At a discrete index parameter, a finite consecutive set of Jones projections already fills the identity. This eliminates a scalar summand on which all the projections vanish, and makes compression faithful on earlier finite algebras. Those two facts force the entire truncated path diagram.
Construction and proof sources: The canonical weighted path algebra is constructed in Propositions 9.1–9.2 and Theorem 9.3 and Theorems 9.4–9.6 of Paths, local projections and a faithful trace. Lemmas 13.1–13.2 and Corollary 13.3 below prove the full-window support and faithful compression; Theorem 13.4 recognizes the complete discrete sequence through Theorem 8.6 of Matrix inclusions and the Markov trace. Propositions 13.5–13.6 and Theorem 13.7 retain the continuous-range alternative and the faithful Markov-trace recognition. The trace and word reduction are Lemma 11.1 and Proposition 11.2 of Removing a projection produces a subfactor.
Fix ,
Let be selfadjoint projections on a nonzero Hilbert space , satisfying
Assume the specific nondegeneracy condition
This condition concerns the ranges of the projections. It is stronger than merely adjoining the identity to the algebra they generate.
The zero quantum integer closes every finite window
Put and . The recursion from Lemma 7.1 constructs projections through , since its denominators are positive. These projection identities are algebraic and hold in the present representation without assuming any trace.
Lemma 13.1. For every ,
Proof. First use the window starting at one. Let , the largest projection annihilated by . The largest-projection assertion follows inductively from the recursion as in Theorem 12.7, and does not use the trace.
The same expansion used to prove that recursion gives the useful identity
For completeness, write , where . The operator commutes with . Thus the two terms in become and , using . Their difference is (13.4).
At , its coefficient is zero because . Hence . Since this equals , we have . The projection already annihilates the earlier generators and commutes with all generators at distance at least two from its last generator. It therefore commutes with the entire sequence. The adjacent relation then propagates its zero product to each later generator: if , then
Condition (13.2) forces . This proves (13.3) for the first window.
For a window starting at , its recursively constructed is again the largest joint-kernel projection. The preceding argument makes it annihilate the generator just to the right of the window. Applying the recursion with the window order reversed constructs the same largest joint-kernel projection, so it also annihilates the generator just to the left, if that generator exists. All remaining generators commute with it by distance. Thus it is central relative to the whole sequence and annihilates every generator by propagation in both directions. Nondegeneracy again gives . This proves every window identity.
In particular, the C*-algebra generated by the , even without explicitly adjoining an identity, is already unital. A finite join of projections belongs to their finite-dimensional generated algebra; Lemma 11.1 ensures that this algebra is finite dimensional.
The joint-kernel projection also specifies the frontier precisely. In the algebra generated by canonical projections, namely ,
is the identity of the new scalar block at vertex , when , and is zero once . Every old block is in the ideal generated by the last projection, by (9.11), while the unique strictly increasing frontier path is annihilated by all projections. This proves the description. The projection is not the identity of the rightmost surviving block after the cutoff. This qualification, together with the generator-to-path-level convention (9.22), makes precise the “far right vertex” assertion in Takesaki's Lemma XIX.3.13.
A later projection cannot lose an earlier block
Let , with .
Lemma 13.2. If is at distance at least two from every generator of , then is faithful on .
Proof. The projection commutes with , so this map is a *-homomorphism into its corner. Its kernel is an ideal of the finite-dimensional algebra , say for a central projection . If the kernel is nonzero, then . The projection commutes with all generators for , because every word defining it uses earlier generators at distance at least two from them. The adjacent relations propagate to all . But the join of is one by Lemma 13.1, so . The kernel is therefore zero.
Corollary 13.3. The initial joint-kernel projections satisfy
Proof. The last assertion is Lemma 13.1. For the earlier ones, . If for , identity (13.4), whose coefficient is nonzero, gives . The projection uses only . Lemma 13.2 therefore gives . Descending reaches the contradiction .
The extension at each stage is forced
Denote the finite path algebras by , with their canonical projections .
Theorem 13.4. For every , there is a unique isomorphism
carrying each , , to . These isomorphisms respect the inclusions. Consequently the C*-algebra generated by the nondegenerate sequence is isomorphic to the truncated path algebra, by an isomorphism preserving all the generators.
Proof. Start with the scalar levels. Suppose the generator-preserving identifications have been proved through . Apply the word reduction to : its elements are sums of and with . Since commutes with ,
In the canonical path model the same formulas describe the trace-preserving expectation . To verify this last assertion, use the same word reduction and compress by . Multiplication by on is faithful, because Theorem 9.5 gives . Thus compression determines a unital positive bimodular map taking to and to . It preserves the trace, again by Theorem 9.5, so its trace pairings characterize it as . For , the expectation is the identity on scalars and (13.6) has only its first case.
Transport that expectation to using the induction hypothesis. Equations (13.6) show the compression identity needed by Theorem 8.6. Lemma 13.2 supplies faithfulness of on . The recognition theorem identifies the central-support part of with the finite-dimensional basic construction of , preserving the earlier algebra and the new projection.
Let be that central support, and put . On the complement, . The adjacent relation, multiplied by the central , gives , and then successively for every . Thus the complementary algebra is at most the scalar block . Moreover, is the largest projection annihilating all : any such projection commutes with the generated algebra, kills the ideal of , and so lies below .
If , Corollary 13.3 says that this joint-kernel projection is nonzero. If , Lemma 13.1 says it is zero, because the first projections already join to one. Hence the scalar complementary block is present exactly when the graph still has a new vertex at distance .
The canonical path algebra has exactly the same decomposition. The old blocks are the basic construction, by the matrix-unit identity (9.11); the only possible remaining block is the scalar frontier vertex. Every canonical vanishes on the strictly increasing frontier path, and that block exists precisely when . Thus the isomorphism of the old parts, together with the scalar character on the frontier if present, gives . It respects every generator and the preceding inclusion.
Uniqueness follows because the listed generators generate each finite algebra. The compatible finite-level isomorphisms are isometric, and therefore extend to their norm closures. This proves the theorem.
For , Lemma 13.1 says each , so the theorem gives the scalar algebra. For , the unique trace and II₁ closure follow from Theorem 9.6. The theorem concerns the generated C*-algebra; a given representation may have arbitrary Hilbert-space multiplicity, and is not thereby identified with its tracial representation.
A single finite extension has less information than this infinite nondegenerate sequence. Extending a finite projection algebra through the cutoff, Theorem 44.2, proves its exact statement: full support in the earlier commutant recognizes the basic-construction part, but an extra common-kernel scalar summand must be excluded at the critical step. Lemma 13.1 supplies that exclusion here; Corollary 13.3 also forces the earlier scalar frontiers to survive.
A condition that cannot be dropped
If one drops (13.2), add a Hilbert-space summand on which every . The relations still hold. The algebra with the identity adjoined then has an extra scalar summand, distinguished by the character sending all the projections to zero. Thus the nondegeneracy hypothesis has a concrete algebraic role.
There is also a sharp distinction between the discrete range and . In that latter range, the relations alone allow compact-operator representations as well as the path model.
Proposition 13.5. For every , there is a nondegenerate sequence of rank-one projections satisfying (13.1). Its nonunital generated C*-algebra is the compact operators on an infinite-dimensional Hilbert space, and its unital generated algebra is their unitization. The nonunital algebra has no tracial state.
Proof. On finitely supported sequences, prescribe formal vectors with Gram matrix
This form is positive definite. On a finite support of length , its matrix is , where is the path adjacency matrix. Proposition 9.1 gives its least eigenvalue
, even at . Complete the resulting inner-product space to ; the vectors are unit vectors with dense span, and finite subsets are linearly independent. Thus is infinite dimensional.
Let project onto . Orthogonal lines give at distance at least two. For adjacent lines, sandwiching rank-one projections gives , and likewise in reverse. Their ranges have dense span, proving (13.2).
Products along a consecutive chain are nonzero scalar multiples of the rank-one maps between its endpoint vectors. Their linear span therefore includes the rank-one maps between any two , and is norm dense in the compact operators. Conversely, each generator is compact. This proves the algebra assertions. A bounded positive trace on the compacts has the same value on all rank-one projections. Sums of orthogonal rank-one projections have norm one, so boundedness forces for every , and hence . Such a trace vanishes on the dense finite-rank operators, so is zero. It cannot be a state.
This example explains why a factorial Markov-trace construction, such as the path model, must be specified in the continuous range. One cannot assign a Markov tracial state to every representation merely from its projection relations. In particular, Takesaki's Theorems XIX.3.16 and XIX.3.18 must be read for the canonical path algebra (or a representation retaining that full algebra) with the specified Markov trace. The rank-one sequence of Proposition 13.5 satisfies the same relations and is nondegenerate, yet its finite generated algebras are for projections: their ranges span an -dimensional subspace, all rank-one maps on that subspace are generated, and the identity supplies the scalar complement. This is a proper quotient of the full half-Pascal path system once its other blocks appear. No tracial state on its unitization can give the nonzero Markov value to these compact rank-one projections, by the trace argument above. Existence and factoriality of the canonical trace are independently proved in Theorem 10.5; they are not conclusions for every sequence with these relations.
Positivity without a trace still excludes the gaps
The algebraic positivity argument can also prove the allowed-parameter restriction without assuming a trace.
Proposition 13.6. Suppose a sequence of projections on a nonzero Hilbert space satisfies (13.1) for some and is nondegenerate in the sense of (13.2). Then
Proof. A nonzero projection in the sequence gives , since . Every projection is nonzero: a zero one would force its neighbors, and then the whole sequence, to be zero. Thus . If there is nothing to prove.
Write , with . Suppose is not an integer. Choose with
The algebraic part of Lemma 7.1 constructs the projection , which is the largest joint-kernel projection for . Identity (13.4) gives
The left side is positive. On the right, commutes with the positive , so their product is positive, but its coefficient is negative. Both sides must therefore be zero. In particular and .
The first equality makes annihilate the generator just outside its initial window. It already annihilates the generators inside that window and commutes with all remaining generators by distance. It is consequently central relative to the entire sequence, and the adjacent relations propagate its zero product to all later generators. Nondegeneracy gives .
Exactly the same calculation applies to any consecutive window of generators. For a window away from the left endpoint, reverse its order and use (13.9) again to annihilate the generator just to its left. Its largest joint-kernel projection is independent of that order. The distance relation and propagation now work on both sides, showing that every such window joins to one.
This window identity makes every sufficiently late compression faithful on an earlier finite generated algebra, by the proof of Lemma 13.2. That proof uses only finiteness of the generated algebra from Lemma 11.1, propagation of a kernel central projection, and a later window joining to one; its discrete-parameter assumption is no longer needed once the window identity has been established.
Apply this to , whose last generator is at distance two from . Faithfulness gives . For , (13.4) has nonzero coefficient. Thus implies , and the same faithful compression gives . Descending reaches , a contradiction. Hence for an integer , proving (13.8).
If the original sequence is degenerate but contains a nonzero projection, restrict to the closed span of all its ranges. Its orthogonal complement is the common kernel, so this span reduces every generator. The restriction is nondegenerate and preserves the nonzero relations. Therefore the same allowed-parameter conclusion holds. The identically zero sequence is the only exception.
A faithful Markov trace retains every block
In the infinite-line range, a trace condition supplies what the relations alone do not. Here “Markov” specifies its value against every element of the preceding algebra, rather than only the traces of the individual projections.
Theorem 13.7. Let , , and let be projections in a unital C*-algebra satisfying (13.1). Put
Suppose has a faithful tracial state satisfying
Then there are unique compatible isomorphisms onto the infinite-line path algebras, preserving every generator and the trace. They extend to a trace-preserving isomorphism of with the canonical path AF algebra. Its tracial GNS closure is an approximately finite-dimensional II₁ factor. No nondegeneracy condition on an independently prescribed Hilbert-space representation is assumed.
Proof. Work in the faithful GNS representation of . Every is finite dimensional by the word reduction of Lemma 11.1. Define quantum integers by , , and . They are all positive for : when they equal ; when , they equal . Thus the algebraic recursion of Lemma 7.1 defines at every level, with . It is the largest projection annihilating . To check this last assertion, an annihilating projection satisfies ; the recursion then gives inductively, since it annihilates the new . Hence .
The Markov hypothesis proves its trace formula directly. With , traciality and (13.10) give
Starting from , the recursion therefore gives
The identity with follows from (9.18), or from its rescaled vertex-weight proof. In particular, every finite initial list has a nonzero joint-kernel projection.
We now inductively identify the algebras and their traces. The two scalar levels agree. Suppose has been identified with , preserving its generators and trace. A reduced word in is a sum of and terms , with . Commutation and the sandwich relation give
Here the expectation is transported from the canonical path inclusion; its values are and , as verified in the proof of Theorem 13.4. For , only the scalar term occurs. This compression is faithful on the earlier algebra. Indeed, if , with , then
and faithfulness of gives .
The finite recognition theorem, Theorem 8.6, now identifies the ideal of in with the full basic construction of , preserving the old algebra and the new projection. If is its central support and , the adjacent relations successively force . Hence . Conversely a joint-kernel projection kills the ideal of , so this is precisely . Equation (13.11) gives .
The canonical infinite path algebra has exactly the same decomposition. Identity (9.11) gives its old basic-construction blocks. Its additional block is the scalar frontier at vertex , which exists at every level. Matching these two parts gives a generator-preserving isomorphism , compatible with the preceding one.
To check the trace at the new level, reduce a word there to , with coefficients in . Its trace is
The canonical trace has the same expression by Theorem 9.5 and agrees on by induction. Thus the isomorphism preserves the new trace. This closes both inductive assertions.
The generators give uniqueness at every level. Compatible finite-level isomorphisms are isometric, so they extend to the norm closures. On tracial GNS spaces the map is unitary, because it preserves ; it intertwines the left actions and hence the von Neumann closures. Theorem 10.5 proves that the canonical closure is an AFD II₁ factor, including . This conclusion concerns the tracial GNS closure. It makes no assertion about the weak closure in a different representation.
In Commuting projections need not be maximal abelian, the faithful tensor-product trace satisfies (13.10) on every preceding finite tensor algebra, hence on , with . Theorem 13.7 therefore identifies that generated algebra and trace with the canonical infinite path model. The noncommuting operators in Theorem 5.2 transport to its tracial GNS closure, preserving their nonzero commutator and their commutation with every even generator. Thus that counterexample applies to the cyclic Markov representation named in Takesaki's Remark XIX.3.17, rather than only to an unrelated representation of the projection relations.
Exercises
Exercise 13.1 — introductory. For , how many consecutive projections must join to one? What happens at ?
Solution. The first value corresponds to , so every two consecutive projections join to one. The second corresponds to , so every single projection is one.
Exercise 13.2 — intermediate. In Lemma 13.2, why is distance one insufficient for the stated proof?
Solution. The map need not be multiplicative if does not commute with the earlier algebra. The kernel then need not be an ideal represented by a central projection. At distance at least two, commutation follows directly from the relations and supplies both steps of the proof.
Exercise 13.3 — intermediate. For the model, at which level does the scalar frontier last appear?
Solution. The graph's greatest vertex distance is . Thus the scalar frontier last appears in , at vertex three. At level four every reachable vertex appeared two levels earlier, and is wholly the old basic-construction part. Equivalently and .
Exercise 13.4 — advanced. At , realize the first three Gram vectors in , and compute the least eigenvalue of their Gram matrix.
Solution. Take
Their norms are one; the consecutive inner products are and the other is zero. The Gram eigenvalues are , , so its least eigenvalue is . This verifies the finite window at the critical parameter without assuming a positive lower spectral bound for the entire infinite Gram operator.
Exercise 13.5 — advanced. At , compute the trace of the common kernel of the first projections in a faithful Markov model. Explain both why every finite scalar frontier survives and why this does not give a nonzero common kernel of the entire infinite sequence in the tracial closure.
Solution. The common-kernel projection is , and (13.11) gives
Thus the scalar block cannot vanish at any finite stage. The projections decrease as the list of annihilated generators increases. Their strong limit is the common kernel of the whole sequence. Normality of the tracial closure's trace gives ; faithfulness gives . A decreasing sequence of nonzero projections can therefore have zero limit. This argument uses the normal tracial closure and does not infer the same conclusion for an arbitrary external representation.
References
Vaughan F. R. Jones, Index for subfactors, Inventiones Mathematicae 72 (1983), 1–25.