Decomposition theorems for unital graph C*-algebras

Guillaume Bellier; Tatiana Shulman

REVIEW 4 major objections 6 minor 2 cited by

Decomposition theorems for unital graph C*-algebras

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Operator norm stability of a unital graph C*-algebra is equivalent to finiteness of a reachability subgraph \tilde G, and residual finite-dimensionality is equivalent to the graph being finite with no cycle that has an entry.

desk verdict A substantial paper that delivers a new decomposition tool and uses it to settle RFD and matricial semiprojectivity for unital graph C*-algebras, though one key lemma has a repairable proof gap. read the letter →

arxiv 2505.12769 v3 pith:57RQO4Y3 submitted 2025-05-19 math.OA

Guillaume Bellier , Tatiana Shulman This is my paper · ORCID

classification math.OA MSC 46L05

keywords graphC*-algebrasamalgamatedfreeproductsresidualfinite-dimensionalitymatricialsemiprojectivityoperatornormstabilityCuntz-Kriegerrelationsreachabilityconditions

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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves complete graph-theoretic characterizations of two approximation properties for unital graph C*-algebras. It shows that $C^*(G)$ is operator norm stable (equivalently, matricially semiprojective) exactly when a certain subgraph $\tilde G$, built from paths leading to cycles and four reachability conditions, is finite. It also shows that $C^*(G)$ is residually finite-dimensional exactly when $G$ is finite and no cycle has an entry. The engine is a decomposition theorem: whenever a graph is split into two subgraphs with no edge of the second part entering the first, $C^*(G)$ is an amalgamated free product of the two subalgebras over a finite-dimensional C*-algebra. This reduces global approximation questions to questions about the pieces and a finite-dimensional amalgam.

What carries the argument

The central mechanism is the decomposition theorem (Theorems 3.2 and 3.3). If $G = G_1 \cup G_2$ is a graph with finitely many vertices, no edge of $G_2$ enters $G_1$, and the two pieces share $n$ vertices, then $C^*(G)$ is isomorphic to $(C^*(G_1) \oplus \mathbb{C}) *_{\mathbb{C}^{n+1}} C^*(G_2)$ when $G_2$ contains all vertices, and to $(C^*(G_1) \oplus \mathbb{C}) *_{\mathbb{C}^{n+2}} (C^*(G_2) \oplus \mathbb{C})$ otherwise. Amalgamating over a finite-dimensional C*-algebra lets the paper transfer the RFD criterion for amalgamated free products and the permanence of semiprojectivity under such products. For stability, the load-bearing object is the subgraph $\tilde G$, whose edges are those reachable from a cycle not contained in $G'$, unreachable from $G'$, reachable from such an unreachable edge, or reachable from the common range of an infinite family of edges outside $G'$; the main theorem equates finiteness of $\tilde G$ with matricial semiprojectivity of $C^*(G)$.

What would settle it

The theorem asserts two exact equivalences, so a single counterexample would settle it. Look for a graph with finitely many vertices whose $\tilde G$ is infinite but whose $C^*(G)$ still lifts every approximate matrix representation; the paper predicts no such graph exists. Equally, a graph with finite $\tilde G$ whose $C^*(G)$ admits an approximate matrix representation with no genuine lift would refute the other direction.

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Extended reading notes

Core claim

The central claim is that for a graph with finitely many vertices, both the RFD property and operator norm stability are decided by the graph alone. Theorem 4.3 states that $C^*(G)$ is residually finite-dimensional if and only if $G$ is finite and no cycle has an entry; the proof decomposes $G$ into a union of disjoint cycles plus an acyclic forest, realizes $C^*(G)$ as an amalgamated free product over $\mathbb{C}^{n+1}$ or $\mathbb{C}^{n+2}$, and applies a known criterion for such products to be RFD. Theorem 5.14 states that $C^*(G)$ is matricially semiprojective if and only if the subgraph $\tilde G$ is finite, where $\tilde G$ is defined by four reachability conditions starting from the subgraph $G'$ of all paths that lead to cycles. The forward direction shows that any homomorphism from $C^*(G)$ into a finite C*-algebra kills every edge outside $\tilde G$, so lifting problems descend to $C^*(\tilde G)$; the reverse direction builds the required lifts directly by induction on vertices, successively removing edges that enter a distinguished vertex. The result is a complete, graph-readable description of operator norm stability for all unital graph C*-algebras.

Load-bearing premise

Everything rests on Lemma 5.6, the claim that any homomorphism from $C^*(G)$ into a finite C*-algebra sends every edge outside $\tilde G$ to zero; the finiteness argument in its proof depends on condition (4) of $\tilde G$'s definition, which rules out an infinite family of edges entering a vertex from outside the path-to-cycle subgraph without forcing every edge reachable from their range into $\tilde G$.

Editorial extensions

If this is right

Every unital graph C*-algebra has a graph-only test for two properties: RFD means finite graph with no entered cycles, and operator norm stability means finite $\tilde G$.
The decomposition theorem turns a one-way split of a graph into an amalgamated free product over a finite-dimensional algebra, so any property preserved by such products can be studied piece by piece.
Theorem 4.3 sharpens the known equivalence between "no cycle has an entry" and quasidiagonality: adding finiteness of the graph upgrades quasidiagonality to residual finite-dimensionality.
Since $C^*(\tilde G)$ is matricially semiprojective exactly when $\tilde G$ is finite, stability failures are visible inside $\tilde G$; no larger portion of the graph can hide them.
The inductive proof of Theorem 5.13 gives a concrete construction that lifts approximate matrix representations whenever $\tilde G$ is finite, so the characterization is effective rather than merely existential.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The one-way no-entry split used in Theorems 3.2 and 3.3 is purely combinatorial, so the same decomposition should apply to any universal algebra generated by projections and partial isometries with local relations of Cuntz-Krieger type.
Condition (4) in the definition of $\tilde G$ is a finiteness guard against edge multiplicity; one could implement the four conditions algorithmically and read operator norm stability directly off a finitely presented graph.
The contrast between the two theorems suggests a hierarchy: operator norm stability tolerates infinite acyclic parts of the graph, while residual finite-dimensionality requires the whole graph to be finite.
Remark 3.4 indicates where the unital assumption is essential, so the boundary of the theorem is already visible: extending to non-unital graph C*-algebras would need a genuinely non-unital amalgamated free product construction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline: this paper actually delivers. The amalgamated free product decomposition (Theorems 3.2 and 3.3) is new, the proof via universal properties is clean, and the paper uses it to give complete characterizations: C*(G) is RFD iff G is finite with no cycle entry (Theorem 4.3), and matricially semiprojective iff the subgraph G-tilde is finite (Theorem 5.14). These were open for the class, and the results are likely correct.

The RFD proof is a nice use of Li-Shen: decompose the graph into cycles plus a forest, embed the pieces into matrix algebras over a common finite-dimensional amalgam, use a dense family of circle representations for the cycles, then apply Li-Shen. The Section 5 machinery is dense but mostly careful, and the induction in Theorem 5.13 is a clever way to handle infinite multiplicities. The citation pattern is sensible: Raeburn-Tomforde for graph algebra basics, Li-Shen for RFD of amalgams, Eilers-Katsura for semiprojectivity.

There are three soft spots, none of which sink the central claims. First, two supporting facts are used without proof or citation: an RFD C*-algebra with finitely many ideals is finite-dimensional, and quasidiagonal plus matricially semiprojective implies RFD. Both are plausible and probably known, but the authors should provide references or short proofs.

Second, and more important, Lemma 5.4 contains a false equality. In the proof, for a path nu=(nu_n,...,nu_1) leading to a cycle, the paper writes nu_{n-1}nu_{n-1}^* = r(nu_{n-1}) = s(nu_n) = nu_n^*nu_n. In a graph C*-algebra, e e^* is only a subprojection of p_{r(e)}, equal to p_{r(e)} only when r(e) receives exactly one edge. The equality fails in general. The written proof of the vanishing lemma is therefore invalid. The repair is immediate: use e e^* <= p_{r(e)}. If pi(nu_n^*nu_n)=0 then pi(p_{s(nu_n)})=0, hence pi(nu_{n-1}nu_{n-1}^*)=0, and the same backward induction works. Since Lemma 5.4 underlies Lemma 5.5, Claim 1 of Theorem 5.6, and Corollary 5.7, this gap must be fixed explicitly, but it does not appear to falsify Theorem 5.14.

Third, the minor typos and the terse proof of Lemma 5.1 are not serious.

Who this is for: anyone working on graph C*-algebras, RFD, or semiprojectivity. The paper deserves a serious referee. I would send it to peer review and ask for a corrected proof of Lemma 5.4 plus the missing references.

Referee Report

4 major / 6 minor

Summary. The paper studies unital graph C*-algebras and gives two structural results. First, it proves decomposition theorems (Theorems 3.2 and 3.3) expressing C*(G) as an amalgamated free product over a finite-dimensional algebra when the graph is split into two subgraphs with no edge from the second entering the first. These decompositions are then used to characterize residually finite-dimensionality: Theorem 4.3 states that C*(G) is RFD if and only if G is finite and no cycle has an entry. The main new result is Theorem 5.14: C*(G) is matricially semiprojective (operator norm stable) if and only if a canonically defined subgraph \tilde G is finite, where \tilde G is built from four reachability conditions relative to the subgraph G' of paths leading to cycles. The proof introduces a vanishing lemma for homomorphisms into finite C*-algebras and uses it to reduce matricial semiprojectivity of C*(G) to that of C*(\tilde G), together with an induction argument for the converse.

Significance. If the main results are correct, this is a substantial contribution. The characterization of RFD graph C*-algebras in Theorem 4.3 is clean and complete for the unital case, and the characterization of matricial semiprojectivity in Theorem 5.14 is a genuinely new and nontrivial graph condition that answers a natural question left open by earlier work on semiprojectivity. The decomposition theorems are of independent interest and are used in a coherent way through the Li--Shen criterion for RFD amalgamated free products. The paper also gives an explicit and checkable graph-theoretic condition, which is a strength. However, the proofs of several key lemmas in Section 5 contain localized errors, and one lemma used in the induction for the main theorem has a gap that needs a different argument. These issues are repairable, and the central claims appear defensible, but the manuscript requires careful revision.

major comments (4)

[Section 5, Lemma 5.4] The proof of Lemma 5.4 contains the false equality ν_{n-1}ν_{n-1}^* = p_{s(ν_n)}. In a graph C*-algebra the CK relation only gives ν_{n-1}ν_{n-1}^* ≤ p_{r(ν_{n-1})} = p_{s(ν_n)}, with equality only when s(ν_n) receives exactly one edge. Since this lemma is used in both directions of Theorem 5.14, the written proof is incomplete. The repair is immediate: from π(ν_n^*ν_n)=0 one obtains π(p_{s(ν_n)})=0, hence π(ν_{n-1}ν_{n-1}^*)=0, and the backward induction goes through. Please correct the argument.
[Section 5, Theorem 5.6, Claim 2] In the proof of Claim 2 the displayed equality "0 ≠ π(e)π(e)^* = π(s(e))" is false; the CK relation is e^*e = p_{s(e)}, not ee^* = p_{s(e)}. The intended argument works with π(e)^*π(e) = π(s(e)) and then uses the CK relation s(e) = Σ_{h∈E} hh^* to find e_1 with π(e_1)≠0, but as written this is an invalid step in the central vanishing lemma. This should be corrected.
[Section 5, Lemma 5.12] The proof of Lemma 5.12 tries to rule out the case where p is the range of an infinite-multiplicity edge belonging to G'. It asserts that then "any edge with range p lies in G'", which is false when p is on a cycle: the incoming cycle edge is in \tilde G and is not in G'. The claimed contradiction therefore does not follow, and the subsequent conclusion that there are only finitely many edges of G\tilde G with range p is not justified in that case. This is load-bearing because it is used to obtain the CK sum relation p = Σ_{r(e)=p} ee^* in the proof of Lemma 5.12. The lemma may still be true, but it requires a different argument for vertices satisfying condition 2), for instance using Corollary 5.7 to show that the infinite entry edges vanish in any representation into the finite C*-algebra ΠM_n/⊕M_n.
[Section 5, Theorem 5.13] The statement of Theorem 5.13 as printed says "If \tilde G is finite, then C*(\tilde G) is matricially semiprojective." This is already Lemma 5.10, while the proof actually proves that C*(G) is matricially semiprojective under the assumption that \tilde G is finite. As printed, the theorem does not provide the converse direction needed in Theorem 5.14. The statement should be corrected to "If \tilde G is finite, then C*(G) is matricially semiprojective."

minor comments (6)

[Section 5, Lemma 5.10] In the displayed decomposition of C*(\tilde G), the second factor should be C*(H_2)(⊕ possibly C), not a second copy of C*(H_1).
[Section 4, Theorem 4.3] The fact that an RFD C*-algebra with finitely many ideals is finite-dimensional is used without proof or citation. This is true, but it should be justified or referenced, especially because it is essential for showing that G_2 is finite.
[Section 5, Lemma 5.10] The implication "Matricial semiprojectivity implies then that C*(\tilde G) is RFD" relies on the standard fact that a quasidiagonal C*-algebra embeds into ΠM_n/⊕M_n, and that an MSP lift of this embedding gives an embedding into ΠM_n. This should be stated explicitly or cited.
[Section 5, Theorem 5.6, final paragraph] The cycle constructed in the final paragraph has an off-by-one indexing: if s(e_N)=s(e_i), the cycle should be (e_N,...,e_{i+1}) (or the indices chosen so that s(e_N)=s(e_{i-1})). The idea is clear, but the indexing should be fixed.
[Section 5, Theorem 5.13, Claim 2] The notation u_i^*u_i = \tilde\psi(\bar f_i^*\bar f_i) is not meaningful because f_i is not an edge of H after removal; it should be u_i^*u_i = \tilde\psi(\overline{s(f_i)}), the lift of the source projection.
[Section 3, Theorem 3.2] The assertion "Since there is no edge of G_2 entering G_1, {p_β}≠∅" is false when G_2 has no edges, e.g., when G is a single cycle and G_2 is the same vertex set with no edges. The decomposition still holds in this case, but the argument should treat it explicitly or remove the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the characterization theorems are derived from Cuntz-Krieger universal properties and independent external theorems.

full rationale

The derivation chain is self-contained. The decomposition theorems (3.2, 3.3) are proven directly from the CK relations and the universal property of amalgamated free products; Lemma 3.1 is a direct identification of CK relations, not an assumption of the target result. The RFD characterization (Theorem 4.3) uses the external Li-Shen theorem (Theorem 2.5), Raeburn's finite-forest and cycle representations (Theorems 2.1 and 2.2), and standard facts on AF algebras and ideals; no fitted parameter is renamed as a prediction. The matricial semiprojectivity characterization (Theorem 5.14) introduces the subgraph tilde-G purely graph-theoretically, then proves the two implications separately: Theorem 5.11 uses the vanishing result Theorem 5.6 (proved from finiteness of finite C*-algebras and CK relations) to descend from C*(G) to C*(tilde-G), and Theorems 5.13 and 5.12 build a lift after removing finitely many edges, relying on Blackadar's external semiprojectivity facts and Lemma 5.10. Lemma 5.10's second direction invokes the already-established Theorem 4.3 after proving quasidiagonality from Schafhauser's external result; this is legitimate reuse of an earlier theorem, not circularity. The only questionable step noted by a skeptical reader, the equality nu_{n-1}nu_{n-1}^* = p_{s(nu_n)} in Lemma 5.4, would be a mathematical gap (and the paper itself uses the CK inequality elsewhere), but a proof gap is not a circular reduction: it does not make the conclusion an input of the argument. No self-citation is load-bearing; the sole self-citation [6] is only an alternative proof and non-unital version of Li-Shen, while the main arguments use external independently published results.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters or empirical inputs appear. The paper's new object \tilde{G} is a precise construction, not a postulated entity. The arguments rely on standard C*-algebra theorems, all cited, plus one uncited standard lemma about RFD algebras with finitely many ideals.

assumptions (8)

standard math ZFC set theory and standard operator algebra theory
All proofs are within standard ZFC and accepted C*-algebra theory; no new foundational axioms.
standard math Universal property of graph C*-algebras with Cuntz-Krieger relations
Section 2.1; C*(G) is the universal C*-algebra generated by projections and partial isometries satisfying the CK-relations, used throughout.
standard math Li-Shen theorem on RFD amalgamated free products
Theorem 2.5, used in Theorem 4.3 to prove RFD via agreeing finite-dimensional inclusions into products of matrix algebras.
standard math Raeburn/Tomforde structure theorems for finite acyclic graphs and single cycles
Theorems 2.1 and 2.2; used to build embeddings in Theorem 4.3 and Lemma 5.10.
standard math Schafhauser: no cycle has an entry iff quasidiagonal for arbitrary graphs
Cited in the introduction and used in Lemma 5.10 to get quasidiagonality of C*(\tilde{G}).
standard math Eilers-Katsura-Ruiz-Tomforde: AF graph C*-algebras have finitely many ideals
Used in Theorem 4.3 (only-if direction) to deduce G2 finite from C*(G2) being RFD with finitely many ideals.
standard math Blackadar: semiprojectivity passes to amalgamated free products over finite-dimensional subalgebras and lifting properties
Used in Lemma 5.10 and Theorem 5.13; specifically Prop 2.23 and 2.32.
domain assumption An RFD C*-algebra with finitely many ideals is finite-dimensional
Stated without proof or citation in the proof of Theorem 4.3; a standard fact but should be cited or proved.

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Pith. "Pith review of Decomposition theorems for unital graph C*-algebras." pith.science (2026). https://pith.science/paper/57RQO4Y3

@misc{pith2026250512769,
  author       = {Pith},
  title        = {Pith review of: Decomposition theorems for unital graph C*-algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/57RQO4Y3}},
  note         = {Machine review of arXiv:2505.12769}
}

read the original abstract

We prove that unital graph C*-algebras often admit a convenient decomposition into amalgamated free products. We use this to give a complete characterization of when a unital graph C*-algebra is residually finite-dimensional and when it is operator norm stable (that is, matricially semiprojective).

Figures

Figures reproduced from arXiv: 2505.12769 by the authors.

**Figure 1.** Examples 3.1. Case G0 2 = G0 . Theorem 3.2. Let G = G1 S G2 be a graph with finitely many vertices such that G0 2 = G0 and no edge of G2 enters G1. Let n be the number of vertices shared by G1 and G2. Then C ∗ (G) ∼= (C ∗ (G1) ⊕ C) ∗Cn+1 C ∗ (G2), where amalgamation is done via the unital embeddings θ1 : C n+1 → C ∗ (G1) ⊕ C and θ2 : C n+1 → C ∗ (G2) defined by θ1(λ1, . . . , λn+1) = (Xn i=1 λip¯i , λn+1), θ2(λ1, . … view at source ↗

**Figure 2.** Embeddings Now we are going to construct representations π1,z of C ∗ (G1) ⊕ C and ˜π2 of C ∗ (G2) (C ∗ (G2) ⊕ C, in the second case respectively) on the space of dimension (k + P µ Nµ −I). (We note that in the first case k + P µ Nµ −I = k). We will view Mk and MP µ Nµ as C*-subalgebras of Mk+ P µ Nµ−I via the embeddings Mk ,→ Mk+ P µ Nµ−I , T 7→ T 0PNµ−I , and MP µ Nµ ,→ Mk+ P µ Nµ−I , T 7→ 0k−I T . For each… view at source ↗

**Figure 3.** Example of G and G′ . Let S ⊂ G and e ∈ G1 . We will say that e can be reached from S if there is a path ν = (e, νn, . . . , ν1) with s(ν) ∈ S. (Here we allow the case ν = e). We define a subgraph G˜ of G. First, we specify what are edges of G˜. An edge e belongs to G˜ if one of the following four conditions holds : (1) e can be reached from a vertex of a cycle whose all edges are not in G′ . On the picture on the r… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Examples of G with G′ in red and G˜ in blue Lemma 5.3. Let G be a graph, A a finite C*-algebra, and π : C ∗ (G) → A a ∗-homomorphism. Let f ∈ G be an edge that enters a cycle. Then π(f) = 0. Proof. Suppose π(f) ̸= 0. We can assume that f enters a cycle (e1, . . . , en)…

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Forward citations

Cited by 2 Pith papers

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For several residually finite-dimensional operator algebras, including the non-commutative disc algebra, no minimal residually finite-dimensional C*-cover exists.
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Graph-theoretic RFD conditions on ultragraphs imply RFD for their Leavitt path algebras and C*-algebras, with equivalences under RFUM2 via branching systems and groupoid models.

Reference graph

Works this paper leans on

12 extracted references · 11 canonical work pages · cited by 2 Pith papers

[1]

Blackadar, Shape theory for C*-algebras, Math

B. Blackadar, Shape theory for C*-algebras, Math. Scand. 56 (1985)

work page 1985
[2]

Dadarlat, Some remarks on the universal coefficient theorem in KK-theory, Operator algebras and mathematical physics (Constanta, 2001), Theta, Bucharest, 65-74, 2003

M. Dadarlat, Some remarks on the universal coefficient theorem in KK-theory, Operator algebras and mathematical physics (Constanta, 2001), Theta, Bucharest, 65-74, 2003

work page 2001
[3]

Dadarlat, Nonnuclear Subalgebras of AF Algebras, American Journal of Mathematics 122(3): 581–597, 2000

M. Dadarlat, Nonnuclear Subalgebras of AF Algebras, American Journal of Mathematics 122(3): 581–597, 2000

work page 2000
[4]

Semiprojectivity and properly infinite projections in graph C*-algebras, Advances in Mathe- matics Volume 317: 108–156, 2017

work page 2017
[5]

Eilers, T

S. Eilers, T. Katsura, E. Ruiz, M. Tomforde, Identifying AF-algebras that are graph C*- algebras, Journal of Functional Analysis, Volume 266, Issue 6, 15 March 2014

work page 2014
[6]

Enders and T

D. Enders and T. Shulman, On the (Local) Lifting Property, arXiv:2403.12224

arXiv
[7]

P. M. Hajac, Sarah Reznikoff, M. Tobolski, Banach Center Publications 31, 2018

work page 2018
[8]

Li and J

Q. Li and J. Shen, A note on unital full amalgamated free products of RFD C*-algebras, Illinois J. Math. 56 (2): 647-659, 2012

work page 2012

Show all 12 references

[9]

Ozawa, About the QWEP conjecture

N. Ozawa, About the QWEP conjecture. Internat. J. Math. 15(5): 501–530, 2004

2004
[10]

Raeburn, Graph algebras, CBMS Regional Conference Series in Mathematics 103 Published for the Conference Board of the Mathematical Sciences, Washington D.C

I. Raeburn, Graph algebras, CBMS Regional Conference Series in Mathematics 103 Published for the Conference Board of the Mathematical Sciences, Washington D.C. by the AMS, Providence, RI, 2005

2005
[11]

Schafhauser, AF-embeddings of graph C*-algebras, J

C. Schafhauser, AF-embeddings of graph C*-algebras, J. Operator Theory, 74: 177–182, 2015

2015
[12]

Tomforde, Graph C*-algebras, Theory, Technique, and Examples, 2011

M. Tomforde, Graph C*-algebras, Theory, Technique, and Examples, 2011

2011

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Blackadar, Shape theory for C*-algebras, Math

B. Blackadar, Shape theory for C*-algebras, Math. Scand. 56 (1985)

work page 1985

[2] [2]

Dadarlat, Some remarks on the universal coefficient theorem in KK-theory, Operator algebras and mathematical physics (Constanta, 2001), Theta, Bucharest, 65-74, 2003

M. Dadarlat, Some remarks on the universal coefficient theorem in KK-theory, Operator algebras and mathematical physics (Constanta, 2001), Theta, Bucharest, 65-74, 2003

work page 2001

[3] [3]

Dadarlat, Nonnuclear Subalgebras of AF Algebras, American Journal of Mathematics 122(3): 581–597, 2000

M. Dadarlat, Nonnuclear Subalgebras of AF Algebras, American Journal of Mathematics 122(3): 581–597, 2000

work page 2000

[4] [4]

Semiprojectivity and properly infinite projections in graph C*-algebras, Advances in Mathe- matics Volume 317: 108–156, 2017

work page 2017

[5] [5]

Eilers, T

S. Eilers, T. Katsura, E. Ruiz, M. Tomforde, Identifying AF-algebras that are graph C*- algebras, Journal of Functional Analysis, Volume 266, Issue 6, 15 March 2014

work page 2014

[6] [6]

Enders and T

D. Enders and T. Shulman, On the (Local) Lifting Property, arXiv:2403.12224

arXiv

[7] [7]

P. M. Hajac, Sarah Reznikoff, M. Tobolski, Banach Center Publications 31, 2018

work page 2018

[8] [8]

Li and J

Q. Li and J. Shen, A note on unital full amalgamated free products of RFD C*-algebras, Illinois J. Math. 56 (2): 647-659, 2012

work page 2012