aa r X i v : . [ m a t h . OA ] J u l CURIOUS PROPERTIES OF FREE HYPERGRAPH C*-ALGEBRAS
TOBIAS FRITZ
Abstract.
A finite hypergraph H consists of a finite set of vertices V ( H ) and a collection ofsubsets E ( H ) ⊆ V ( H ) which we consider as partition of unity relations between projectionoperators. These partition of unity relations freely generate a universal C*-algebra, which we callthe free hypergraph C*-algebra C ∗ ( H ). General free hypergraph C*-algebras were first studied inthe context of quantum contextuality. As special cases, the class of free hypergraph C*-algebrascomprises quantum permutation groups, maximal group C*-algebras of graph products of finitecyclic groups, and the C*-algebras associated to quantum graph homomorphism, isomorphism,and colouring.Here, we conduct the first systematic study of aspects of free hypergraph C*-algebras. Weshow that they coincide with the class of finite colimits of finite-dimensional commutative C*-algebras, and also with the class of C*-algebras associated to synchronous nonlocal games. Wehad previously shown that it is undecidable to determine whether C ∗ ( H ) is nonzero for given H . We now show that it is also undecidable to determine whether a given C ∗ ( H ) is residuallyfinite-dimensional, and similarly whether it only has infinite-dimensional representations, andwhether it has a tracial state. It follows that for each one of these properties, there is H suchthat the question whether C ∗ ( H ) has this property is independent of the ZFC axioms, assumingthat these are consistent. We clarify some of the subtleties associated with such independenceresults in an appendix. Contents
1. Introduction 12. The class of free hypergraph C*-algebras and alternative characterizations 43. Undecidable properties and independence of ZFC 13Appendix A. A Turing machine whose halting is independent of ZFC 17References 191.
Introduction
The quantum permutation group [1, 2] of a finite set of cardinality n is the universal unital C*-algebra generated by a matrix of projections ( p ij ) ni,j =1 satisfying a certain system of equations, which Mathematics Subject Classification.
Primary: 46L99, 03D80; Secondary: 81P13, 03F40.
Key words and phrases. free hypergraph C*-algebra, undecidability, Connes Embedding Problem, nonlocal games,first incompleteness theorem.We thank William Slofstra and Andreas Thom for discussions, the anonymous referee for helpful feedback, WilhelmWinter for suggesting the term “free hypergraph C*-algebra”, and especially Nicholas Gauguin Houghton-Larsen forcopious help with set theory and incompleteness and for detailed feedback on a draft. Without his assistance andpatience, we probably would have stayed completely oblivious to the subtleties discussed in Appendix A. We alsothank David Roberson and William Slofstra for helpful comments on a draft. · · ·· · ·· · ·· · · ... ... ... .... . .
Figure 1.
Quantum permutation groups are free hypergraph C*-algebras gener-ated by partition of unity relations between projections, where these relations areencoded in hypergraphs of this particular form. Each vertex represents a generatingprojection, and each hyperedge stands for a partition of unity relation.state exactly that this matrix is formally unitary. Due to the idempotency and self-adjointness ofeach matrix entry, this unitarity turns out to be equivalent to the condition that the projectionsmaking up each column should form a partition of unity, n X i =1 p ij = 1 , ∀ j = 1 , . . . , n, (1)and similarly for each row, n X j =1 p ij = 1 , ∀ i = 1 , . . . , n. (2)In order to obtain the quantum permutation group, one also needs to equip this C*-algebra witha suitable comultiplication and antipode. But since we are concerned only with the C*-algebrastructure in this paper, we will not discuss these additional structures, but rather focus on theabove partition of unity relations only. These can be conveniently illustrated by a hypergraph asin Figure 1.It is easy to generalize this example to arbitrary hypergraphs, which leads to the definition offree hypergraph C*-algebras. We had introduced free hypergraph C*-algebras previously in [3] inorder to study the phenomenon of quantum contextuality in the foundations of quantum mechanics.There, we had asked whether it is Turing decidable to determine whether C ∗ ( H ) = 0 for a given H or not. We noted that the decidability would follow if every C ∗ ( H ) was residually finite-dimensional.Questions of this type are of interest due to Kirchberg’s QWEP conjecture—or equivalently theConnes Embedding Problem—which can be formulated as asking whether the free hypergraph C*-algebra of the hypergraph shown in Figure 2 is residually finite-dimensional. Building on resultsof Slofstra [4], we had subsequently shown that the problem “Is C ∗ ( H ) = 0 for given H ?” is URIOUS PROPERTIES OF FREE HYPERGRAPH C*-ALGEBRAS 3
Figure 2.
The C*-algebra of this hypergraph is residually finite-dimensional ifand only if the Connes Embedding Eroblem has a positive answer.undecidable, thereby resolving the inverse sandwich conjecture of [3]. It follows that there are freehypergraph C*-algebras which are not residually finite-dimensional.Here, we will go further and also prove that the residual finite-dimensionality itself is undecidable(Theorem 3.4). And similarly for two other finiteness properties, namely the existence of an infinite-dimensional representation without a finite-dimensional one (Theorem 3.5), and the existence ofa tracial state (Theorem 3.6). We also discuss the implications of such undecidability results forquestions concerning independence from the ZFC axioms: for each of these undecidable properties,there are hypergraphs H for which the question whether C ∗ ( H ) has this property of independentof ZFC, assuming that ZFC is consistent; and likewise for any other consistent and recursivelyaxiomatizable formal system which contains arithmetic (Corollary 3.7). There are some treacheroussubtleties hidden in the logical aspects here which we discuss in Appendix A.We also show that the class of free hypergraph C*-algebras is quite natural in itself, from two verydifferent perspective: first, they are exactly the finite colimits of finite-dimensional commutativeC*-algebras (Theorem 2.16); second, they are exactly the C*-algebras associated to synchronousnonlocal games [5] (Theorem 2.19). Also the large number of examples in Section 2 underlines thenaturality.We hope that this paper will provide a unifying perspective on the different contexts in whichfree hypergraph C*-algebras have been used, and also that our undecidability results will indicatethat the general study of free hypergraph C*-algebras may be as interesting as the study of finitelypresented groups. Conventions.
We work with unital
C*-algebras and unital ∗ -homomorphisms throughout, evenwhen we do not explicitly say so; in particular, all the finitely presented C*-algebras that weconsider are defined in terms of the corresponding universal property in C ∗ alg , the category ofunital C*-algebras and unital ∗ -homomorphisms. Caveat.
As far as we can see, our free hypergraph C*-algebras have no direct relation with themore widely studied graph C*-algebras . We welcome suggestions for a different terminology thatwill avoid the possibility for confusion here. Depending on what is easier to parse in each case, we either draw hyperedges either as closed curves that surroundvertices, or as non-closed curves that connect vertices. Also, the colours that we use have no meaning beyond aidingthe human eye.
TOBIAS FRITZ The class of free hypergraph C*-algebras and alternative characterizations
For our purposes, a hypergraph H consists of a finite set of vertices V ( H ) and a collectionof edges E ( H ) ⊆ V ( H ) such that each vertex is contained in at least one edge, or equivalently S E ( H ) = V ( H ). Definition 2.1.
Given a hypergraph H , the free hypergraph C*-algebra is the finitely presentedC*-algebra C ∗ ( H ) := * ( p v ) v ∈ V ( H ) (cid:12)(cid:12)(cid:12)(cid:12) p v = p ∗ v = p v , X v ∈ e p v = 1 ∀ e ∈ E ( H ) + C ∗ alg (3) where the subscript indicates that this is a presentation in the category of (unital) C*-algebras. Since all generators are projections, there is a preexisting bound on the norm of each ∗ -polynomialin the generators, resulting in an Archimedean quadratic module in the ∗ -algebra of noncommutativepolynomials. This guarantees that the so defined finitely presented C*-algebra indeed exists, andit can be constructed as the completion of the complex ∗ -algebra C [ H ] of (4) with respect to theseminorm k x k := sup π : C [ H ] →B ( H ) k π ( x ) k , where π : C [ H ] → B ( H ) ranges over all ∗ -representations, and the process of taking the completionalso comprises taking the quotient with respect to the null ideal of all elements x ∈ C [ H ] with π ( x ) = 0 first. See e.g. [6, p. 3] for a review of this type of construction. Remark . Since free hypergraph C*-algebras are defined as universal C*-algebras given by afinite presentation, we generally do not have an explicit concrete form for them, in the sense ofan explicitly described embedding into some B ( H ). A notable and interesting exception is the freehypergraph C*-algebra associated to the hypergraph consisting of two non-overlapping hyperedgescontaining two vertices each. Equivalently, this is universal C*-algebra generated by two projections,or the group C*-algebra C ∗ ( Z ∗ Z ); see Example 2.6. Since this group is amenable (e.g. since itis isomorphic to Z ⋊ Z ), the maximal and reduced group C*-algebras coincide, and we thereforehave a concrete description available [7].However, in general we are not aware of a precise definition of what would make a finitelypresented C*-algebra “concrete”. It would be plausible to define concreteness as the existenceof a countable family of states which are recursively enumerable in the sense of an algorithmwhich computes the value of any of these states on any ∗ -polynomial in the generators up to anydesired accuracy; or equivalently in terms of operators on ℓ ( Z ) which represent the generators andare computable in the sense of an algorithm which returns any matrix entry with respect to thestandard basis with any desired accuracy. However, we will not pursue this concreteness questionany further in this work.One way think that working with the C*-algebra C ∗ ( H ) is overkill, since we are merely analyzinga combinatorial structure, and might therefore just as well do something entirely algebraic. Forthis reason, it may also be interesting to study the analogously defined ∗ -algebra C [ H ] := * ( p v ) v ∈ V ( H ) (cid:12)(cid:12)(cid:12)(cid:12) p v = p ∗ v = p v , X v ∈ e p v = 1 ∀ e ∈ E ( H ) + ∗ - alg . (4) URIOUS PROPERTIES OF FREE HYPERGRAPH C*-ALGEBRAS 5
Here, the subscript ∗ - alg indicates that this presentation is now understood to encode the corre-sponding universal property in the category of complex unital ∗ -algebras, in order to keep track ofthe difference to (3). So concretely, C [ H ] is the free noncommutative algebra over C generated byelements p v modulo the two-sided ideal generated by the elements p v − p v for all v and 1 − P v ∈ e p v for all e . Upon declaring each generator to be self-adjoint, C [ H ] becomes a ∗ -algebra, since theideal is automatically self-adjoint.The problem with this is that the canonical ∗ -homomorphism C [ H ] → C ∗ ( H ) is generally notinjective, since the orthogonality relations p v p w = 0 for distinct v, w ∈ e fail to be satisfied evenin the trivial case where H only consists of a single edge [5, p. 17], but they necessarily hold in C ∗ ( H ) per Example 2.5. We can try to fix this by quotienting by these relations as well, meaningthat we consider C [ H ] /I for the two-sided ideal generated by the elements p v p w , where v and w range over all distinct v, w ∈ V ( H ) occurring in some common edge. This is again a self-adjointideal by definition, so that C [ H ] /I is another ∗ -algebra. One may now hope to have a purelyalgebraic description of a dense subset of C ∗ ( H ), in the sense that the canonical ∗ -homomorphism C [ H ] /I → C ∗ ( H ) is injective. Unfortunately, this is not the case, in a very extreme sense: Theorem 2.3.
There are hypergraphs H for which C ∗ ( H ) = 0 despite C [ H ] /I = 0 . As the proof shows, this is a simple consequence of results of Helton, Meyer, Paulsen and Satri-ano [5] and Ortiz and Paulsen [8]. Our specific H which displays this behaviour is of the form ofExample 2.12, arising from quantum graph 4-colouring. Proof.
We use the free hypergraph C*-algebras of synchronous nonlocal games and their ∗ -algebras C [ H ] /I , as discussed before the upcoming Theorem 2.19. Then we consider the game of 4-colouringthe complete graph K . Surprisingly, a ∗ -algebraic 4-colouring exists [5, Theorem 6.2], meaningthat C [ H ] /I = 0. However, no C*-algebraic 4-colouring can exist [8, Proposition 4.10], so that C ∗ ( H ) = 0. (cid:3) It is possible to extract from the proof a concrete description of additional algebraic relationsarising between projections together with partition of unity relations, such that these relations areenforced in any Hilbert space representation, but do not follow from the standard orthogonalityrelations discussed above.Thus understanding the algebra of C ∗ ( H ) remains mostly an open problem: Problem 2.4.
Is there a concrete description of the kernel of the canonical ∗ -homomorphism C [ H ] → C ∗ ( H ) ? In particular, is there an algorithm to determine whether a given x ∈ Q [ H ] is inthe null ideal, meaning that π ( x ) = 0 for all ∗ -representations π : C [ H ] → B ( H ) ? Examples of free hypergraph C*-algebras.
Example . The most trivial examples are those hypergraphs H which consist of only a singleedge, say with n vertices. In this case, the free hypergraph C*-algebra is the C*-algebra generatedby n projections p , . . . , p n satisfying P i p = 1. The universal property of C ∗ ( H ) states exactlythat, for any C*-algebra A , the partitions of unity in A , given by tuples of projections p , . . . , p n satisfying P i p i = 1, must be in bijection with their classifying homomorphisms C ∗ ( H ) → A .Since the orthogonality relations p i p j = δ ij are automatic , it is easy to see that this freehypergraph C*-algebra is isomorphic to C n with p i the i -th standard basis vector, or by Fourier It is well-known that if p + q ≤ p and q on a Hilbert space, then p and q are orthogonal,meaning that pq = 0. TOBIAS FRITZ transform equivalently to the group C*-algebra C ∗ ( Z n ). As a degenerate special case, we alsoobtain the zero algebra C ∼ = 0 (associated to the hypergraph consisting of the empty edge). For n = 2, the partitions of unity are necessarily of the form ( p, − p ) for a single projection p , so thatthe homomorphisms C → A are precisely the classifying homomorphisms of projections in A . Example . Given two hypergraphs H and H , we can take their disjoint union H ∐ H , whichresults in the free product C ∗ ( H ∐ H ) ∼ = C ∗ ( H ) ∗ C ∗ ( H ) , where the isomorphism is easy to see since both sides have the same universal property. (Here, ∗ denotes the coproduct in C ∗ alg , which is the free product amalgamated over 1.) In combinationwith the previous example, we obtain that all C*-algebras of the form C n ∗ . . . ∗ C n k ∼ = C ∗ ( Z n ∗ . . . ∗ Z n k )are free hypergraph C*-algebras, corresponding to the class of hypergraphs having only disjointedges. Example . As we will see in Lemma 2.14, imposing commutation between the generating pro-jections of a free hypergraph C*-algebra results in another free hypergraph C*-algebra. Thereforealso any graph product [9] of finite cyclic group has a maximal group C*-algebra which is a freehypergraph C*-algebra. The proof of Proposition 2.13 gives a concrete example.
Example . Continuing with the theme of maximal group C*-algebras C ∗ (Γ) for finitely presentedgroups Γ, one also obtains a free hypergraph C*-algebra whenever Γ is a solution group in the senseof [10]; the construction of the hypergraph is essentially that of [11, Lemma 10], which shows thatone also obtains a free hypergraph C*-algebra upon additionally enforcing J = − C ∗ (Γ) is (isomorphic to) a free hypergraph C*-algebra. Example . There are many nontrivial hypergraphs H with C ∗ ( H ) ∼ = 0. For example if H consistsof three vertices such that any two of them make up an edge, then C ∗ ( H ) is generated by threeprojections p , p , p with p + p = p + p = p + p = 1. This implies 2 p i = 1 for every i . Thisleads to 0 = 1, since this is the only way for p i = to be a projection. Therefore C ∗ ( H ) is the zeroalgebra. Theorem 3.1 can be interpreted as stating that it is impossible to classify all the ways inwhich C ∗ ( H ) = 0 can occur. Example . As outlined in the introduction, the quantum permutation group on n elements [1,2]is the free hypergraph C*-algebra generated by projections ( p ij ) ni,j =1 satisfying the relations (1,2).These are equivalent to postulating that the matrix of projections U := ( p ij ) ni,j =1 must be formallyunitary, n X j =1 p ∗ ji p jk = δ jk , which is a set of equations easily seen to be equivalent to the requirement that each row andeach column of the matrix must be a partition of unity, assuming that each matrix entry p ij is aprojection. Example . A generalization of the previous example is that of quantum graph isomorphism [12,13]. We fix a number of vertices n and consider the quantum permutation group as above. Given a URIOUS PROPERTIES OF FREE HYPERGRAPH C*-ALGEBRAS 7 graph G on n vertices, its adjacency matrix is the matrix A G ∈ { , } n × n with A ij = 1 for vertices i, j ∈ G if and only if i ∼ j , meaning that i and j are adjacent. Now given graphs G and G ′ , letus consider the quantum permutation group generated by U = ( p ij ) ni,j =1 subject to the additionalrelation U A G = A G ′ U . The ( j, k )-entry of this matrix equation is X k : k ∼ j p ik = X k : i ∼ ′ k p kj , (5)where ∼ ′ denotes adjacency in G ′ . We now argue that this set of relations can be encoded in aset of orthogonality relations between the generating projections, so that we again obtain a freehypergraph C*-algebra by Lemma 2.14; see also the discussion in the second half of [13, Section 2.1].The crucial observation is that we have X k : k ∼ j p ik + X k : k j p ik = 1 , X k : i ∼ ′ k p kj + X k : i ′ k p kj = 1 . Thanks to these relations, it follows that (5) is equivalent to the orthogonality relations X k : k j p ik ⊥ X k : i ∼ ′ k p kj , X k : k ∼ j p ik ⊥ X k : i ′ k p kj . And each one of these in turn is equivalent to each projection appearing in the sum on the left to beorthogonal to each projection appearing in the sum on the right. This reproduces the presentationin terms of orthogonality relations as in [13, Section 2.1]. (See also the earlier [14, Lemma 3.1.1] inthe case G = G ′ .) This defines the quantum isomorphism free hypergraph C*-algebra associatedto G and G ′ . Also [13, Theorems 2.4 and 2.5] follow, since the abstract C*-algebra stays the same;it’s only the presentation which changes upon exchanging the relations (5) with the orthogonalityrelations.[13] defines graphs G and G ′ to be quantum isomorphic if this free hypergraph C*-algebra isnonzero. A stronger notion of quantum isomorphism was considered in the prior work [12], whereit was required in addition that the free hypergraph C*-algebra must have a finite-dimensionalrepresentation. These two notions of quantum isomorphism are not equivalent [12, Result 4],although the latter is trivially stronger. There also is an explicit example of two graphs which arenot isomorphic, but quantum isomorphic in both senses [12, Theorem 6.4].In the special case G = G ′ , the above free hypergraph C*-algebra becomes the quantum auto-morphism group of the graph G as introduced by Banica [15]. Further specializing to G being thecomplete or the empty graph on n vertices recovers the quantum permutation group itself, since inthis case the commutation U A G = A G U holds trivially. Example . A closely related class of examples is given by quantum graph homomorphism ratherthan isomorphism. Given graphs G and G ′ , the C*-algebra is quantum graph homomorphisms isthe free hypergraph C*-algebra generated by a family of projections ( p ij ) indexed by i ∈ G and j ∈ G ′ , such that if i ∼ i and j ′ j , then again we have orthogonality p i j p i j = 0 [8]. In thiscase, we are not aware of an alternative formulation in terms of the adjacency matrices analogous tothe one from Example 2.11. Then again there are various possible definitions for when one says thata quantum graph homomorphism exists, depending on whether one requires the free hypergraph Here, all graphs are finite and do not contain parallel edges. Although [12, 13] only consider undirected graphswithout loops, the definition and basic properties—such as the equivalence of (5) with the orthogonality relations—donot need this assumption and apply to all directed graphs, potentially with loops. This is not exactly how the definition is phrased, but is known to be equivalent [13, Theorem 2.1].
TOBIAS FRITZ
C*-algebra to have a finite-dimensional representation as in [16], or a representation with a tracialstate, or just to be nonzero [8].In the special case where G ′ = K n is the complete graph, the existence of a quantum graphhomomorphism G → K n means (by definition) that G is quantum n -colourable , for which one canagain consider the same variations [5].The following is a variation on Kirchberg’s formulation of the Connes Embedding Problem,i.e. the QWEP conjecture [17]. Although—as we will see in the proof—the corresponding freehypergraph C*-algebra is really just a group C*-algebra, we nevertheless present the argument inthe hope that future work will be able to reduce the size of the relevant hypergraph (Figure 2)further, possibly resulting in a free hypergraph C*-algebra that is not a group C*-algebra. Proposition 2.13.
The Connes Embedding Problem has a positive answer if and only if the freehypergraph C*-algebra associated to Figure 2 is residually finite-dimensional.Proof.
The Connes Embedding Problem is equivalent to the question whether the maximal groupC*-algebra C ∗ ( F × F ) is residually finite-dimensional [6]. Now in place of F × F , one can justas well take any other group which contains this one as a subgroup of finite index; such a group ise.g. G := ( Z ∗ Z ) × ∼ = P SL ( Z ) × .Now the free hypergraph C*-algebra associated to the hypergraph depicted in Figure 2 is isomor-phic to C ∗ ( G ), as one can see by first drawing it in a way analogous to [3, Figure 7(g)], and thennoting that some of the edges are redundant, in the sense that the partition of unity relations whichthey implement are implied by the others via simple linear combinations [3, Appendix C]. (cid:3) We will encounter further ways to construct free hypergraph C*-algebras in Theorems 2.16and 2.19.
Permanence properties.
As we have seen in the above examples, one often wants to imposeadditional relations on the generating projections of a free hypergraph C*-algebra. We now showthat one frequently obtains another free hypergraph C*-algebra.
Lemma 2.14.
Let C ∗ ( H ) be a free hypergraph C*-algebra and v, w ∈ V ( H ) . Then imposing eitherof the following additional relations results in another free hypergraph C*-algebra: (a) p v = 0 ; (b) p v = p w ; (c) p v p w = 0 ; (d) p v ≤ p w . (e) p v p w = p w p v . (f) p v + p w ≤ .Proof. (a) We simply remove vertex v from H .(b) We add one additional vertex u and the two edges { u, v } and { u, w } , implementing therelations p u + p v = 1 and p u + p w = 1. Clearly such a projection p u exists if and only if p v = p w , and then it is unique.(c) We add one additional vertex u and the edge { u, v, w } . Then a projection p u satisfyingthe relation p u + p v + p w = 1 exists if and only if p v and p w are orthogonal, which means p v p w = 0. In this case, the projection p u is clearly unique.(d) Let e ∈ E ( H ) be any edge with w ∈ e . Then p v ≤ p w is equivalent to p v p w ′ = 0 for all w ′ ∈ e \ { w } , so that we can apply (c). URIOUS PROPERTIES OF FREE HYPERGRAPH C*-ALGEBRAS 9 (e) We add four additional vertices vw , ¯ vw , v ¯ w and ¯ v ¯ w , together with edges { v, ¯ vw, ¯ v ¯ w } , { w, v ¯ w, ¯ v ¯ w } , { vw, ¯ vw, v ¯ w, ¯ v ¯ w } . Then we obtain the relations p v = p vw + p v ¯ w and p w = p vw + p ¯ vw , which implies com-mutativity thanks to p vw + p v ¯ w + p ¯ vw + p ¯ v ¯ w = 1, which implies pairwise orthogonality.Conversely, given p v p w = p w p v , the new projections are uniquely determined as products,such as p ¯ vw = (1 − p v ) p w . (cid:3) (f) This is as in (e), but with the vertex vw removed. Lemma 2.15.
Every free hypergraph C*-algebra is isomorphic to some other free hypergraph C*-algebra C ∗ ( H ) , where all edges of H have cardinality , and any two edges intersect in at most onevertex. This is similar to the reduction of SAT to 3-SAT in computational complexity. See Remark 3.3for more on this.
Proof.
Let e ∈ E ( H ) be an edge of the largest cardinality, and suppose that | e | >
3. Write thisedge as a disjoint union e = e ∪ e , such that | e | ≥ | e | ≥
2. Then introduce two newvertices s, t and three new edges e ′ := e ∪ { s } , e ′ := e ∪ { t } , ˆ e := { s, t } . The resulting partition of unity relations are P v ∈ e p v + p s = 1 and P v ∈ e p v + p t = 1 as well as p s + p t = 1. It is easy to see that such p s and p t exist if and only if P v p v = 1, which is the originalpartition of unity relation. We thus obtain a new hypergraph with an isomorphic C*-algebra,but with one edge of cardinality | e | less. Hence iterating this process will result in an equivalenthypergraph where all edges have cardinality at most 3.We next show that one can achieve cardinality equal to 3. If there is an edge of cardinality 0,then we have C ∗ ( H ) = 0 anyway, in which case can be replaced by any H ′ with C ∗ ( H ′ ) and suchthat all edges have cardinality 3, which happens e.g. for the hypergraph consisting of the four facesof a tetrahedron. Otherwise, we take all edges of cardinality less than 3 and add new vertices untilthey all have cardinality 3. For each new vertex v , we need to guarantee that P v = 0; this can beachieved e.g. by attaching a copy of the hypergraph depicted in Figure 3.Finally, for all pairs of edges that intersect in two vertices, say e = { t, u, v } and e ′ = { u, v, w } , weremove the edge e ′ and replace it by two new vertices x and y and edges { t, x, y } and { x, y, w } . Thisworks because under the assumption p t + p u + p v = 1, there exist projections p x and p y satisfying p t + p x + p y = 1 = p x + p y + p w if and only if p t = p w , or equivalently p u + p v + p w = 1. (cid:3) Hypergraph C*-algebras via finite colimits.
As we saw in Example 2.5, the finite-dimensionalcommutative C*-algebras C n are trivially all free hypergraph C*-algebras. Theorem 2.16.
Up to isomorphism, the class of free hypergraph C*-algebras coincides with theclass of finite colimits in C ∗ alg of finite-dimensional commutative C*-algebras. Intuitively, the reason is that every such colimit has a presentation given by a finite set ofprojections as generators, and relations of two types: one saying that a certain subset of gener-ating projections should form a partition of unity; and one type saying that some two generatingprojections should be equal. The latter type reduces to the former by Lemma 2.14(b). v Figure 3.
Hypergraph H with the property that p v = 0 in C ∗ ( H ). Proof.
We first show how to express C ∗ ( H ) as such a colimit for a given hypergraph H = ( V, E ).We can present C ∗ ( H ) as the unital C*-algebra with a family of generators ( p v,e ) where v ∈ V and e ∈ E are such that v ∈ e , subject to the relations that X v ∈ e p v,e = 1 (6)for all e ∈ E , as well as p v,e = p v,e ′ for all v ∈ V and e, e ′ ∈ E with v ∈ e, e ′ . In other words, C ∗ ( H ) is the unital free product of the C e for all e ∈ E , subject to suitable identifications of theirgenerating projections. This means that C ∗ ( H ) is canonically isomorphic to the colimit C*-algebraof a diagram looking like C C e ... ... C C e ′ where the copies of C on the left are indexed by the set of vertices v ∈ V , and the objectson the right by the edges e ∈ E and are given by the C*-algebras freely generated by the abovepartitions of unity (6). There is an arrow between v and e if and only if we have incidence v ∈ e . Thecorresponding ∗ -homomorphism C → C n is the classifying homomorphism of p v,e . By construction,the colimit C*-algebra of this diagram has the same universal property as C ∗ ( H ) in the presentationabove.The converse direction is a bit trickier. Let J be a finite category indexing a diagram D : J → C ∗ alg such that every D ( J ) for J ∈ J is commutative finite-dimensional. By definition of thecolimit, homomorphisms colim D → A classify tuples of homomorphisms ( α J : D ( J ) → A ) J ∈ J suchthat for every f : J → J ′ , we have α J ′ ◦ D ( f ) = α J . Homomorphisms D ( J ) → A are the same thingas partitions of unity in A indexed by Spec( D ( J )), and the D ( f ) : D ( J ) → D ( J ′ ) are determined URIOUS PROPERTIES OF FREE HYPERGRAPH C*-ALGEBRAS 11 by Spec( f ) : Spec( D ( J ′ )) → Spec( D ( J )). Now consider the hypergraph H with vertices V ( H ) := a J ∈ J Spec( D ( J )) , and edges E ( H ) := { e f,v } indexed by all morphisms f : J → J ′ in J and all v ∈ Spec( D ( J )), where e f,v := { v } ∪ { v ′ ∈ Spec( D ( J ′ )) | Spec( D ( f )) = v } . In particular for f = id J and any v ∈ Spec( D ( J )), the associated edge contains precisely theelements of Spec( D ( J )).We now verify that C ∗ ( H ) has the same universal property as the original colimit. By con-struction, into every A ∈ C ∗ alg , C ∗ ( H ) classifies families of homomorphisms ( α J : D ( J ) → A ) J ∈ J satisfying the additional compatibility requirement that for every morphism f : J → J ′ and element v ∈ Spec( D ( J )), we have α J ( e v ) + X v ′ ∈ J ′ : Spec( D ( f ))( v ′ ) = v α J ′ ( v ′ ) = 1 . But since the α J ′ ( v ′ ) are assumed to form a partition of unity as v ′ ∈ Spec( D ( J ′ )) varies, thisequation is equivalent to α J ( e v ) = α J ′ (Spec( D ( f ))( v )), or equivalently α J = α J ′ ◦ D ( f ) as describedabove. (cid:3) Remark . The above result does not imply that the class of free hypergraph C*-algebras isclosed under finite colimits, since a finite diagram of free hypergraph C*-algebras does not need toarise from a diagram of diagrams.
Problem 2.18.
What is an explicit example of a finite colimit of free hypergraph C*-algebras thatis not itself isomorphic to a free hypergraph C*-algebra?
Hypergraph C*-algebras via synchronous nonlocal games. A nonlocal game for two players A and B is a cooperative game specified by the following data: finite sets I and O of inputs and outputs for each player, and a map λ : I × I × O × O → { , } which specifies which combinations of inputs and outputs are considered winning. A round of thegame consists of a referee choosing x ∈ I and y ∈ I and communicating the value to player A andplayer B , respectively; the two players must then return respective output values a ∈ O and b ∈ O to the referee. The players should try to coordinate these values in such a way that the winningcondition λ ( x, y, a, b ) = 1 is satisfied for any choice of input values a, b ∈ O . The game is winnable if there is a strategy which achieves this. It is synchronous [5, p. 4] if λ ( x, x, a, b ) = δ a,b holds forevery input x ∈ I . Intuitively, this means that the two players must definitely agree on the answerwhen asked the same question, because otherwise they lose.Nonlocal games are studied in particular in quantum information theory, where the coordinationof the two players may be aided by making use of shared quantum entanglement, which allows theplayers to win certain nonlocal games that would not be winnable without quantum entanglement.In general it is a difficult question to determine whether a given game is winnable with the helpof quantum entanglement, and there also are various slightly different definitions of what types ofquantum entanglement are allowed, concerning e.g. whether infinite-dimensional Hilbert spaces are In general, one can also take different sets (of different cardinalities) for the two players. But as far as thestandard properties of nonlocal games are concerned, there is no loss of generality in assuming that both playershave the same sets of inputs and outputs. permitted or not [4]. In order to address subtleties of this kind, and also as a general tool for thestudy of synchronous nonlocal games, [5, p. 9] has introduced a ∗ -algebra associated to each suchgame. If we take its C*-completion, then it is the universal C*-algebra generated by projections { p x,a : x ∈ I, a ∈ O } subject to the relations X a ∈ O p x,a = 1 ∀ x ∈ I,p x,a p y,b = 0 whenever λ ( x, y, a, b ) = 0 . (7)In order to write it as a free hypergraph C*-algebra, we use the proof of Lemma 2.14(c), whichresults in the following hypergraph: the set of vertices is O × I , plus one additional vertex for everyquadruple ( x, y, a, b ) which satisfies λ ( x, y, a, b ) = 0. There is one edge for every a ∈ I , and it is givenby { ( x, a ) : a ∈ O } ; furthermore, there is one edge for every ( x, y, a, b ) with λ ( x, y, a, b ) = 0, andit contains that extra vertex together with ( x, a ) and ( y, b ). It is noteworthy that the associated ∗ -algebra C [ H ] /I defined after (4) is canonically isomorphic to the ∗ -algebra associated to the gameintroduced at [5, p. 9]: the synchronicity condition guarantees that the orthogonality relationswhich follow from the partition of unity relations in (7) are already contained in the orthogonalityrelations of (7). Using this fact, it is easy to check that the two ∗ -algebras enjoy equivalent universalproperties. Theorem 2.19.
Up to isomorphism, the class of free hypergraph C*-algebras coincides with theclass of C*-algebras of synchronous nonlocal games. Furthermore, it is enough to restrict to syn-chronous nonlocal games with only | O | ≤ outputs. In quantum information terms, the proof is a translation of contextuality into a nonlocal game,which is conceptually not a new idea [18]. However, our translation achieves this for the hypergraphapproach to contextuality [3], which results in a more general translation than what seems to beknown.
Proof.
We have shown above that the relations (7) define a free hypergraph C*-algebra.Constructing a 3-outcome synchronous nonlocal game for a given hypergraph H is less obvious.We can assume by Lemma 2.15 that H only has edges of cardinality 3 which pairwise intersect inat most one vertex. We enumerate the vertices in each edge e as v e, , v e, , v e, . We then need toconstruct our nonlocal game such that the winning condition λ encodes which vertices coincide,i.e. when v e,i = v e ′ ,j for edges e, e ′ ∈ E ( H ) and i, j ∈ { , , } . Although it is not clear how toimplement this directly, we can equivalently encode the relations v e,i ⊥ v e ′ ,j ′ for all j ′ = j in theform of the orthogonality relations of (7) as follows. Let the set of inputs by given by the edges, I := E ( H ), and put λ ( x, y, a, b ) := ( v x,a = v y,b ′ for b ′ = b, , for the winning condition. It is easy to see that this defines a synchronous nonlocal game. ItsC*-algebra is given by generating projections p e,i satisfying partition of unity relations P i p e,i = 1,as well as orthogonality relations p e,i ⊥ p e ′ ,j whenever there is j ′ = j with v e,i = v e ′ ,j ′ . (cid:3) The λ constructed in the proof has the following alternative definition: if a ∩ b = ∅ , then theplayers always win; if a = b , then the players win if and only if they choose the same vertex;otherwise | a ∩ b | = 1, in which case the players win either if they both choose the common vertex URIOUS PROPERTIES OF FREE HYPERGRAPH C*-ALGEBRAS 13 in a ∩ b , or both do not choose it. In particular, this shows that the winning condition λ enjoys thesymmetry property λ ( x, y, a, b ) = λ ( y, x, b, a ). Problem 2.20.
Is every free hypergraph C*-algebra isomorphic to one arising from quantum graphisomorphism as in Example 2.11? From quantum graph homomorphism as in Example 2.12?Remark . The C*-algebras associated to synchronous nonlocal games have also been generalizedto imitation games in [19, Definition 4.2]. By Lemma 2.14(c), it is easy to see that every imitationgame C*-algebra is also a free hypergraph C*-algebra. Conversely, every synchronous nonlocalgame C*-algebra, and therefore also every free hypergraph C*-algebra, is an imitation game C*-algebra [19, Example 4.4]. In conclusion, the class of imitation game C*-algebras also coincideswith the class of free hypergraph C*-algebras.3.
Undecidable properties and independence of ZFC
By the definition of C ∗ ( H ) in terms of generators and relations, the unital representations C ∗ ( H ) → B ( H ) are in bijection with families of closed subspaces in H indexed by the vertices V ( H ), such that certain subsets of the family correspond to pairwise orthogonal subspaces thatspan H . As in [11], we call such a configuration of subspaces a quantum representation or just representation of H , assuming that H 6 = 0.
Nontriviality.
We have C ∗ ( H ) = 0 if and only if H has a quantum representation in some Hilbertspace H ; it is enough to assume H to be separable without loss of generality. Based on results ofSlofstra [4], we had proven in [11, Corollary 11] the inverse sandwich conjecture from [3]: Theorem 3.1.
There is no algorithm to determine whether C ∗ ( H ) ? = 0 for a given H . In fact, the proof of [11, Collary 11] based on the methods of [20] shows that the problem isnevertheless semi-decidable, since there is a non-terminating algorithm. Since the undecidabilityproof is via reduction from the word problem for groups, which in turned is undecidable by reductionfrom the halting problem, we can therefore conclude that C ∗ ( H ) ? = 0 has the Turing degree of thehalting problem. Remark . The commutative analogue of the decision problem C ∗ ( H ) ? = 0 asks whether there isan assignment of a truth value to each vertex, such that exactly one vertex in every edge is true.Since all our hypergraphs are finite, this is a Boolean satisfiability problem and therefore triviallydecidable (but NP-complete, by NP-completeness of 1-in-3-SAT). Remark . Determining whether C ∗ ( H ) ? = 0 can also be understood as a quantum satisfiabilityproblem [11,21]. Due to the equivalence of Lemma 2.15, where we had shown that every free hyper-graph C*-algebra is computably isomorphic to one where all edges have cardinality 3, determiningwhether C ∗ ( H ) ? = 0 is undecidable even when all edges of H have cardinality at most 3, which is aresult of [21]. Residual finite-dimensionality.
In [11, Corollary 13], we had also shown that there are C ∗ ( H )which are not residually finite-dimensional. In fact, we can now say something stronger: Theorem 3.4.
There is no algorithm to determine whether C ∗ ( H ) is residually finite-dimensionalfor a given H . This result matches the empirical difficulty of answering the Connes Embedding Problem, knownto be equivalent to the residual finite-dimensionality of the free hypergraph C*-algebra of Figure 2.
Proof.
Suppose that we had such an algorithm. Then we can solve the decision problem C ∗ ( H ) ? = 0as follows: deciding whether C ∗ ( H ) is nontrivial is equivalent to deciding whether k k = 1 or k k =0 in C ∗ ( H ). Using ideas from semidefinite programming and enumeration of finite-dimensionalrepresentations, it was shown in [20] that the norm of every polynomial in the generators of afinitely presented (or recursively presented) C*-algebra is computable if the C*-algebra is residuallyfinite-dimensional. So for given H , we first run the assumed residual finite-dimensionality oracle. If H is residually finite-dimensional, then we run the algorithm of [20] to determine whether k k = 1or k k = 0. In the other case, if H is not residually finite-dimensional, then we already know thatit must be nontrivial. Hence we have an algorithm that decides C ∗ ( H ) ? = 0. (cid:3) In contrast to the situation with Theorem 3.1, we do not even know whether residual finite-dimensionality of C ∗ ( H ) is semi-decidable. Infinite-dimensional representations.
A result which is stronger than the existence of C ∗ ( H )which is not residually finite-dimensional is the following: Theorem 3.5.
There are H such that C ∗ ( H ) is nonzero, but has no finite-dimensional represen-tation. Moreover, there is no algorithm to determine whether this is the case for a given H . As per Remark 3.9, the existence of H which only has infinite-dimensional representations is notnew. It means that there are certain finite configurations of closed subspaces which can only berealized in infinite-dimensional Hilbert spaces. Proof.
This is very much analogous to the proof of Theorem 3.4. The only difference is that thefinite-dimensional representations may now not be dense in the dual. However if k k = 1, then thiswill still be attained in any finite-dimensional representation, which is enough for the algorithmof [20] to work.This proves the undecidability. The existence of H for which the decision problem has a positiveanswer trivially follows. (cid:3) Again, we do not know whether the problem of Theorem 3.5 is semi-decidable.
Existence of tracial states.
Another interesting property of free hypergraph C*-algebras is theexistence of a tracial state.
Theorem 3.6.
There is no algorithm to determine whether C ∗ ( H ) has a tracial state for a given H .Proof. By Theorem 2.19 and the fact that the proof is constructive, it is enough to prove this for C*-algebras of synchronous nonlocal games. But then it is known that such a C*-algebra has a tracialstate if and only if the game has a perfect commuting-operator strategy [5, Theorem 3.2(3)]. Usingthe construction of synchronous games associated to binary constraint systems [22, Corollary 4.4(1)],the claim follows from Slofstra’s undecidability result for binary constraint systems [4, Corollary 3.3]. (cid:3)
In this case, we know that the problem is semi-decidable, and therefore also Turing equivalentto the halting problem: if there is no tracial state, then there is again a hierarchy of semidefiniteprograms which will detect this [23].
URIOUS PROPERTIES OF FREE HYPERGRAPH C*-ALGEBRAS 15
Independence from the ZFC axioms.
The relation between undecidability and independenceguaranteed by Theorem A.1 allows us to translate our undecidability results into independenceresults.
Corollary 3.7.
Let F be a formal system which is recursively axiomatizable and contains elemen-tary arithmetic. Then if F is consistent, there are hypergraphs H to H such that each of thefollowing sentences is independent of F : (a) C ∗ ( H ) = 0 ; (b) C ∗ ( H ) is residually finite-dimensional; (c) H has infinite-dimensional representations but no finite-dimensional ones; (d) C ∗ ( H ) has a tracial state.In particular, this holds true when F is given by the standard ZFC axioms.Proof. All four relevant undecidability proofs are ultimately by reduction from the halting problem.We can therefore apply Theorem A.1. (cid:3)
In each case, what we mean by “independent of F ” is the metamathematical statement “canneither be proven nor disproven in F ”. Since the proof of Theorem A.1 is constructive, and soare all the computational reductions involved in reducing the halting problem to our undecidabledecision problems, we conclude that it is possible in principle to write down these hypergraphsexplicitly. However, we have not attempted to do so in the case where F is ZFC, since we expecttheir descriptions to be prohibitively large, and we do not anticipate to gain much additional insightfrom doing so. Although it is possible that one can take H to be the hypergraph of Figure 2 forZFC, so that the Connes Embedding Problem becomes independent of the ZFC axioms, we haveno particular reason to believe that this is the case. Remark . Finding explicit examples of H for which C ∗ ( H ) = 0 only has infinite-dimensionalrepresentations should be easier than finding examples of the independence from ZFC, since nowone needs to retrace the computational reductions only starting with any Turing machine that doesnot halt, or more directly any word in the generators of a finitely presented group that does notrepresent the unit element. This will give an explicit example of such H . Concluding remarks.
In the previous three subsections, we have considered three different prop-erties of free hypergraph C*-algebras. For C ∗ ( H ) = 0, the negation of the second property statesthat C ∗ ( H ) has some finite-dimensional representation. In terms of this property, we have thefollowing implications:residually finite-dimensional = ⇒ ∃ finite-dimensional representation = ⇒ ∃ tracial state . Remark . As was pointed out to us by William Slofstra, it is not hard to see that the firstimplication is strict: take a solution group Γ which is not residually finite; such a group is knownto exist by Slofstra’s embedding theorem [4]. Then C ∗ (Γ) is a free hypergraph C*-algebra byExample 2.8, but is not residually finite-dimensional. However, C ∗ (Γ) has a finite-dimensionalrepresentation since Γ has at least the trivial representation.As was pointed out to us by David Roberson, also the second implication is strict: it is knownthat a synchronous nonlocal game has a perfect commuting-operator strategy if and only if theassociated free hypergraph C*-algebra C ∗ ( H ) has a tracial state, and a perfect finite-dimensionalstrategy if and only it has a finite-dimensional representation [5, Theorem 3.2]. Now there aresynchronous nonlocal games for which the former type of strategy exists, but the latter does not [22,Corollary 4.6]. Embarrassingly, we not know whether every nontrivial free hypergraph C*-algebra has a tracialstate. This is a question due to Andreas Thom:
Problem 3.10 (Thom) . Is there H such that C ∗ ( H ) is nonzero, but has no tracial state? If the answer is positive, then the decision problems addressed by Theorems 3.1 and 3.6 aredistinct. As was pointed out to us by David Roberson, the answer is known to be negative forall free hypergraph C*-algebras arising from quantum graph isomorphism as in Example 2.11 [13,Theorem 4.4].Our results indicate that many properties of free hypergraph C*-algebras are undecidable. Thisis reminiscent of the
Adian–Rabin theorem for finitely presented groups, which gives extremelygeneral sufficient conditions for a property of group presentations to be undecidable.
Problem 3.11.
Is there some analog of the Adian–Rabin theorem for free hypergraph C*-algebras?
In particular, we expect a negative answer to the following interesting question:
Problem 3.12.
Is there an algorithm to determine whether C ∗ ( H ) is commutative for a given H ? URIOUS PROPERTIES OF FREE HYPERGRAPH C*-ALGEBRAS 17
Appendix A. A Turing machine whose halting is independent of ZFC
This section is based on a series of fruitful discussions with Nicholas Gauguin Houghton-Larsen.We do not make any claims of originality here, but merely include this appendix for the sake ofcompleteness. Indeed Theorem A.1 below seems to be a well-known folklore result, but not readilyavailable in the literature in this form.It is a standard observation that if ZFC is consistent, then there is a Turing machine T ind forwhich the question whether it halts on a blank tape is independent of the ZFC axioms of set theory.This is not specific to ZFC: just like G¨odel’s incompleteness theorems, this holds for any sufficientlyexpressive and recursively axiomatizable formal system F . This is Theorem A.1 below.There are two closely related standard arguments employed when trying to prove this [24–26], which we now sketch. However, the subtlety with these arguments is that both actuallyrequire stronger assumptions than consistency. While the following exposition is not intended tobe completely precise in the details, we keep it formal enough to illustrate the difficulties involved,which are easy to overlook. A reader not interested in these difficulties may proceed directly to thestatement and proof of Theorem A.1.In either kind of argument, one reasons about Turing machines in F by encoding every Turingmachine T via its description number ˆ T . One also uses a natural number predicate Halt in F suchthat Halt ( ˆ T ) is a sentence expressing the halting of T , in the sense that T halts on the blank tape = ⇒ F ⊢ Halt ( ˆ T ) . (8)For many reasonable F , such as e.g. the ZFC axioms, it is natural to assume that also the converseimplication holds: if F proves Halt ( ˆ T ), then T does indeed halt. Similarly, if F is consistent, thenthe contrapositive of (8) shows that if proves ¬ Halt ( ˆ T ), then T does indeed not halt. So usingthe converse of (8) as a soundness assumption, we can reason as follows: if F is such that either F ⊢ Halt ( ˆ T ) or F ⊢ ¬ Halt ( ˆ T ) for all T , then we can define T halt-solver to be the Turing machinewhich takes another Turing machine T as input and enumerates all consequences of the axioms of F until it finds a proof of Halt ( ˆ T ) or a proof of ¬ Halt ( ˆ T ). Then by the assumption that F decidesall instances of Halt ( ˆ T ), we conclude that T halt-solver always terminates, thereby solving the haltingproblem. Since this is absurd, one of our assumptions must have been false, meaning either that T is inconsistent, or that some instance Halt ( ˆ T ) is independent of F .In the case of ZFC (and a suitable metatheory), the relevant soundness requirement—namelythe converse of (8)—would follow e.g. from the existence of a standard model, which is a naturalenough additional assumption. Nevertheless, it is also perfectly possible that F is some kind of settheory like ZFC such that the set of natural numbers in F contains nonstandard numbers, and thata given Turing machine T halts in F after nonstandard many steps; it has been argued that thisis a real possibility even in the case of ZFC [27]. If this happens, then the proof of its halting canclearly not be externalized, and our hypothetical Turing machine T ind does not solve the haltingproblem correctly on all instances.The other standard argument for proving the existence of a Turing machine whose halting isindependent of F is closely related, but requires ω -consistency; although this is a weaker assumptionthan soundness, it is still strictly stronger that consistency. G¨odel’s second incompleteness theoremis concerned with a sentence Con( F ) which expresses the consistency of F , and states that if F is consistent, then F Con( F ). The informal statement that Con( F ) expresses the consistencyof F means formally that Con( F ) is a sentence of the form ∀ n. Con(
F, n ), where Con(
F, n ) states This is certainly not a new observation, see e.g. https://mathoverflow.net/questions/130789/are-the-two-meanings-of-undecidable-related? that the first n consequences of the axioms of F do not contain a contradiction, in the sense thatif n ∈ N is an external natural number, then F ⊢ Con(
F, n ) if and only if the first n consequencesof F do indeed not contain a contradiction. So we now take T ind to be the Turing machine whichenumerates all consequences of the axioms of F and halts as soon as it encounters a contradiction.Since the proof of the equivalence between halting of T ind and consistency of F can be internalized,we conclude that F decides Halt ( ˆ T ind ) if and only if it decides Con( F ). But since we alreadyknow that F Con( F ), it is enough to argue that F
6⊢ ¬
Con( F ) as well, which is equivalentto F
6⊢ ∃ n. ¬ Con(
F, n ). And this is because if F ⊢ ∃ n. ¬ Con(
F, n ) were the case, then we wouldhave an external n ∈ N with F ⊢ ¬ Con(
F, n ) by ω -consistency, in which case F would actually beinconsistent. Thus Halt ( ˆ T ind ) is independent of F .In order to see that ω -consistency is necessary in order to make this argument, it is enoughto note that there are consistent F such that F ⊢ ¬ Con( F ). Such an F also proves Halt ( ˆ T ind ),although T ind would not actually halt, thereby violating soundness as well.In conclusion, the standard arguments for proving the existence of a Turing machine whosehalting is independent require an assumption stronger than mere consistency. Nevertheless, thereis another argument which proves that consistency is enough after all. In contrast to the firstargument above, it is even constructive. Theorem A.1.
Let F be a formal system which is recursively axiomatizable and contains elemen-tary arithmetic. Then if F is consistent, there is an explicit Turing machine T ind whose halting isindependent of F , F Halt ( ˆ T ind ) , F
6⊢ ¬
Halt ( ˆ T ind ) . Proof.
We use Kleene’s symmetric version of the first incompleteness theorem [26, p. 69]. The sets K : = { ˆ T | T halts and accepts on the blank tape } ,K : = { ˆ T | T halts and rejects on the blank tape } are strongly separable, since simply running a Turing machine T until it halts provides a proof ofmembership in K or K , and this proof can be formalized in F .Now for given T , let T ′ be the Turing machine which runs T and accepts if T halts and accepts,but branches into an infinite loop if T halts and rejects. Then the function ˆ T ˆ T ′ is recursive.The predicate ˆ T Halt ( ˆ T ′ ) expresses that for a given description number ˆ T , the modified Turingmachine T ′ halts, and it strongly separates K and K . Therefore by Kleene’s symmetric versionof the first incompleteness theorem, we obtain an explicit Turing machine T such that F Halt ( ˆ T ′ )and F
6⊢ ¬
Halt ( ˆ T ′ ). Therefore we can take T ind := T ′ . (cid:3) See e.g. mathoverflow.net/a/256862/27013.
URIOUS PROPERTIES OF FREE HYPERGRAPH C*-ALGEBRAS 19
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