ) by C( X ) Throughout the present section, A denotes a C ∗ -algebra fitting into an exact sequence / / K ( H ) / / A π / / C / / (3-1)where • C = C( X ) for a (non-empty and for us always Hausdorff) compact space X , • the ideal K ( H ) is as in (1-1), where H = { H λ } λ ∈ Λ , dim H λ ≥ , ∀ λ ∈ Λ (3-2)is a finite, non-empty family of Hilbert spaces.Note that such A is automatically of type I and K ( H ) is its CCR-ideal. We denote H := M λ ∈ Λ H λ . (3-3)We list some examples of interest. Example 3.1.
For any finite family (3-2), the unitization K ( H ) + satisfies the hypotheses. Example 3.2.
The
Toeplitz C ∗ -algebra T ( ∂D ) ([HR00, Definition 2.8.4]) associated to a strongly(or strictly) pseudoconvex domain Ω ⊂ C n ([Kra01, §3.2] or [Upm96, Definition 1.2.18]) is of theform above, with H a singleton.This applies in particular to the case when D is the open unit disk in C . T ( ∂D ) is then theuniversal C ∗ -algebra generated by an isometry, and Theorem 3.5 below specializes to the mainresult of [KS20b]. Example 3.3.
The non-quotient
Podleś spheres introduced in [Pod87] and surveyed for instancein [Dą03, §2.5, point 5]. According to [She91, Proposition 1.2] those algebras (denoted herecollectively by A ) are all isomorphic to the pullback of two copies of the symbol map T → C( S ) .It follows that the C ∗ -algebra in question fits into an extension / / K ( ℓ ) ⊕ K ( ℓ ) / / A / / C( S ) / / i.e. of the form (3-1) for a two-element family H = { H λ } λ ∈ Λ of Hilbert spaces. Example 3.4.
As recalled in [Les91, Example, p.123], the algebra
CZ( M ) of Calderón-Zygmundoperators (i.e. pseudo-differential operators of order zero; cf. e.g. [Ste93, §VI.1]) on a smoothcompact manifold M fits into an exact sequence / / K ( L ( M )) / / CZ( M ) / / C( S ∗ M ) / / where S ∗ M denotes the unit sphere bundle attached to the cotangent bundle of M .With all of this in place, the main result of this section is Theorem 3.5. If G is a compact quantum group such that a unital C ∗ -algebra A = C( G ) fits intoan exact sequence (3-1) as above, then A is finite-dimensional. OMPACT QUANTUM GROUP STRUCTURES ON TYPE-I C ∗ -ALGEBRAS 7 Remark 3.6.
The discreteness hypothesis on the ideal K ( H ) in Theorem 3.5 is crucial: accordingto [Wor87b, Appendix 2], for deformation parameters µ of absolute value < the function algebra C(SU µ (2)) fits into an exact sequence / / C( S ) ⊗ K ( ℓ ) / / C(SU µ (2)) / / C( S ) / / . Remark 3.7.
Let us also note that the fact that we are dealing with a unital C ∗ -algebra A is essential for Theorem 3.5 as well. Indeed the C ∗ -algebras associated with the non-compactquantum “ az + b ” groups ([Wor01, Soł05b]) are extensions of K ( H ) by C for an infinite dimensionalseparable Hilbert space H .Theorem 3.5 will require some preparation. We will first address the issue of faithfulness of theHaar measure. Proposition 3.8.
Let A be an extension of K ( H ) by C( X ) as in (3-1) and suppose A = C( G ) forsome compact quantum group G . Then G is co-amenable. In particular A = C( G ) is reduced.Proof. This is a direct application of Theorem 2.7, since our C ∗ -algebra A satisfies the hypothesesof that earlier result. (cid:3) We henceforth write A = C r ( G ) to emphasize the faithfulness of the Haar measure, as allowedby Proposition 3.8.Recall from Section 1 that for each λ we have the irreducible representation ρ λ : A → B ( H λ ) obtained via the canonical extension of the embedding K ( H λ ) ֒ → B ( H λ ) . Now in the present case { ρ λ } λ ∈ Λ is precisely the subset of those irreducible representations of A which are of dimensionstrictly greater than one. It follows that the subset { ρ λ } λ ∈ Λ ⊂ b A of the spectrum is invariantunder every automorphism of A . On the other hand, because that set is discrete in our case, eachindividual ρ λ is invariant under every one-parameter automorphism group of A . In other words,every one-parameter automorphism group of A (e.g. the modular group ( σ t ) t ∈ R or the scalinggroup ( τ s ) s ∈ R coming from the CQG structure, for instance) • restricts to a one-parameter automorphism group of each ideal K ( H λ ) of A , and also • induces a one-parameter automorphism group of the image A λ of ρ λ .In this context, we have Lemma 3.9.
On each A λ ⊂ B ( H λ ) , the modular automorphism σ t of the Haar measure on A acts as conjugation by a itλ for some non-singular, positive, trace-class operator a λ .Proof. The restriction of the Haar measure h to K ( H λ ) is of the form Tr (cid:0) d · d (cid:1) for some positive,trace-class operator d on H λ . It follows that σ t (cid:12)(cid:12) K ( H λ ) = conjugation by d it . On the other hand, we know from Lemma 1.4 that σ t (cid:12)(cid:12) A λ = conjugation by a itλ for a possibly-unbounded non-singular positive self-adjoint operator a λ on H λ . Since conjugationby a itλ and d it agree on K ( H λ ) , the operators a λ and d must be mutual scalar multiples. Finally,since d is trace-class, so is a λ . (cid:3) Recall that by Lemma 1.4, on each H λ τ s (cid:12)(cid:12) A λ = conjugation by b isλ for a possibly-unbounded positive self-adjoint non-singular operator b λ on H λ . Moreover, becausefor each s, t ∈ R the automorphisms τ s and σ t commute, conjugation by b isλ and a itλ do too (with a λ as in Lemma 3.9). This also follows from a reasoning similar to the one in the proof of Lemma 1.3.
ALEXANDRU CHIRVASITU, JACEK KRAJCZOK, AND PIOTR M. SOŁTAN
Proof of Theorem 3.5.
If at least one of the spaces H λ is finite-dimensional then A is finite-dimensional. Indeed, assume that dim( H λ ) < + ∞ for some λ ∈ Λ and let p ∈ A be the centralprojection corresponding to the unit of K ( H λ ) . Then p A is a finite dimensional ideal in A isomor-phic to K ( H λ ) = B ( H λ ) . It follows that it is also a weakly closed ideal in L ∞ ( G ) ⊆ B ( L ( G )) ,hence the claim is a consequence of [DCKSS18, Theorem 3.4].Due to the above observation, we assume all H λ are infinite-dimensional throughout the restof the proof, and derive a contradiction. Observe that G cannot be of Kac type as K ( H ) has nofaithful bounded traces. Claim.
The operator L λ ∈ Λ b λ implementing the scaling group has finite spectrum. Assuming the claim for now, we can conclude by noting that since ρ (cid:0) τ s ( z ) (cid:1) = M λ ∈ Λ b isλ ρ λ ( z ) b − isλ for all z ∈ A and s ∈ R , and operators L λ ∈ Λ b λ , L λ ∈ Λ b − λ are bounded, the analytic generator τ − i hasbounded extension to all of A . It follows that G is of Kac type (see [NT13, discussion followingExample 1.7.10]), hence we arrive at a contradiction.It thus remains to prove the claim. We will do this with an argument similar to the one usedin the proof of [KS20b, Theorem 13]. Let us for each α ∈ Irr( G ) choose a unitary representation U α ∈ α together with an orthonormal basis in the corresponding Hilbert space in which thepositive operator ρ α is diagonal with entries ρ α, , . . . , ρ α,n α . (cf. [NT13, Section 1.4]) Moreover, let U αu,v ( u, v ∈ { , . . . , n α } ) be the corresponding matrixelements of U α . Recall that we have a quotient map π : A −→ C = A / K ( H ) . Clearly it factors through the canonical Kac quotient A kac ([Soł05a, Appendix]), hence thanks tothe Lemma 1.5 we have π ◦ τ s = π for all s ∈ R . On the other hand, τ s scales U αu,v by ρ − isα,u ρ isα,v ,and hence non-trivially whenever ρ α,u = ρ α,v . Consequently π ( U αu,v ) = 0 whenever ρ α,u = ρ α,v . This means that upon applying π : A → C , the matrix U α = U α , · · · U α ,n α ... . . . ... U αn α , · · · U αn α ,n α becomes block-diagonal, with one block for each distinct value in the spectrum of ρ α . Havingrelabeled that spectrum we can assume that ρ α, , . . . , ρ α,d are all of the instances of a specific eigenvalue ρ > in that spectrum. Now, define x = U α , · · · U α ,d ... . . . ... U αd, · · · U αd,d ∈ B ( C d ) ⊗ B ( H ) = B ( C d ⊗ H ) (3-4)to be the block of U α corresponding to ρ .The fact that the original matrix U α was unitary and the above remark that off-diagonal U αu,v are annihilated by π now imply that (3-4) is unitary mod K ( C d ⊗ H ) . In particular, the operator x has finite-dimensional kernel by Atkinson’s theorem (e.g. [Arv02, Theorem 3.3.2]).Consider the operators A = ⊗ (cid:0)M λ ∈ Λ a λ (cid:1) and B = ⊗ (cid:0)M λ ∈ Λ b λ (cid:1) OMPACT QUANTUM GROUP STRUCTURES ON TYPE-I C ∗ -ALGEBRAS 9 acting on C d ⊗ H . Clearly they are positive, self-adjoint and non-singular, A is bounded and x , B is commute for all s ∈ R . Furthermore, A it , B is commute for all t, s ∈ R . Indeed, it suffices toargue that a itλ , b isλ commute for each λ ∈ Λ . As conjugation by these unitary operators implementsthe modular and the scaling group on A λ , we have a itλ b isλ a − itλ b − isλ = e i ~ st , ∀ s, t ∈ R for some fixed ~ ∈ R (see e.g. [Hal13, p.5 and Definition 14.2]). If ~ = 0 then according to theStone-von Neumann theorem ([Hal13, Theorem 14.8]) there is a unitary operator from H λ onto L ( R ) ⊗ H (for some non-zero Hilbert space H ) and identifying a λ exp (cid:0) − i ~ ddx (cid:1) ⊗ H ,b λ (cid:0) multiplication by e x (cid:1) ⊗ H . Neither of these operators is bounded, hence we get a contradiction. It follows that ~ must vanish,so we can henceforth assume that A and B strongly commute. The following lemma gives us theclaim and ends the proof. (cid:3) Lemma 3.10.
On a Hilbert space H , let • a and b be strongly commuting positive self-adjoint non-singular operators with a bounded, • x be a bounded operator with finite-dimensional kernel, commuting with b is for all s ∈ R ,and such that a it xa − it = ρ it x, ∀ t ∈ R (3-5) for some ρ > .Then, b has finite spectrum.Proof. Naturally, it suffices to assume H is infinite-dimensional (otherwise there is nothing toprove). Let us denote byBorel subsets of R ∋ Ω E Ω ∈ Projections on H the spectral resolution of b . If the latter has infinite spectrum, we could partition R into infinitelymany Ω n , n ∈ Z ≥ with E n := E Ω n non-zero.Because a and b strongly commute, a preserves the subspaces H n := Im( E n ) and thus admitsa spectral resolution Ω P n, Ω thereon. By (3-5) and the fact that x and b strongly commute, x maps each range Im( P n, Ω ) to P n, ρ Ω . The boundedness of a means that we cannot scale by ρ > indefinitely, so the kernel of x (cid:12)(cid:12) H n is non-zero for all n . Since there are infinitely many summands H n , we are contradicting theassumption on the finite-dimensionality of ker x . (cid:3) Remark 3.11.
Due to the argument in the proof of Theorem 3.5 showing that A it and B is com-mute, Lemma 3.10 in fact goes through under the formally weaker assumption that the conjugationactions by a it and b is commute on the algebra of compact operators. References [Arv76] W. Arveson,
An invitation to C ∗ -algebras , Springer-Verlag, New York-Heidelberg, 1976, GraduateTexts in Mathematics, No. 39. MR 0512360[Arv02] , A short course on spectral theory , Graduate Texts in Mathematics, vol. 209, Springer-Verlag,New York, 2002. MR 1865513[BMT01] E. Bédos, G. J. Murphy, and L. Tuset,
Co-amenability of compact quantum groups , J. Geom. Phys. (2001), no. 2, 130–153. MR 1862084[BMT02] E. Bédos, G. J. Murphy, and L. Tuset, Amenability and coamenability of algebraic quantum groups ,Int. J. Math. Math. Sci. (2002), no. 10, 577–601. MR 1931751[Boc95] F. P. Boca, Ergodic actions of compact matrix pseudogroups on C ∗ -algebras , no. 232, 1995, Recentadvances in operator algebras (Orléans, 1992), pp. 93–109. MR 1372527[BS18] T. Brzeziński and W. Szymański, The C ∗ -algebras of quantum lens and weighted projective spaces , J.Noncommut. Geom. (2018), no. 1, 195–215. MR 3782057[BT03] E. Bédos and L. Tuset, Amenability and co-amenability for locally compact quantum groups , Internat.J. Math. (2003), no. 8, 865–884. MR 2013149 [CN16] J. Crann and M. Neufang, Amenability and covariant injectivity of locally compact quantum groups ,Trans. Amer. Math. Soc. (2016), no. 1, 495–513. MR 3413871[CS19] M. Caspers and A. Skalski, On C ∗ -completions of discrete quantum group rings , Bull. Lond. Math.Soc. (2019), no. 4, 691–704. MR 3990385[Daw13] M. Daws, Remarks on the quantum Bohr compactification , Illinois J. Math. (2013), no. 4, 1131–1171.MR 3285870[DCKSS18] K. De Commer, P. Kasprzak, A. Skalski, and P. M. Sołtan, Quantum actions on discrete quantumspaces and a generalization of Clifford’s theory of representations , Israel J. Math. (2018), no. 1,475–503. MR 3819700[DDS19] B. Das, M. Daws, and P. Salmi,
Admissibility conjecture and Kazhdan’s property (T) for quantumgroups , J. Funct. Anal. (2019), no. 11, 3484–3510. MR 3944302[Dix77] J. Dixmier, C ∗ -algebras , North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977, Trans-lated from the French by Francis Jellett, North-Holland Mathematical Library, Vol. 15. MR 0458185[DLRZ02] S. Doplicher, R. Longo, J. E. Roberts, and L. Zsidó, A remark on quantum group actions and nuclearity ,vol. 14, 2002, Dedicated to Professor Huzihiro Araki on the occasion of his 70th birthday, pp. 787–796.MR 1932666[Dą03] L. Dąbrowski,
The garden of quantum spheres , Noncommutative geometry and quantum groups (War-saw, 2001), Banach Center Publ., vol. 61, Polish Acad. Sci. Inst. Math., Warsaw, 2003, pp. 37–48.MR 2024420[Gli61] J. Glimm,
Type I C ∗ -algebras , Ann. of Math. (2) (1961), 572–612. MR 124756[Hal13] B. C. Hall, Quantum theory for mathematicians , Graduate Texts in Mathematics, vol. 267, Springer,New York, 2013. MR 3112817[HR00] N. Higson and J. Roe,
Analytic K -homology , Oxford Mathematical Monographs, Oxford UniversityPress, Oxford, 2000, Oxford Science Publications. MR 1817560[HS02] J. H. Hong and W. Szymański, Quantum spheres and projective spaces as graph algebras , Comm. Math.Phys. (2002), no. 1, 157–188. MR 1942860[Kra01] S. G. Krantz,
Function theory of several complex variables , AMS Chelsea Publishing, Providence, RI,2001, Reprint of the 1992 edition. MR 1846625[KS20a] J. Krajczok and P. M. Sołtan,
The quantum disk is not a quantum group , arXiv e-prints (2020),arXiv:2005.02967.[KS20b] J. Krajczok and P. M. Sołtan,
The quantum disk is not a quantum group , 2020, arXiv:2005.02967.[Kus01] J. Kustermans,
Locally compact quantum groups in the universal setting , Internat. J. Math. (2001),no. 3, 289–338. MR 1841517[Lan73] C. Lance, On nuclear C ∗ -algebras , J. Functional Analysis (1973), 157–176. MR 0344901[Les91] M. Lesch, K -theory and Toeplitz C ∗ -algebras—a survey , Séminaire de Théorie Spectrale et Géométrie,No. 9, Année 1990–1991, Sémin. Théor. Spectr. Géom., vol. 9, Univ. Grenoble I, Saint-Martin-d’Hères,1991, pp. 119–132. MR 1715935[MRW12] R. Meyer, S. Roy, and S. L. Woronowicz, Homomorphisms of quantum groups , Münster J. Math. (2012), 1–24. MR 3047623[NT13] S. Neshveyev and L. Tuset, Compact quantum groups and their representation categories , Cours Spé-cialisés [Specialized Courses], vol. 20, Société Mathématique de France, Paris, 2013. MR 3204665[Pod87] P. Podleś,
Quantum spheres , Lett. Math. Phys. (1987), no. 3, 193–202. MR 919322[PW90] P. Podleś and S. L. Woronowicz, Quantum deformation of Lorentz group , Comm. Math. Phys. (1990), no. 2, 381–431. MR 1059324[Sak66] S. Sakai,
On a characterization of type I C ∗ -algebras , Bull. Amer. Math. Soc. (1966), 508–512.MR 192363[She91] A. J.-L. Sheu, Quantization of the Poisson
SU(2) and its Poisson homogeneous space—the -sphere ,Comm. Math. Phys. (1991), no. 2, 217–232, With an appendix by Jiang-Hua Lu and Alan Wein-stein. MR 1087382[Soł05a] P. M. Sołtan, Quantum Bohr compactification , Illinois J. Math. (2005), no. 4, 1245–1270.MR 2210362[Soł05b] P. M. Sołtan, New quantum “ az + b ” groups , Rev. Math. Phys. (2005), no. 3, 313–364. MR 2144675[Soł10a] P. M. Sołtan, Quantum spaces without group structure , Proc. Amer. Math. Soc. (2010), no. 6,2079–2086. MR 2596045[Soł10b] P. M. Sołtan,
When is a quantum space not a group? , Banach algebras 2009, Banach Center Publ.,vol. 91, Polish Acad. Sci. Inst. Math., Warsaw, 2010, pp. 353–364. MR 2777489[Soł14] P. M. Sołtan,
On quantum maps into quantum semigroups , Houston J. Math. (2014), no. 3, 779–790.MR 3275623[Ste93] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals , Prince-ton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993, With the assistanceof Timothy S. Murphy, Monographs in Harmonic Analysis, III. MR 1232192[SV14] P. M. Sołtan and A. Viselter,
A note on amenability of locally compact quantum groups , Canad. Math.Bull. (2014), no. 2, 424–430. MR 3194189 OMPACT QUANTUM GROUP STRUCTURES ON TYPE-I C ∗ -ALGEBRAS 11 [SW07] P. M. Sołtan and S. L. Woronowicz, From multiplicative unitaries to quantum groups. II , J. Funct.Anal. (2007), no. 1, 42–67. MR 2357350[Tom06] R. Tomatsu,
Amenable discrete quantum groups , J. Math. Soc. Japan (2006), no. 4, 949–964.MR 2276175[Upm96] H. Upmeier, Toeplitz operators and index theory in several complex variables , Operator Theory: Ad-vances and Applications, vol. 81, Birkhäuser Verlag, Basel, 1996. MR 1384981[Var85] V. S. Varadarajan,
Geometry of quantum theory , second ed., Springer-Verlag, New York, 1985.MR 805158[Vis17] A. Viselter,
Weak mixing for locally compact quantum groups , Ergodic Theory Dynam. Systems (2017), no. 5, 1657–1680. MR 3668004[Wor80] S. L. Woronowicz, Pseudospaces, pseudogroups and Pontriagin duality , Mathematical problems intheoretical physics (Proc. Internat. Conf. Math. Phys., Lausanne, 1979), Lecture Notes in Phys., vol.116, Springer, Berlin-New York, 1980, pp. 407–412. MR 582650[Wor87a] ,
Compact matrix pseudogroups , Comm. Math. Phys. (1987), no. 4, 613–665. MR 901157[Wor87b] ,
Twisted
SU(2) group. An example of a noncommutative differential calculus , Publ. Res. Inst.Math. Sci. (1987), no. 1, 117–181. MR 890482[Wor01] , Quantum “ az + b ” group on complex plane , Internat. J. Math. (2001), no. 4, 461–503.MR 1841400 Department of Mathematics, University at Buffalo
E-mail address : [email protected] Institute of Mathematics of the Polish Academy of Sciences, Warsaw
E-mail address : [email protected] Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw
E-mail address ::