OA-Problems – Seite 2

The Blackadar-Handelman conjectures

Von hannesthiel | November 1, 2021

In [1], Blackadar and Handelman made two conjectures:

Conjecture 1: (Below Theorem I.2.4 in [1]) Let $A$ be a unital C*-algebra. Then the set $\mathrm{LDF}(A)$ of lower-semicontinuous dimension functions is dense in the set $\mathrm{DF}(A)$ of dimension functions.

Conjecture 2: (Below Theorem II.4.4 in [1]) Let $A$ be a unital C*-algebra. Then the compact, convex set $\mathrm{DF}(A)$ is a Choquet simplex.

Here, a dimension function on a unital C*-algebra $A$ is a map $d\colon M_\infty(A)\to[0,\infty]$ that associates to every matrix over $A$ a positive number and satisfying properties that generalize the classical properties of the rank of complex matrices:

If $a$ and $b$ are orthogonal matrices (that is, $ab=a^*b=ab^*=a^*b^*=0$ ), then $d(a+b)=d(a)+d(b)$ .
If $a$ is Cuntz-dominated by $b$ (that is, there exist sequences $(r_n)_n$ and $(s_n)_n$ such that $\|a-r_nbs_n\|\to 0$ ), then $d(a)\leq d(b)$ .
$d(1)=1$ .

Such a dimension function is said to be lower-semicontinuous if it is lower-semicontinuous with respect to the norm-topology, that is, whenever $(a_n)_n$ is a sequence in $M_\infty(A)$ converging to some $a$ , then $d(a)\leq\liminf_n d(a_n)$ . We equip $\mathrm{DF}(A)$ with the topology of pointwise convergence. This gives $\mathrm{DF}(A)$ the structure of a compact, convex set. We note that the subset $\mathrm{LDF}(A)\subseteq\mathrm{DF}(A)$ is usually not closed (indeed, the conjecture is that it is dense). There is another (natural) topology on $\mathrm{LDF}(A)$ giving it the structure of a compact, convex set that is even a Choquet simplex. Here, a compact, convex $K$ set is a Choquet simplex if the set $\mathrm{Aff}(K)$ of continuous, affine functions $K\to\mathbb{R}$ satisfies Riesz interpolation, that is, given $f_1,f_2,g_1,g_2\in\mathrm{Aff}(K)$ satisfying $f_j\leq g_k$ for $j,k\in\{1,2\}$ there exists $h\in\mathrm{Aff}(K)$ such that $f_j\leq h\leq g_k$ for $j,k\in\{1,2\}$ . Choquet simplices have the property that every element can be represented in a unique way by a boundary measure. Important examples of Choquet simplices are the Bauer simplices: Given a compact, Hausdorff space $X$ , the set $M_1(X)$ of positive, Borel probability measures on $X$ is a Choquet simplex with boundary $\partial_e M_1(X)\cong X$ , and $\mathrm{Aff}(M_1(X))\cong C(X,\mathbb{R})$ .

If $A=C(X)$ is a commutative C*-algebra, then $\mathrm{DF}(C(X))$ naturally corresponds to the set of finitely-additive probability measures on $X$ , while $\mathrm{LDF}(C(X))$ naturally corresponds to the set of ( $\sigma$ -additive) probability measures on $X$ . For a general C*-algebra $A$ , we therefore consider the dimension functions on $A$ as „noncommutative, finitely-additive probability measures“, and similarly $\mathrm{LDF}(A)$ are the „noncommutative probability measures“ on $A$ . It is easy to see that the probability measures on a compact, Hausdorff space are dense in the set of finitely-additive probability measures (see the proof of Theorem I.2.4 in [1]), and it is a classical result that the finitely-additive probability measures on a compact, Hausdorff space form a Choquet simplex. The Blackadar-Handelman conjectures predict that theses results generalize to the noncommutative setting.

The first Blackadar-Handelman conjecture has been verified in the following cases: if $A$ is commutative (Theorem I.2.4 in [1]); if $A$ is simple, exact, stably finite and has strict comparison of positive elements (Theorem B and 6.4, and Remark 6.5 in [2]); if $A$ has finite radius of comparison (Theorem 3.3 in [3]).

The second Blackadar-Handelman conjecture has been verified in the following cases: if $A$ is commutative; if $A$ is simple, exact, stably finite and $\mathcal{Z}$ -stable (Theorem B in [2]); if $A$ has real rank zero and stable rank one (Corollary 4.4 in [4]); in [5], the assumption of real rank zero was removed from the result in [4], thus verifying the second Blackadar-Handelman conjecture for all C*-algebras of stable rank one.

With view to the results in [5] it is natural to ask if the first Blackadar-Handelmann conjecture can be verified for all C*-algebras of stable rank one.

[1]
B. Blackadar, D. Handelman, Dimension functions and traces on C∗-algebras, Journal of Functional Analysis. (1982) 297–340. https://doi.org/10.1016/0022-1236(82)90009-x.
[2]
N.P. Brown, F. Perera, A.S. Toms, The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras, Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal). (2008). https://doi.org/10.1515/crelle.2008.062.
[3]
K. De Silva, A note on two Conjectures on Dimension funcitons of C*-algebras, ArXiv. (2016) 1601.03475.
[4]
F. Perera, The Structure of Positive Elements for C*-Algebras with Real Rank Zero, Int. J. Math. (1997) 383–405. https://doi.org/10.1142/s0129167x97000196.
[5]
R. Antoine, F. Perera, L. Robert, H. Thiel, C*-algebras of stable rank one and their Cuntz semigroups, Duke Math. J. (2022) (to appear).

Realizing Cuntz classes in commutative subalgebras

Von hannesthiel | Juli 18, 2021

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Let $A$ be a unital, simple C*-algebra of stable rank one. Does there exist a commutative sub-C*-algebra $C(X)\subseteq A$ such that for every lower-semicontinuous function $f\colon\mathrm{QT}_1(A)\to[0,1]$ there exists an open subset $U\subseteq X$ such that $f(\tau)=\mu_\tau(U)$ for $\tau\in\mathrm{QT}_1(A)$ ? Here, $\mathrm{QT}_1(A)$ denotes the Choquet simplex of normalized $2$ -quasitraces on $A$ (if $A$ is exact, then this is just the Choquet simplex of tracial states on $A$ ), and $\mu_\tau$ denotes the probability measure on $X$ induced by the restriction of $\tau$ to $C(X)$ .

More specifically, one may ask if this is always the case for a Cartan subalgebra of $A$ .

Inductive limits of semiprojective C*-algebras

Von hannesthiel | Dezember 20, 2020

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Question: Is every separable C*-algebra an inductive limit of semiprojective C*-algebras?

This question was first raised by Blackadar in [1]. If we think of C*-algebras as noncommutative topological spaces, then semiprojective C*-algebras are noncommutative absolute neighborhood retracts (ANRs). It is a classical result from shape theory that every metrizable space $X$ is homeomorphic to an inverse limit of metrizable ANRs $X_n$ , that is, $X\cong\varprojlim_n X_n$ . This means that $C(X)$ is isomorphic to an inductive limit of the $C(X_n)$ , that is, $C(X)\cong\varinjlim_n C(X_n)$ . The question is if the noncommutative analog of this result also holds.

It has been verified for many classes of C*-algebras that they are inductive limits of semiprojective C*-algebras. For example, Enders showed in [2] that every UCT-Kirchberg algebra is an inductive limit of semiprojective C*-algebras. In [3], it was shown that the class of C*-algebras that are inductive limits of semiprojective C*-algebras is closed under shape domination, and in particular under homotopy equivalence. One deduces, for example, that if $X$ is a contractible, compact, metrizable space, and if $A$ is an inducitve limit of semiprojective C*-algebras, then so is $C(X,A)$ . It also follows that every contractible C*-algebra is an inductive limit of semiprojective C*-algebras – in fact, even of projective C*-algebras, as was shown in [4].

The commutative C*-algebra $C(S^2)$ is probably the easiest C*-algebra where it is currently unknown if it is an inductive limit of semiprojective C*-algebras. Equivalently, it is unknown if $C_0(\mathbb{R}^2)$ is an inductive limit of semiprojective C*-algebras. By Example 4.6 in [3], we know that the stabilization $C_0(\mathbb{R}^2)\otimes\mathcal{K}$ is an inductive limit of semiprojective C*-algebras.

[1]
B. Blackadar, Shape theory for $C^*$ -algebras, MATH. SCAND. 56 (1985) 249–275. https://doi.org/10.7146/math.scand.a-12100.
[2]
D. Enders, Semiprojectivity for Kirchberg algebras, ArXiv Preprint ArXiv:1507.06091. (2015).
[3]
H. Thiel, Inductive limits of semiprojective $C^*$ -algebras, Advances in Mathematics. 347 (2019) 597–618. https://doi.org/10.1016/j.aim.2019.02.030.
[4]
H. Thiel, Inductive limits of projective $C$ *-algebras, J. Noncommut. Geom. 13 (2020) 1435–1462. https://doi.org/10.4171/jncg/350.

C*-algebras complemented in their biduals

Von hannesthiel | Dezember 20, 2020

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Question: Let $A$ be a C*-algebra that is complemented in its bidual $A^{**}$ by a *-homomorphism, that is, there exists a *-homomorphism $\pi\colon A^{**}\to A$ such that $\pi(a)=a$ for all $a\in A$ . Is $A$ a von Neumann algebra?

The converse is true: Let $A$ be a von Neumann algebra. Then $A$ has a (unique) isometric predual $A_*$ . Let $\kappa_{A_*}\colon A_*\to (A_*)^{**}$ be the natural inclusion of the Banach space $A_*$ in its bidual. We naturally identify the dual of $A_*$ with $A$ , and the dual of $(A_*)^{**}$ with $A^{**}$ . Then the transpose $\kappa_{A_*}^*\colon A^{**}=(A_*)^{***}\to (A_*)^*=A$ is a *-homomorphism that complements $A$ in $A^{**}$ .

Contractibility of unitary groups of II-1 factors

Von hannesthiel | Oktober 7, 2020

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Question: Let $M$ be a $\mathrm{II}_1$ -factor. Is the unitary group $\mathcal{U}(M)$ contractible when equipped with the strong operator topology?

Update (August 2025): A positive answer to the question was recently announced:

Jekel. The unitary group of a II1 factor is SOT-contractible. preprint arXiv:2508.05834

Background/Motivation: By Kuiper’s theorem, if $H$ is an infinite-dimensional Hilbert space, then the unitary group of the $\mathrm{I}_\infty$ -factor $\mathcal{B}(H)$ is contractible in the norm topology. This was generalized by Breuer [1] to (certain) $\mathrm{I}_\infty$ and $\mathrm{II}_\infty$ von Neumann algebras, and eventually Brüning-Willgerodt [2] showed that the unitary group of every properly infinite von Neumann algebra is contractible in the norm topology. This is no longer true for finite von Neumann algebras: The unitary group of a finite matrix algebra $M_n(\mathbb{C})$ is not contractible. Similarly, the unitary group of a $\mathrm{II}_1$ factor is not contractible in the norm topology since its fundamental group does not vanish – in fact, it was shown in [3] that $\pi_1(\mathcal{U}(M))\cong\mathbb{R}$ for every $\mathrm{II}_1$ factor $M$ .

As noted in the introduction of [4], the unitary group of every properly infinite von Neumann algebra is also contractible in the strong operator topology. This naturally leads to the above question, which was considered by Popa-Takesaki in [4]. They showed that $\mathcal{U}(M)$ is contractible in the strong operator topology if $M$ is a separable $\mathrm{II}_1$ factor such that the associated $\mathrm{II}_\infty$ factor $M\bar{\otimes} \mathcal{B}(H)$ admits a trace scaling one-parameter group of automorphisms. This includes all McDuff factors ( $M$ is McDuff if $M\cong M\bar{\otimes}\mathcal{R}$ for the hyperfinite $\mathrm{II}_\infty$ factor $\mathcal{R}$ ) and all factors that satisfy $M\cong M\bar{\otimes} L(\mathbb{F}_\infty)$ , where $L(\mathbb{F}_\infty)$ is the group von Neumann algebra of the free group on infinitely many generators.

[1]
M. Breuer, On the homotopy type of the group of regular elements of semifinite von Neumann algebras, Math. Ann. (1970) 61–74. https://doi.org/10.1007/bf01350761.
[2]
J. Brüning, W. Willgerodt, Eine Verallgemeinerung eines Satzes von N. Kuiper, Math. Ann. (1976) 47–58. https://doi.org/10.1007/bf01354528.
[3]
H. Araki, M.-S.B. Smith, L. Smith, On the homotopical significance of the type of von Neumann algebra factors, Commun.Math. Phys. (1971) 71–88. https://doi.org/10.1007/bf01651585.
[4]
S. Popa, M. Takesaki, The topological structure of the unitary and automorphism groups of a factor, Commun.Math. Phys. (1993) 93–101. https://doi.org/10.1007/bf02100051.

Scottish Book Problem 166

Von hannesthiel | Oktober 7, 2020

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This is one of the few problems from the Scottish book that are still open. In slightly modernized form, and correcting the typo (in the book, $f$ and $f_0$ should be switched in the last sentence) the problem is:

Let $M$ be a topological manifold, and let $f\colon M\to\mathbb{R}$ be a continuous function. Let $G^M_f$ denote the subgroup of homeomorphisms $T\colon M\to M$ that satisfy $f\circ T=f$ . Let $N$ be another manifold that is not homeomorphic to $M$ . Does there exist a continuous function $f_0\colon N\to\mathbb{R}$ such that $G^M_f$ is not isomorphic to $G^N_{f_0}$ ?

Scottish Book Problem 155

Von hannesthiel | Oktober 7, 2020

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This is one of the few problems from the Scottish book that are still open. In modern terminology, the problem is:

[Correction May 2026: Corrected sphere to closed ball]

Let $X$ and $Y$ be Banach spaces, and let $U\colon X\to Y$ be a bijective map with the following property: For every $x_0\in X$ there exists $\varepsilon>0$ such that for the closed ball $B(x_0,\varepsilon) := \{ x\in X : \|x-x_0\| \leq \varepsilon \}$ , the restriction $U|_{B(x_0,\varepsilon)}$ is isometric. Does it follow that $U$ is isometric?

It is noted in the Scottish Book that the answer is „yes“ whenever $U^{-1}$ is continuous, which is automatic if $Y$ is finite-dimensional, or if $Y$ has the property that for any two elements $y_1,y_2\in Y$ satisfying $y_2\neq 0$ and $\|y_1+y_2\|=\|y_1\|+\|y_2\|$ there exists $\lambda\geq 0$ such that $y_1=\lambda y_2$ .

[Update May 2026] In [1], Mori solved the problem under the additional assumption that X is separable, and even under the weaker assumption of surjectivity instead of bijectivity. He also explains that sphere in the Scottish book [2] means closed ball. (What one would call a sphere nowadays, namely the set $\{ x\in X : \|x-x_0\| = \varepsilon \}$ , is called the surface of a sphere in the Scottish book.)

[1] Mori. On the Scottish book problem 155 by Mazur and Sternbach. C. R. Math. Acad. Sci. Paris 362 (2024), 813–816.

[2] The Scottish Book. Mathematics from the Scottish Café with selected problems from the new Scottish Book. Second edition. Including selected papers presented at the Scottish Book Conference held at North Texas University, Denton, TX, May 1979. Edited by R. Daniel Mauldin. Birkhäuser/Springer, Cham, 2015. xvii+322 pp.

Conjugacy of pointwise conjugate homomorphisms

Von hannesthiel | Oktober 7, 2020

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Given groups $H$ and $G$ , let us say that the pair $(H,G)$ has property (*) if any two injective homomorphisms $\alpha_0,\alpha_1\colon H\to G$ are conjugate if (and only if) they are pointwise conjugate.

Problem 1: Describe the class $C_1$ of groups $H$ such that $(H,G)$ has (*) for every group $G$ .
Problem 2: Describe the class $C_2$ of groups $G$ such that $(H,G)$ has (*) for every group $H$ .

Definitions: Given a group $G$ , two elements $a,b\in G$ are conjugate if there exists $x\in G$ such that $a=xbx^{-1}$ . Two homomorphisms $\alpha_0,\alpha_1\colon H\to G$ are pointwise conjugate if $\alpha_0(a)$ is conjugate to $\alpha_1(a)$ in $G$ for every $a\in H$ . (Thus, for each $a\in H$ there exists $x_a\in G$ such that $\alpha_0(a)=x_a\alpha_1(a)x_a^{-1}$ .) Further, $\alpha_0$ and $\alpha_1$ are conjugate if there exists $x\in G$ such that $\alpha_0(a)=x\alpha_1(a)x^{-1}$ for every $a\in H$ . If $\alpha_0$ and $\alpha_1$ are conjugate, then they are locally conjugate. Property (*) records that the converse holds (for injective homomorphisms).

Motivation: Problem 30 of the Scottish Book [1] (which is still open) asks to determine which groups $G$ have the following property: Pairs $(a,a')$ and $(b,b')$ in $G$ are conjugate (there exists $x\in G$ such that $a=xbx^{-1}$ and $a'=xb'x^{-1}$ ) if (and only if) for every word $w$ in two noncommuting variables the elements $w(a,a')$ and $w(b,b')$ are conjugate. Using property (*) formulated above, Problem 30 asks to determine which groups $G$ have the property that $(H,G)$ satisfies (*) for every subgroup $H$ of $G$ that is generated by two elements. The above Problem 2 is a more general (and possibly more natural) version of this problem.

An automorphism $\alpha\colon G\to G$ is class-preserving if $\alpha(a)$ is conjuagte to $a$ for every $a\in G$ . If $(G,G)$ has (*), then every class-preserving automorphism of $G$ is inner. The study of groups with (or without) outer class-preserving automorphisms has a long history; see for instance [2] and [3]. There exist finite groups with outer class-preserving automorphisms. In particular, there exist finite groups that belong neither to $C_1$ nor to $C_2$ . Note also that $C_2$ contains all abelian groups and that $C_1$ contains all cyclic groups.

[1]
R.D. Mauldin, The Scottish Book, Springer International Publishing, 2015. https://doi.org/10.1007/978-3-319-22897-6.
[2]
C.-H. Sah, Automorphisms of finite groups, Journal of Algebra. (1968) 47–68. https://doi.org/10.1016/0021-8693(68)90104-x.
[3]
M.K. Yadav, Class preserving automorphisms of finite p-groups: a survey, in: C.M. Campbell, M.R. Quick, E.F. Robertson, C.M. Roney-Dougal, G.C. Smith, G. Traustason (Eds.), Groups St Andrews 2009 in Bath, Cambridge University Press, 2007: pp. 569–579. https://doi.org/10.1017/cbo9780511842474.019.

Purely infinite rings and C*-algebras

Von hannesthiel | September 23, 2020

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Question 1: If $A_0\subseteq\mathcal{B}(H)$ is a *-subalgebra of bounded linear operators on a separable Hilbert space $H$ such that $A_0$ is purely infinite as a ring, is the norm-closure $\overline{A_0}$ purely infinite as a C*-algebra?

This is Problem 8.4 in [1]. As noted in Problem 8.6 in [1], this is even unclear if $A_0$ is a unital, simple, purely infinite ring. In the converse direction, it seems natural to ask:

Question 2: Given a purely infinite C*-algebra $A$ , is there a dense *-subalgebra $A_0\subseteq A$ that is purely infinite as a ring?

Definitions: The relation $\precsim$ on a ring $R$ is defined by setting $x\precsim y$ if there exist $a,b\in R$ such that $x=ayb$ ; see Definition 2.1 in [1]. A ring $R$ is purely infinite if no quotient of $R$ is a division ring, and if any $x,y\in R$ satisfy $x\precsim y$ if (and only if) $x\in RyR$ ; see Definition 3.1 in [1]. (Here, $RyR$ denotes the two-sided ideal generated by $y$ , that is, $RyR=\{a_1yb_1+\ldots+a_nyb_n : n\geq 1, a_i,b_i\in R\}$ .)

Given a C*-algebra $A$ , the Cuntz subequivalence relation $\precsim$ is defined by setting $x\precsim y$ if there exist sequences $(a_n)_n$ and $(b_n)_n$ in $A$ such that $\lim_{n\to\infty} \| x - a_nyb_n \| = 0$ . A C*-algebra $A$ is purely infinite if it admits no nonzero one-dimensional representations and if any $x,y\in A$ satisfy $x\precsim y$ if and only if $x$ belongs to $\overline{\mathrm{span}}AyA$ , the closed, two-sided ideal generated by $y$ ; see Definition 4.1 in [2]. (The definition in [2] only considers positive elements in $A$ , but it equivalent to the definition given here.)

Background: By Proposition 3.17 in [1], if a C*-algebra is purely infinite as a ring, then it is purely infinite as a C*-algebra. The converse does probably not hold (Remark 3.18 in [1]), which is why we ask Question 2 above. A unital, simple C*-algebra is purely infinite as a C*-algebra if and only if it is purely infinite as a ring. Thus, Question 2 has a positive answer in this case.

Given a C*-algebra $A$ , let $\mathrm{Ped}(A)\subseteq A$ denote its Pedersen ideal (the minimal dense ideal in $A$ ). By Proposition 8.5 in [1], if $\mathrm{Ped}(A)$ is purely infinite as a ring, then $A$ is purely infinite. Thus, in this particular instance, Question 1 has a positive answer.

[1]
G. Aranda Pino, K.R. Goodearl, F. Perera, M. Siles Molina, Non-simple purely infinite rings, American Journal of Mathematics. (2010) 563–610. https://doi.org/10.1353/ajm.0.0119.
[2]
E. Kirchberg, M. Rordam, Non-simple purely infinite C*-algebras, American Journal of Mathematics. (2000) 637–666. https://doi.org/10.1353/ajm.2000.0021.

Liftable normal elements

Von hannesthiel | September 22, 2020

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Given a C*-algebra $A$ and a closed, two-sided ideal $I\subseteq A$ , is the image of the normal elements in $A$ under the quotient map $A\to A/I$ a closed subset of $A/I$ ?

Equivalently, if $(x_n)_n$ is a sequence of normal elements in $A/I$ that converge to $x$ , and if each $x_n$ admits a normal lift in $A$ , does $x$ admit a normal lift?

This question is raised in Example 6.5 in [1]. It is equivalent to the question of whether the commutative C*-algebra of continuous functions on the disc vanishing at zero is $\ell$ -closed in the sense of Definition 6.1 in [1].

Of related interest is the class of normal elements that are universally liftable: We say that a normal element $a$ in a C*-algebra $A$ is universally liftable if for every C*-algebra $B$ and every surjective *-homomorphism $\pi\colon B\to A$ there exists a normal element $b\in B$ with $\pi(b)=a$ . One can show that a normal element $a\in A$ is universally liftable if and only if there exists a projective C*-algebra $P$ , a normal element $b\in P$ and a *-homomorphism $\varphi\colon P\to A$ with $\varphi(b)=a$ . (A C*-algebra $P$ is projective if for every C*-algebra $D$ , every closed, two-sided ideal $I\subseteq D$ , and every *-homomorphism $\psi\colon P\to D/I$ there exists a *-homomorphism $\tilde{\psi}\colon P\to D$ such that $\pi\circ\tilde{\psi}=\psi$ , where $\pi\colon D\to D/I$ is the quotient map.) Indeed, for the forward direct, one uses that every C*-algebra is the quotient of a projective C*-algebra, and for the converse direction one applies the definition of projectivity. In particular, every normal element in a projective C*-algebra is universally liftable.

Question: Given a C*-algebra $A$ , is the set of universally liftable normal elements in $A$ closed? Is it open (relative to the set of normal elements)?

Question: Is there a projective C*-algebra $P$ and a normal element $b\in P$ such that for every C*-algebra $A$ , a normal element $a\in A$ is universally liftable if and only if there exists a *-homomorphism $\varphi\colon P\to A$ with $\varphi(b)=a$ ?

[1]
B. Blackadar, The Homotopy Lifting Theorem for Semiprojective $C^*$ -Algebras, MATH. SCAND. (2016) 291. https://doi.org/10.7146/math.scand.a-23691.