# On unital absorbing extensions of C∗-algebras of stable rank one and real
rank zero

Qingnan An  School of Mathematics and Statistics, Northeast Normal University,
Changchun, 130024, China<EMAIL_ADDRESS>and  Zhichao Liu  School
of Mathematical Sciences, Dalian University of Technology, Dalian, 116024,
China<EMAIL_ADDRESS>
###### Abstract.

Suppose that $A,B$ are nuclear, separable ${\rm C}^{*}$-algebras of stable
rank one and real rank zero, $A$ is unital simple, $B$ is stable and $({\rm
K}_{0}(B),{\rm K}_{0}^{+}(B))$ is weakly unperforated in the sense of Elliott
[4]. We show that any unital extension with trivial index maps of $A$ by $B$
is absorbing.

###### Key words and phrases:

Absorbing extension; Stable rank one; Real rank zero; Stable.

###### 2000 Mathematics Subject Classification:

Primary 46L80, Secondary 46K35 19K35

## Introduction

The theory of ${\rm C}^{*}$-algebra extensions began with the work of Brown,
Douglas and Fillmore on classifying essentially normal operators [3]. But soon
it develops into a theory related to many fields of mathematics. Absorbing
extensions play an important role in the theory of ${\rm C}^{*}$-algebras. The
noncommutative Weyl-von Neumann theorem of Voiculescu [13] shows that for any
separable ${\rm C}^{*}$-algebra $A$, every (unital) essential extension of $A$
by $\mathcal{K}$ is (unital) absorbing. For the more general separable case,
the essential extensions of $A$ by $B\otimes\mathcal{K}$, Thomsen shows that
there always exists a (unital) trival extension which is (unital) absorbing
[12]. It is a non-trivial task to decide that under which conditions any
essential extension of separable ${\rm C}^{*}$-algebras is absorbing.

In this paper, we consider the ${\rm C}^{*}$-algebra extensions of $A$ by $B$.
We say an extension $e:0\to B\to E\to A\to 0$ has $trivial$ $index$ $maps$, if
the induced six-term exact sequence

$\textstyle{{\,\,\mathrm{K}_{0}(B)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\,\,\mathrm{K}_{0}(E)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\,\,\mathrm{K}_{0}(A)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{0}}$$\textstyle{{\,\,\mathrm{K}_{1}(A)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{1}}$$\textstyle{{\,\,\mathrm{K}_{1}(E)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\,\,\mathrm{K}_{1}(B)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

satisfies that both $\delta_{0}$ and $\delta_{1}$ are the zero maps. Using the
intrinsic characterizations established by Elliott and Kucerovsky [5] and
techniques for stable C*-algebras, we prove the following theorem.

######  Theorem 0.1.

Let $A,B$ be nuclear, separable ${\rm C}^{*}$-algebras of stable rank one and
real rank zero. Suppose that $A$ is simple, $B$ is stable and $({\rm
K}_{0}(B),{\rm K}_{0}^{+}(B))$ is weakly unperforated. Let $e$ be a unital
extension with trivial index maps:

$0\to B\to E\to A\to 0.$

Then $e$ is absorbing.

We apply this result to the stably strong Ext-groups and obtain the
correspondence from the isomorphic classes of extensions of ${\rm
K}_{*}$-groups to subgroup of strong Ext-groups.

##  1\. Preliminaries

Let $A$ be a unital $\mathrm{C}^{*}$-algebra. $A$ is said to have stable rank
one, written $sr(A)=1$, if the set of invertible elements of $A$ is dense. $A$
is said to have real rank zero, written $rr(A)=0$, if the set of invertible
self-adjoint elements is dense in the set $A_{sa}$ of self-adjoint elements of
$A$. If $A$ is not unital, let us denote the minimal unitization of $A$ by
$\widetilde{A}$. A non-unital $\mathrm{C}^{*}$-algebra is said to have stable
rank one (or real rank zero) if its unitization has stable rank one (or real
rank zero).

Denote $\mathcal{P}(A)$ the set consisting of all the projections in $A$. Let
$p,$ $q\in\mathcal{P}(A)$. One says that $p$ is $Murray$–$von$ $Neumann$
$equivalent$ to $q$ in $A$ and writes $p\sim q$ if there exists $x\in A$ such
that $x^{*}x=p$ and $xx^{*}=q$. We will write $p\preceq q$ if $p$ is
equivalent to some subprojection of $q$; we will write $p\leq q$ if $p$ is a
subprojection of $q$. Denote $\mathcal{K}$ the ${\rm C}^{*}$-algebra of the
compact operators on an infinite-dimensional separable Hilbert space. The
classes of projections $p,q\in A\otimes\mathcal{K}$ in $\mathrm{K}_{0}(A)$
will be denoted by $[p],[q]$, and we say $[p]\leq[q]$, if $p\preceq q$ in
$A\otimes\mathcal{K}$.

$A$ is said to have cancellation of projections, if for any projections
$p,q,e,f\in A$ with $pe=0$, $qf=0$, $e\sim f$, and $p+e\sim q+f$, then $p\sim
q$. $A$ has cancellation of projections if and only if $p\sim q$ implies that
there exists a unitary $u\in\widetilde{A}$ such that $u^{*}pu=q$ (see [8,
3.1]). Every unital $\mathrm{C}^{*}$-algebra of stable rank one has
cancellation of projections, hence, any two projections with the same ${\rm
K}$-theory generate the same ideal. We will use $I_{p}$ to represent the ideal
generated by $p$ in $A$.

A ${\rm C}^{*}$-algebra $A$ is $stable$ if it is isomorphic to its tensor
product with the $\mathrm{C}^{*}$-algebra $\mathcal{K}$, i.e., $A\cong
A\otimes\mathcal{K}$.

A ${\rm C}^{*}$-subalgebra $B$ of a ${\rm C}^{*}$-algebra $A$ is said to be
$full$ if it is not contained in any proper two-sided closed ideal of $A$.
Denote ${\mathcal{M}}(A)$ the multiplier algebra of $A$. Given a projection
$p$ in ${\mathcal{M}}(A)$, $pAp$ is a hereditary subalgebra of $A$, which will
be called a corner. A projection $p$ in ${\mathcal{M}}(A)$ is full if the
corner $pAp$ is full in $A$.

######  Definition 1.1.

Let $A$, $B$ be ${\rm C}^{*}$-algebras and

$e\,:\,0\to B\to E\xrightarrow{\pi}A\to 0$

be an extension of $A$ by $B$ with Busby invariant
$\tau:\,A\to{\mathcal{M}}(B)/B$. We say the extension $e$ is $essential$, if
$\tau$ is injective. When $A$ is unital, we say $e$ is $unital$, if $\tau$ is
unital; we say a unital extension $e$ is $unital$ $trivial$, if there is a
unital $*$-homomorphism $\rho\,:A\to E$ such that $\pi\circ\rho=id_{A}$.

Note that if $A$ is unital simple and $B\neq 0$, any unital extension $e$ of
$A$ by $B$ is essential.

######  Definition 1.2.

Let $e_{i}\,:\,0\to B\to E_{i}\to A\to 0$ be two extensions with Busby
invariants $\tau_{i}$ for $i=1,2$. Then $e_{1}$ and $e_{2}$ are called
$strongly$ $unitarily$ $equivalent$, denoted by $e_{1}\sim_{s}e_{2}$, if there
exists a unitary $u\in{\mathcal{M}}(B)$ such that
$\tau_{2}(a)=\pi(u)\tau_{1}(a)\pi(u)^{*}$ for all $a\in A$, where
$\pi:\,{\mathcal{M}}(B)\to{\mathcal{M}}(B)/B.$

Assume $B$ is stable. Fix an isomorphism of $\mathcal{K}$ with
$M_{2}(\mathcal{K})$; this isomorphism induces an isomorphism $B\cong
M_{2}(B)$ and hence isomorphisms ${\mathcal{M}}(B)\cong
M_{2}({\mathcal{M}}(B))$ and ${\mathcal{M}}(B)/B\cong
M_{2}({\mathcal{M}}(B)/B)$. These isomorphisms are uniquely determined up to
unitary equivalence.

Let $s_{1},s_{2}$ be two isometries in ${\mathcal{M}}(B\otimes\mathcal{K})$
with $s_{1}s_{1}^{*}+s_{2}s_{2}^{*}=1$. Define the addition $e_{1}\oplus
e_{2}$ to be the extension of $A$ by $B$ with Busby invariant

$\tau_{1}\oplus\tau_{2}:=s_{1}\tau_{1}s_{1}^{*}+s_{2}\tau_{2}s_{2}^{*}.$

We say that a unital extension $e$ with Busby invariant $\tau$ is $absorbing$
if it is strongly unitarily equivalent to its sum with any unital trivial
extension, i.e., $e\sim_{s}e\oplus\sigma$ for any unital trivial extension
$\sigma$.

######  Definition 1.3.

([5]) Let $E$ be a $\mathrm{C}^{*}$-algebra and $B$ be a closed two-sided
ideal of $E$. Let us say that $E$ is $purely$ $large$ with respect to $B$ if
for every element $c$ in $E\backslash B$, the $\mathrm{C}^{*}$-algebra
$\overline{cBc^{*}}$ (the intersection with $B$ of the hereditary
sub-$\mathrm{C}^{*}$-algebra of $E$ generated by $cc^{*}$) contains a
sub-$\mathrm{C}^{*}$-algebra which is stable and is full in $B$.

Let $e\,:\,0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0$ be an
extension of $A$ by $B$, we say that $e$ is $purely\,\,large$ if the
$\mathrm{C}^{*}$-algebra $E$ is purely large with respect to the image of $B$.

The following is the main result in [5] under the nuclear setting.

######  Theorem 1.4.

([5, Theorem 6]) Let $A$ and $B$ be nuclear, separable
$\mathrm{C}^{*}$-algebras, with $B$ stable and $A$ unital. A unital extension
of $A$ by $B$ is absorbing if and only if it is purely large.

We will use the following theorem frequently.

######  Theorem 1.5.

([2, Theorem 2.8]). If $B$ is a full hereditary ${\rm C}^{*}$-subalgebra of
$A$ and if each of $A$ and $B$ has a strictly positive element, then
$B\otimes\mathcal{K}$ is canonically isomorphic to $A\otimes\mathcal{K}$.

######  Remark 1.6.

Let $A$ be a separable ${\rm C}^{*}$-algebra of stable rank one and $p$ be a
full projection in $\mathcal{M}(A)$. Denote $\iota:\,pAp\to A$ the natural
embedding map. By Theorem 1.5, $pAp\otimes\mathcal{K}$ is isomorphic to
$A\otimes\mathcal{K}$, and the isomorphism is induced by a partial isometry
$v\in{\mathcal{M}}(A\otimes\mathcal{K})$ which takes $x$ to $v^{*}xv$.

We note that the inclusion $\iota$ induces the same order on ${\rm K}_{0}$
with $v^{*}(\cdot)v$. For any projection $q\in pAp\otimes\mathcal{K}$, we
identify $(\iota\otimes id_{\mathcal{K}})(q)$ with $q$. Note that $v^{*}q\in
A\otimes\mathcal{K}$ satisfies

$v^{*}q(v^{*}q)^{*}=v^{*}qv\quad{\rm and}\quad(v^{*}q)^{*}v^{*}q=q,$

which implies that ${\rm K}_{0}(\iota)([q])=[v^{*}qv]$. (A more generalised
result can be found in [6, Proposition 5.1].)

For any ${\rm C}^{*}$-algebra $A$, the positive cone of ${\rm K}_{0}$ and the
scale are given by

${\rm
K}_{0}^{+}(A)=\\{[p]:\,p\in\mathcal{P}(A\otimes\mathcal{K})\\},\quad\Sigma\,A=\\{[p]:\,p\in\mathcal{P}(A)\\}.$

Now we list some results for stable algebras due to Rørdam.

######  Proposition 1.7.

([10, Proposition 3.1]) Let $A$ be a ${\rm C}^{*}$-algebra with the
cancellation property and with a countable approximate unit consisting of
projections. Then $A$ is stable if and only if $\Sigma\,A={\rm K}_{0}^{+}(A).$

######  Definition 1.8.

([10, Definition 3.2]) A triple $\left(G,G^{+},\Sigma\right)$ will be called a
scaled, ordered abelian group if $\left(G,G^{+}\right)$ is an ordered abelian
group, and $\Sigma$ is an upper directed, hereditary, full subset of $G^{+}$,
i.e.,

(i) $\forall\,x_{1},x_{2}\in\Sigma,\,\exists\,x\in\Sigma:x_{1}\leq x,x_{2}\leq
x$,

(ii) $\forall\,x\in G^{+},\,\forall\,y\in\Sigma:x\leq y\Rightarrow
x\in\Sigma$,

(iii) $\forall\,x\in
G^{+},\,\exists\,y\in\Sigma,\,\exists\,k\in\mathbb{N}:x\leq ky$.

Let $\left(G,G^{+}\right)$ be an ordered abelian group, and let $\Sigma_{1}$
and $\Sigma_{2}$ be upper directed, hereditary, full subsets of $G^{+}$.
Define $\Sigma_{1}\,\hat{+}\,\Sigma_{2}$ to be the set of all elements $x\in
G^{+}$ for which there exist $x_{1}\in\Sigma_{1}$ and $x_{2}\in\Sigma_{2}$
with $x\leq x_{1}+x_{2}$. Observe that $\Sigma_{1}\,\hat{+}\,\Sigma_{2}$ is an
upper directed, hereditary, full subset of $G^{+}$.

Denote the $k$-fold sum $\Sigma\,\hat{+}\,\Sigma\hat{+}\cdots\hat{+}\Sigma$ by
$k\cdot\Sigma$. Using that $\Sigma$ is upper directed, we see that $y\in
k\cdot\Sigma$ if and only if $0\leq y\leq kx$ for some $x\in\Sigma$.

If $A$ is a ${\rm C}^{*}$-algebra of stable rank one and has an approximate
unit consisting of projections, then $\left({\rm K}_{0}(A),{\rm
K}^{+}_{0}(A),\Sigma\,A\right)$ is a scaled, ordered abelian group and

$\left({\rm K}_{0}\left(M_{k}(A)\right),{\rm
K}_{0}^{+}\left(M_{k}(A)\right),\Sigma\left(M_{k}(A)\right)\right)\cong\left({\rm
K}_{0}(A),{\rm K}^{+}_{0}(A),k\,\hat{\cdot}\,\Sigma\,A\right).$

######  Definition 1.9.

([4],[7, 8.1]) An ordered abelian group $\left(G,G_{+}\right)$ is called
weakly unperforated if it satisfies both of the following two conditions.

(i) Given $x\in G$ and $m\in\mathbb{N}$ such that $mx\in G_{+}$, there exists
$t\in\operatorname{tor}(G)$ such that $x+t\in G_{+}$ and $mt=0$.

(ii) Given $y\in G_{+},t\in\operatorname{tor}(G)$, and $n\in\mathbb{N}$ such
that $ny+t\in G_{+}$, then $y\pm t\in G_{+}$.

Now we prove the following result for the nonsimple ordered group case, which
generalizes [10, Proposition 3.3].

######  Proposition 1.10.

Let $(G,G_{+},\Sigma)$ be a weakly unperforated, scaled, ordered, abelian
group, and suppose that $n\,\,\hat{\cdot}\,\Sigma=G_{+}$ for some
$n\in\mathbb{N}$. Then $\Sigma=G_{+}$.

###### Proof.

Let $g\in G_{+}$, then $(n+1)g\in G_{+}$. Since
$n\,\hat{\cdot}\,\Sigma=G_{+}$, there is an element $x\in\Sigma$ with
$nx\geq(n+1)g$. From $n(x-g)\in G_{+}$ and Definition 1.9 (i), there exists
$t\in{\rm tor}(G)$ such that $x-g+t\in G^{+}$ and $nt=0$. Then $x+t\in G_{+}$,
and hence,

$n(x+t)+t=(n-1)x+x+t\in G_{+}.$

By Definition 1.9 (ii), we have $x+2t\in G^{+}$. Then,

$(nx-(n+1)g)+(x+2t)+nt=(n+1)(x-g+t)+t\in G^{+}.$

Using Definition 1.9 (ii) again, we have

$x-g=x-g+t-t\in G^{+}.$

By the hereditary property of $\Sigma$, we obtain $g\in\Sigma$. Thus
$\Sigma=G^{+}$.

∎

The following is a straight result from above.

######  Corollary 1.11.

Let $A$ be a separable ${\rm C}^{*}$-algebra of stable rank one and real rank
zero and with a weakly unperforated $({\rm K}_{0}(A),{\rm K}_{0}^{+}(A))$. If
$M_{n}(A)$ is stable for some integer $n\geq 1$, then $A$ is stable.

###### Proof.

If $M_{n}(A)$ is stable for some integer $n\geq 1$, with the fact that

$\left({\rm K}_{0}\left(M_{n}(A)\right),{\rm
K}_{0}^{+}\left(M_{n}(A)\right),\Sigma\left(M_{n}(A)\right)\right)\cong\left({\rm
K}_{0}(A),{\rm K}^{+}_{0}(A),n\,\hat{\cdot}\,\Sigma\,A\right),$

by Proposition 1.7, we have $n\,\hat{\cdot}\,\Sigma\,A={\rm K}^{+}_{0}(A).$
Then Proposition 1.10 implies $\Sigma\,A={\rm K}^{+}_{0}(A).$ By Proposition
1.7 again, $A$ is stable.

∎

######  Remark 1.12.

We claim that without the assumption of $A$ has a countable approximate unit
consisting of projections, $({\rm K}_{0}(A),{\rm K}_{0}^{+}(A),\Sigma\,A)$ is
not necessary a scaled, ordered, abelian group. We put a short example here.

Take

$A=\\{f\in
M_{2}(C[0,1]):\,f(0)=\left(\\!\\!\begin{array}[]{cc}\lambda&0\\\\[4.2679pt]
0&\lambda\end{array}\\!\\!\right)~{}~{}{\rm
and}~{}~{}f(1)=\left(\\!\\!\begin{array}[]{cc}\mu&0\\\\[4.2679pt]
0&0\end{array}\\!\\!\right),~{}~{}{\rm}~{}~{}\lambda,\mu\in\mathbb{C}\\}.$

Then we have the following exact sequence,

$0\to
M_{2}(C_{0}(0,1))\xrightarrow{\iota}A\xrightarrow{\pi}\mathbb{C}\oplus\mathbb{C}\to
0,$

where $\iota$ is the natural embedding and $\pi(f)=(\lambda,\mu)$ for $f\in
A$. The induced well-known six-term exact sequence goes as follows:

$0\to{\rm
K}_{0}(A)\to\mathbb{Z}\oplus\mathbb{Z}\xrightarrow{(2,-1)}\mathbb{Z}\to{\rm
K}_{1}(A)\to 0,$

in which ${\rm K}_{0}(M_{2}(C_{0}(0,1)))=0$ and ${\rm
K}_{1}(\mathbb{C}\oplus\mathbb{C})=0$. So we obtain that

${\rm
K}_{0}(A)=\\{(a,b)\in\mathbb{Z}\oplus\mathbb{Z}:\,2a-b=0\\}\cong\mathbb{Z}$

and

${\rm
K}_{0}^{+}(A)=\\{(n,2n)\in\mathbb{Z}\oplus\mathbb{Z}:\,n\in\mathbb{N}\\}\cong\mathbb{N}.$

Note that $A$ is a non-unital subhomogenous algebra of stable rank one, hence,
$A$ has cancellation of projections. Since $0$ is the unique projection in
$A$, i.e., $A$ is projectionless, $A$ doesn’t have a countable approximate
unit consisting of projections. Then we can not lift the generator
$(1,2)\in{\rm K}_{0}^{+}(A)$ to any projection in $A$. We have

$\Sigma\,A=\\{(0,0)\\}\subset{\rm K}_{0}^{+}(A),$

which means Definition 1.8 (iii) is not satisfied. Also note that the
generator $(1,2)$ can be lifted to a projection $p$ in $M_{2}(A)$ with

$p(0)=\left(\\!\\!\begin{array}[]{cccc}1&&&\\\\[4.2679pt] &1&&\\\\[4.2679pt]
&&0&\\\\[4.2679pt] &&&0\end{array}\\!\\!\right)\quad{\rm and}\quad
p(1)=\left(\\!\\!\begin{array}[]{cccc}1&&&\\\\[4.2679pt] &0&&\\\\[4.2679pt]
&&1&\\\\[4.2679pt] &&&0\end{array}\\!\\!\right).$

##  2\. Unital Absorbing Extensions

In this section, we consider the unital extensions of $A$ by $B$:

$0\to B\to E\to A\to 0,$

where $A,B,E$ are all separable ${\rm C}^{*}$-algebra of stable rank one and
real rank zero, $B$ is stable and $A$ is unital simple. We will always
identify $B$ as its image in $E$.

The following proposition is well-known, see [9, Proposition 4].

######  Proposition 2.1.

Let

$0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0$

be an extension of $\mathrm{C}^{*}$-algebras. Let
$\delta_{j}:\mathrm{K}_{j}(A)\rightarrow\mathrm{K}_{1-j}(B)$ for $j=0,1$, be
the index maps of the sequence.

(i) Assume that $A$ and $B$ have real rank zero. Then the following three
conditions are equivalent:

(a) $\delta_{0}\equiv 0$,

(b) $rr(E)=0$,

(c) all projections in $A$ are images of projections in $E$.

(ii) Assume that $A$ and $B$ have stable rank one. Then the following are
equivalent:

(a) $\delta_{1}\equiv 0$,

(b) $sr(E)=1$.

If, in addition, $E$ (and $A$) are unital, then (a) and (b) in (ii) are
equivalent to

(c) all unitaries in $A$ are images of unitaries in $E$.

Then we have a direct corollary as follows which indicates that it is
interesting to consider the extension with trivial index maps.

######  Corollary 2.2.

Let $A,B$ be ${\rm C}^{*}$-algebras of stable rank one and real rank zero. Let
$e\,:\,$$0\to B\to E\to A\to 0$ be an extension. Then $e$ has trivial index
maps if and only if $E$ has stable rank one and real rank zero.

Let $A,B$ be separable ${\rm C}^{*}$-algebras of stable rank one and real rank
zero. The lattice of ideals in $A$ is isomorphic to the lattice of order
ideals in ${\rm K}_{0}(A)$ (see [16], [7]). A positive morphism of groups
$\phi:{\rm K}_{0}(A)\to{\rm K}_{0}(B)$ induces a map $I\mapsto I^{\prime}$
from the ideals of $A$ to the ideals of $B$. Here $I^{\prime}$ is the unique
ideal in $B$ such that the image of ${\rm K}_{0}(I^{\prime})$ in ${\rm
K}_{0}(B)$ is the order ideal generated by $\phi({\rm K}_{0}(I))$. The map
${\rm K}_{0}(I)\to{\rm K}_{0}(A)$ is injective, since $A$ has stable rank one.

######  Proposition 2.3.

Let $A$ be a separable ${\rm C}^{*}$-algebra of stable rank one and real rank
zero and $p$ be a projection in $\mathcal{M}(A)$. Denote $\iota:\,pAp\to A$ to
be the natural embedding map. Then ${\rm K}_{0}(\iota):\,{\rm
K}_{0}(pAp)\to{\rm K}_{0}(A)$ is injective.

Moreover, we have the follow statements are equivalent:

(i) $pAp$ is full in $A$;

(ii) ${\rm K}_{0}(\iota):\,{\rm K}_{0}(pAp)\to{\rm K}_{0}(A)$ is an
isomorphism.

###### Proof.

Consider the natural inclusion diagram

---
$\textstyle{{\,\,pAp\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\scriptstyle{\iota_{1}}$$\textstyle{{\,\,A,\,\,}}$$\textstyle{{\,\,I_{p}\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{2}}$

where $I_{p}$ is the ideal of $A$ generated by $pAp$. And it induces the
commutative diagram as follows

---
$\textstyle{{\,\,{\rm
K}_{0}(pAp)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
K}_{0}(\iota)}$$\scriptstyle{{\rm K}_{0}(\iota_{1})}$$\textstyle{{\,\,{\rm
K}_{0}(A).\,\,}}$$\textstyle{{\,\,{\rm
K}_{0}(I_{p})\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
K}_{0}(\iota_{2})}$

By Theorem 1.5 and Remark 1.6, we have $pAp\otimes\mathcal{K}$ and
$I_{p}\otimes\mathcal{K}$ are canonically isomorphic and ${\rm
K}_{0}(\iota_{1})$ is an isomorphism. By Proposition 2.1(ii)(a), we have ${\rm
K}_{0}(\iota_{2})$ is injective. Then the composed map ${\rm K}_{0}(\iota)$ is
injective.

For (i) $\Rightarrow$ (ii), $pAp$ is full in $A$ implies $I_{p}=A$, and hence,
${\rm K}_{0}(\iota_{2})$ is an isomorphism. Thus, the composed map ${\rm
K}_{0}(\iota)$ is an isomorphism.

For (ii) $\Rightarrow$ (i), we have shown that ${\rm K}_{0}(\iota):\,{\rm
K}_{0}(pAp)\to{\rm K}_{0}(A)$ is an injective map. By assumption, ${\rm
K}_{0}(\iota):\,{\rm K}_{0}(pAp)\to{\rm K}_{0}(A)$ is an isomorphism. This is
equal to say that the ideal generated by ${\rm K}_{0}(\iota)({\rm
K}_{0}(pAp))$ in ${\rm K}_{0}(A)$ is ${\rm K}_{0}(A)$ itself. Since $A$ a
separable ${\rm C}^{*}$-algebra of stable rank one and real rank zero, we have
${\rm Lat}(A)\cong{\rm Lat}({\rm K}_{0}(A))$. In addition, the above argument
implies the ideal generated by $pAp$ in $A$ is $A$ itself.

∎

######  Lemma 2.4.

Let $A,B$ be nuclear, separable ${\rm C}^{*}$-algebras of stable rank one and
real rank zero. Suppose that $A$ is unital and $B$ is stable. Let $e$ be a
unital essential extension

$0\to B\to E\to A\to 0.$

For any projection $f$ in $M_{k}(E)$ with $1_{E}\leq f\leq 1_{M_{k}(E)}$ for
some $k\in\mathbb{N}$, we have ${fM_{k}(B)f}\cong B$ is stable and $B$ is full
in $fM_{k}(B)f$.

###### Proof.

Consider the natural inclusion diagram

---
$\textstyle{{\,\,B\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{3}}$$\scriptstyle{\iota_{1}}$$\textstyle{{\,\,M_{k}(B),\,\,}}$$\textstyle{{\,\,fM_{k}(B)f\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{2}}$

where $B$ is regarded as the first corner of $M_{k}(B)$. It induces the
following commutative diagram

---
$\textstyle{{\,\,{\rm
K}_{0}(B)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
K}_{0}(\iota_{3})}$$\scriptstyle{{\rm K}_{0}(\iota_{1})}$$\textstyle{{\,\,{\rm
K}_{0}(M_{k}(B)).\,\,}}$$\textstyle{{\,\,{\rm
K}_{0}(fM_{k}(B)f)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
K}_{0}(\iota_{2})}$

Since $B$ is full in $M_{k}(B)$, then $fM_{k}(B)f$ is also full in $M_{k}(B)$.
By Theorem 1.5 and Remark 1.6, we have both ${\rm K}_{0}(\iota_{2})$ and ${\rm
K}_{0}(\iota_{3})$ are isomorphisms. By commutativity, we have ${\rm
K}_{0}(\iota_{1})$ is also an isomorphism, by Proposition 2.3, we have $B$ is
full in $fM_{k}(B)f$.

For any projection $p$ in $B$, as ${\rm
K}_{0}(\iota_{1})([p])=[\iota_{1}(p)]$, we have

${\rm K}_{0}(\iota_{2})\circ{\rm K}_{0}(\iota_{1})(\Sigma\,B)\subset{\rm
K}_{0}(\iota_{2})\left(\Sigma\,{(fM_{k}(B)f)}\right)\subset\Sigma\,(M_{k}(B)).$

Apply Proposition 1.7 for $B$ and $M_{k}(B)$, then

$\Sigma\,B={\rm K}_{0}^{+}(B)=\Sigma\,(M_{k}(B)).$

Since $B$ is full in $M_{k}(B)$, by Theorem 1.5 and Remark 1.6, we have

${\rm K}_{0}(\iota_{2})\circ{\rm
K}_{0}(\iota_{1})(\Sigma\,B)=\Sigma\,(M_{k}(B)),$

we obtain

${\rm K}_{0}(\iota_{1})(\Sigma\,B)=\Sigma\,{(fM_{k}(B)f)},$

that is,

$\Sigma{(fM_{k}(B)f)}={\rm K}_{0}(\iota_{1})({\rm K}_{0}^{+}(B))={\rm
K}_{0}^{+}(fM_{k}(B)f).$

By Proposition 1.7, $fM_{k}(B)f$ is stable, and hence, by Theorem 1.5,

$fM_{k}(B)f\cong B.$

∎

######  Lemma 2.5.

Let $A,B$ be nuclear, separable ${\rm C}^{*}$-algebras of stable rank one and
real rank zero. Suppose that $A$ is unital, $B$ is stable and $({\rm
K}_{0}(B),{\rm K}_{0}^{+}(B))$ is weakly unperforated. Let $e$ be a unital
essential extension with trivial index maps

$0\to B\to E\to A\to 0.$

Suppose $p,f$ are projections in $M_{k}(E)$ satisfying

$[1_{E}]\leq n\cdot[p]=[f],\,\,p\leq f,\,\,1_{E}\leq f\leq 1_{M_{k}(E)}$

for some $n,k\in\mathbb{N}$. Then ${pM_{k}(B)p}\cong B$ is stable and
${pM_{k}(B)p}$ is full in $fM_{k}(B)f$.

###### Proof.

By Lemma 2.4, we have $fM_{k}(B)f\cong B$ and $B$ is full in $fM_{k}(B)f$.
Since $[f]=n\cdot[p]$, $1_{E}\leq f\leq 1_{M_{k}(E)}$ and $p\leq f$, noting
that $E$ has stable rank one, we have $\underbrace{p\oplus p\oplus\cdots\oplus
p}_{n-1}$ is Murray–von Neumann equivalent to $f-p$ in $M_{l}(E)$ for some
$l$, set $p_{1}=p$, thus we have

$f=p_{1}+p_{2}+\cdots+p_{n},$

where each $p_{i}\in M_{k}(E)$ is a subprojection of $f$, $p_{i}\sim p_{1}$
and $p_{i}p_{j}=0$ for any $i\neq j$. Denote $v_{j}$ the partial isometry in
$M_{k}(E)$ with

$v_{j}v_{j}^{*}=p_{1},\quad
v_{j}^{*}v_{j}=p_{j}\quad{\rm~{}~{}for~{}~{}any}~{}~{}j=1,2,\cdots,n.$

Now we show $pM_{k}(B)p$ is full in $fM_{k}(B)f$. Note that $fM_{k}(B)f$ is
also an ideal of $fM_{k}(E)f$, so we only need to show that $fM_{k}(B)f$ is
the ideal of $fM_{k}(E)f$ generated by $pM_{k}(B)p$. (If $C,D,W$ are ${\rm
C}^{*}$-algebras with $D$ is an ideal of $W$ and $C$ is an ideal of $D$, we
will have $C$ is also an ideal of $W$.)

For any $x\in fM_{k}(B)f$, then $x=\sum_{i=1}^{n}\sum_{j=1}^{n}p_{i}xp_{j}$.
For any $i,j$, $p_{i}xp_{j}\in fM_{k}(B)f$, we have

$v_{i}p_{i}xp_{j}v_{j}^{*}\in pM_{k}(B)p,$

which implies

$v_{i}^{*}v_{i}p_{i}xp_{j}v_{j}^{*}v_{j}=p_{i}xp_{j},$

i.e., $p_{i}xp_{j}$ is contained in the ideal of $fM_{k}(E)f$ generated by
$pM_{k}(B)p$. Then we have $x$ is contained in the ideal of $fM_{k}(B)f$
generated by $pM_{k}(B)p$. In general, we have $pM_{k}(B)p$ is full in
$fM_{k}(B)f$.

Moreover, regarding $\\{v_{i}v_{j}^{*}\\}$ as the matrix units and
$pM_{k}(B)p$ as the first corner, we have

$pM_{k}(B)p\otimes M_{n}\cong fM_{k}(B)f$

naturally.

Since $pM_{k}(B)p$ is full in $fM_{k}(B)f$, by Theorem 1.5, we have

$pM_{k}(B)p\otimes\mathcal{K}\cong fM_{k}(B)f\otimes\mathcal{K}\cong B.$

Then $({\rm K}_{0}(pM_{k}(B)p),{\rm K}_{0}^{+}(pM_{k}(B)p))\cong({\rm
K}_{0}(B),{\rm K}_{0}^{+}(B))$ is weakly unperforated.

Denote the natural embedding map $\iota:\,pM_{k}(B)p\to fM_{k}(B)f$, by
Theorem 1.5 and Remark 1.6, we identify ${\rm K}_{0}^{+}(pM_{k}(B)p)$ with
${\rm K}_{0}^{+}(fM_{k}(B)f)$ through ${\rm K}_{0}(\iota)$.

Apply Proposition 1.7 for $fM_{k}(B)f$, we obtain

$n\,\widehat{\cdot}\,\Sigma\,(pM_{k}(B)p)=\Sigma(fM_{k}(B)f)={\rm
K}_{0}^{+}(fM_{k}(B)f)={\rm K}_{0}^{+}(pM_{k}(B)p).$

Then by Proposition 1.10, we have

$\Sigma\,(pM_{k}(B)p)={\rm K}_{0}^{+}(pM_{k}(B)p).$

By Proposition 1.7, $pM_{k}(B)p$ is stable, and hence, $pM_{k}(B)p\cong B$.

∎

######  Theorem 2.6.

Let $A,B$ be nuclear, separable ${\rm C}^{*}$-algebras of stable rank one and
real rank zero. Suppose that $A$ is unital simple, $B$ is stable and $({\rm
K}_{0}(B),{\rm K}_{0}^{+}(B))$ is weakly unperforated. Let $e$ be a unital
extension with trivial index maps

$0\to B\rightarrow E\xrightarrow{\pi}A\to 0.$

Then $e$ is absorbing.

###### Proof.

We only need to prove $e$ is purely large.

For any $c\in E\backslash B$, $\pi(c)\neq 0$. Since $E$ has real rank zero,
then there exists a nonzero projection $p\in\overline{cEc^{*}}$ such that
$p\in E\backslash B$. Otherwise, if such $p$ doesn’t exist, by
$rr(\overline{cEc^{*}})=0$, we will have $\overline{cEc^{*}}\subset B$, which
implies $c\in B$ and forms a contradiction.

Note that $pBp\subset\overline{cBc^{*}}$, we will show that $pBp$ is stable
and is full in $B$.

Since $p\in E\backslash B$, $\pi(p)$ is a nonzero projection in $A$. As $A$ is
unital simple and has stable rank one, there exists an integer $n$ such that
$n\cdot[\pi(p)]\geq 2[1_{A}]$. Now we have

${\rm K}_{0}(\pi)(n\cdot[p]-2[1_{E}])\geq 0\quad{\rm in}\,\,{\rm
K}_{0}^{+}(A).$

As $E$ has real rank zero, by Proposition 2.1(i)(c), there exists $x\in{\rm
K}_{0}^{+}(E)$ such that

${\rm K}_{0}(\pi)(n\cdot[p]-2[1_{E}])={\rm K}_{0}(\pi)(x).$

Recall that $e$ has trivial index maps, then the following is an exact
sequence

$0\to{\rm K}_{0}(B)\rightarrow{\rm K}_{0}(E)\xrightarrow{{\rm K}_{0}(\pi)}{\rm
K}_{0}(A)\to 0.$

So ${\rm K}_{0}(B)$ can be identified as the subgroup of ${\rm K}_{0}(E)$.
Then from the exactness, there exist projections $d,r$ in $B$ such that

$n[p]-2[1_{E}]+[d]-[r]=x\geq 0\quad{\rm in}\,\,{\rm K}_{0}^{+}(E).$

Moreover, we have

$n[p]+n[d]\geq 2[1_{E}]\quad{\rm in}\,\,{\rm K}_{0}^{+}(E).$

Since $E$ has stable rank one, we can lift $n[p]+n[d]$ to a projection $f$ in
$M_{k}(E)$ ($k\geq 2$) with

$1_{E}\leq 1_{M_{2}(E)}\leq f\leq 1_{M_{k}(E)}.$

Since $p\leq 1_{E}$ and $d\leq 1_{E}$, we regard $p\oplus d$ as a projection
in $M_{2}(E)\subset M_{k}(E)$. Then we have

$p\oplus d\leq 1_{M_{2}(E)}\leq f\quad{\rm in}\,\,M_{k}(E).$

Apply Lemma 2.5, we have $(p\oplus d)M_{k}(B)(p\oplus d)\cong B$ and $(p\oplus
d)M_{k}(B)(p\oplus d)$ is full in $fM_{k}(B)f$.

Consider the following commutative diagram

$\textstyle{{\,\,(p\oplus d)M_{k}(B)(p\oplus
d)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{6}}$$\scriptstyle{\iota_{1}}$$\textstyle{{\,\,M_{k}(E)\,\,}}$$\textstyle{{\,\,E\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{4}}$$\textstyle{{\,\,fM_{k}(B)f\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{2}}$$\textstyle{{\,\,M_{k}(B)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{5}}$$\textstyle{{\,\,B\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\iota_{3}}$$\scriptstyle{\iota_{7}}$

where for each $i$, $\iota_{i}$ is the natural embedding map ($B$ and $E$ are
regarded as the first corners of $M_{k}(B)$ and $M_{k}(E)$, respectively).
Then it induces a commutative diagram as follows

$\textstyle{{\,\,{\rm K}_{0}((p\oplus d)M_{k}(B)(p\oplus
d))\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
K}_{0}(\iota_{6})}$$\scriptstyle{{\rm K}_{0}(\iota_{1})}$$\textstyle{{\,\,{\rm
K}_{0}(M_{k}(E))\,\,}}$$\textstyle{{\,\,{\rm
K}_{0}(E)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
K}_{0}(\iota_{4})}$$\textstyle{{\,\,{\rm
K}_{0}(fM_{k}(B)f)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
K}_{0}(\iota_{2})}$$\textstyle{{\,\,{\rm
K}_{0}(M_{k}(B))\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
K}_{0}(\iota_{5})}$$\textstyle{{\,\,{\rm
K}_{0}(B)\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{{\rm
K}_{0}(\iota_{3})}$$\scriptstyle{{\rm K}_{0}(\iota_{7})}$

Since $E$ has stable rank one and real rank zero, by Proposition 2.3, on one
hand, for each $1\leq i\leq 7$, we have ${\rm K}_{0}(\iota_{i})$ is injective;
on the other hand, by the fullness we have obtained, for each $1\leq i\leq 4$,
we have ${\rm K}_{0}(\iota_{i})$ is an isomorphism.

Note that $(p\oplus d)M_{k}(B)(p\oplus d)$ is stable and also has stable rank
one, then for any projection $q\in B$, we can lift

${\rm K}_{0}(\iota_{1})^{-1}\circ{\rm K}_{0}(\iota_{2})^{-1}\circ{\rm
K}_{0}(\iota_{3})([q])\in{\rm K}_{0}^{+}((p\oplus d)M_{k}(B)(p\oplus d))$

to a projection $q^{\prime}\in(p\oplus d)M_{k}(B)(p\oplus d)$. Then

$\iota_{6}(q^{\prime})\leq p\oplus d\quad{\rm in}\,\,M_{k}(E).$

Combining with the commutativity, we have

${\rm K}_{0}(\iota_{5})\circ{\rm K}_{0}(\iota_{3})([q])={\rm
K}_{0}(\iota_{6})([q^{\prime}])\leq[p\oplus d]=[p]+[d]\quad{\rm in}\,\,{\rm
K}_{0}^{+}(M_{k}(E)).$

From the arbitrariness of $q$, by choosing $\widetilde{q}$ with
$[\widetilde{q}]=[q]+[d]$, we have

${\rm K}_{0}(\iota_{5})\circ{\rm K}_{0}(\iota_{3})([q])\leq[p]\in{\rm
K}_{0}^{+}(M_{k}(E)).\qquad(*)$

Note that we can identify ${\rm K}_{0}(B)$ as the subgroup of ${\rm K}_{0}(E)$
through the injective map ${\rm K}_{0}(\iota_{5})\circ{\rm K}_{0}(\iota_{3})$.

As $p\leq 1_{E}$, $pM_{k}(B)p$ is naturally identified as $pBp$ in $B$. Denote
$\iota:\,pBp\to B$ the natural inclusion. Since $B$ has stable rank one, we
can identify ${\rm K}_{0}^{+}(pBp)$ as a sub-semigroup of ${\rm K}_{0}^{+}(B)$
(Proposition 2.3). Now $(*)$ implies $\Sigma\,(pBp)={\rm K}_{0}^{+}(B)$, which
means ${\rm K}_{0}^{+}(pBp)={\rm K}_{0}^{+}(B)$. Then by Proposition 1.7, we
have $pBp$ is stable, and hence, $pBp\cong B.$ From Proposition 2.3, we get
$pBp$ is full in $B$. Now $e$ is purely large, then by Theorem 1.4, $e$ is
absorbing.

∎

######  Corollary 2.7.

Suppose $A,B,E$ satisfy the assumption of Theorem 2.6. Then for any
$p\in\mathcal{P}(E\backslash B)$ and $q\in\mathcal{P}(B)$, we have
$[q]\leq[p]$ in ${\rm K}_{0}(E)$.

##  3\. Applications

######  Definition 3.1.

Let

$e_{i}:\quad 0\to B\to E_{i}\to A\to 0$

be two extensions with Busby invariants $\tau_{i}$ for $i=1,2$.

Two unital extensions $e_{1}$ and $e_{2}$ are called $stably$ $strongly$
$unitarily$ $equivalent$, denoted by $e_{1}\sim_{ss}e_{2}$, if there exist
unital trivial extensions $\sigma_{1},\sigma_{2}$ such that
$e_{1}\oplus\sigma_{1}\sim_{s}e_{2}\oplus\sigma_{2}$.

If $A$ is unital, denote $\mathrm{Ext}^{u}_{ss}(A,B)$ the set of stably
strongly unitary equivalence classes of unital extensions of $A$ by $B$.

From our main result Theorem 2.6, we have

######  Corollary 3.2.

Let $A,B$ be nuclear, separable ${\rm C}^{*}$-algebras of stable rank one and
real rank zero. Suppose that $A$ is unital simple, $B$ is stable and $({\rm
K}_{0}(B),{\rm K}_{0}^{+}(B))$ is weakly unperforated. Let $e_{1},e_{2}$ be
two unital extensions with trivial index maps:

$e_{i}\,:\,0\to B\to E_{i}\to A\to 0.$

Then $e_{1}\sim_{ss}e_{2}$ if and only if $e_{1}\sim_{s}e_{2}$.

######  Definition 3.3.

Given abelian groups $H,K$ and two extensions

$e_{i}\,\,:\,\,0\to K\to G_{i}\to H\to 0,\quad i=1,2,$

we say $e_{1}$, $e_{2}$ are $equivalent$, if there is a homomorphism $\alpha$
making the following diagram commute:

$\textstyle{{\,\,0\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\,\,K\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{{\,\,G_{1}\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{{\,\,H\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{{\,\,0\,\,}}$$\textstyle{{\,\,0\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\,\,K\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\,\,G_{2}\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\,\,H\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\,\,0\,\,}.}$

Denote $\mathrm{Ext}(H,K)$ the set of all the equivalent classes of extensions
of $H$ by $K$. It is well-known that $\mathrm{Ext}(H,K)$ forms an abelian
group.

Let $h_{0}\in H$. We consider the following extension of $H$ by $K$ with base
point $h_{0}$,

$e:\quad 0\to K\to(G,g_{0})\xrightarrow{\psi}(H,h_{0})\to 0,$

where $g_{0}\in G$ and $\psi(g_{0})=h_{0}$.

Suppose we have extensions

$e_{i}:\quad 0\to K\to(G_{i},g_{i})\xrightarrow{\psi}(H,h_{0})\to 0,\quad
i=1,2$

with $\psi(g_{i})=h_{0}$. We say $e_{1}$ and $e_{2}$ are equivalent if there
is a homomorphism $\phi:G_{1}\to G_{2}$ with $\phi(g_{1})=g_{2}$ such that the
following diagram commutes

$\textstyle{{\,\,0\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\,\,K\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{{\,\,(G_{1},g_{1})\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{{\,\,(H,h_{0})\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{{\,\,0\,\,}}$$\textstyle{{\,\,0\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\,\,K\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\,\,(G_{2},g_{2})\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\,\,(H,h_{0})\,\,}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\,\,0\,\,}.}$

Let $\mathrm{Ext}((H,h_{0}),K)$ be the set of equivalent classes of all
extensions of $H$ by $K$ with base point $h_{0}$.

Let $A$ and $B$ be ${\rm C}^{*}$-algebras. When $A$ is unital, we set

$\mathrm{Ext}_{[1]}(\mathrm{K}_{*}(A),\mathrm{K}_{*}(B))=\mathrm{Ext}((\mathrm{K}_{0}(A),[1_{A}]),\mathrm{K}_{0}(B))\oplus\mathrm{Ext}(\mathrm{K}_{1}(A),\mathrm{K}_{1}(B)),$
${\rm Hom}_{[1]}(\mathrm{K}_{0}(A),\mathrm{K}_{1}(B))=\\{\rho\in{\rm
Hom}((\mathrm{K}_{0}(A),[1_{A}]),\mathrm{K}_{1}(B)):\rho([1_{A}])=0\\},$ ${\rm
Hom}_{[1]}(\mathrm{K}_{*}(A),\mathrm{K}_{*}(B))={\rm
Hom}_{[1]}(\mathrm{K}_{0}(A),\mathrm{K}_{1}(B))\oplus{\rm
Hom}(\mathrm{K}_{1}(A),\mathrm{K}_{0}(B)).$

Denote $\mathcal{N}$ the “bootstrap” class of ${\rm C}^{*}$-algebras in [11].
For the unital version of “UCT”, Wei used the stably strongly unitary
equivalence classes of unital extensions to get the following theorem (a
similar statement can be found in [15, Proposition 3.2]).

######  Theorem 3.4.

([14, Theorem 4.9]) Suppose that $A$ is a unital separable nuclear ${\rm
C}^{*}$-algebra with $A\in\mathcal{N}$ and $B$ is separable stable ${\rm
C}^{*}$-algebra. Then there is a short exact sequence of groups

$0\to\mathrm{Ext}_{[1]}(\mathrm{K}_{*}(A),\mathrm{K}_{*}(B))\to\mathrm{Ext}_{ss}^{u}(A,B)\to{\rm
Hom}_{[1]}(\mathrm{K}_{*}(A),\mathrm{K}_{*}(B))\to 0.$

Combining with Corollary 3.2, we get the following result.

######  Theorem 3.5.

Let $A,B$ be nuclear, separable ${\rm C}^{*}$-algebras of stable rank one and
real rank zero with $A\in\mathcal{N}$. Assume that $A$ is unital simple, $B$
is stable and $({\rm K}_{0}(B),{\rm K}_{0}^{+}(B))$ is weakly unperforated.
Denote ${\rm Text}_{s}^{u}(A,B)$ to be the set of strongly unitary equivalence
classes of all the unital extensions of $A$ by $B$ with trivial index maps.
Then we have

$\mathrm{Ext}_{[1]}(\mathrm{K}_{*}(A),\mathrm{K}_{*}(B))\cong{\rm
Text}_{s}^{u}(A,B).$

In [1], Corollary 2.7 and Theorem 3.5 are applied to construct a
counterexample for Elliott Conjecture with real rank zero and obtain
classification theorems of ${\rm C}^{*}$-algebras in the setting of stable
rank one and real rank zero.

## Acknowledgements

The authors are very grateful to Yifeng Xue and Changguo Wei for many helpful
discussions. The first author was supported by NNSF of China (No.:12101113).
The second author was supported by NNSF of China (No.:12101102).

## References

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††thanks:  0000-0002-4892-2093††thanks:  0000-0002-9118-7354††thanks:
0000-0003-2028-6782††thanks:  0000-0002-2020-7223††thanks:
0000-0001-6749-0022

# Back to (Mass-)Square(d) One:
The Neutrino Mass Ordering in Light of Recent Data

Kevin J. Kelly<EMAIL_ADDRESS>Theoretical Physics Department, Fermi
National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA Pedro
A. N. Machado<EMAIL_ADDRESS>Theoretical Physics Department, Fermi National
Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA Stephen J. Parke
<EMAIL_ADDRESS>Theoretical Physics Department, Fermi National Accelerator
Laboratory, P. O. Box 500, Batavia, IL 60510, USA Yuber F. Perez-Gonzalez
<EMAIL_ADDRESS>Theoretical Physics Department, Fermi National
Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA Department of
Physics & Astronomy, Northwestern University, Evanston, IL 60208, USA Colegio
de Física Fundamental e Interdisciplinaria de las Américas (COFI), 254
Norzagaray Street, San Juan, Puerto Rico 00901 Renata Zukanovich Funchal
<EMAIL_ADDRESS>Instituto de Física, Universidade de São Paulo, C.P. 66.318,
05315-970 São Paulo, Brazil

###### Abstract

We inspect recently updated neutrino oscillation data – specifically coming
from the Tokai to Kamioka and NuMI Off-axis $\nu_{e}$ Appearance experiments –
and how they are analyzed to determine whether the neutrino mass ordering is
normal ($m_{1}<m_{2}<m_{3}$) or inverted ($m_{3}<m_{1}<m_{2}$). We show that,
despite previous results giving a strong preference for the normal ordering,
with the newest data from T2K and NOvA, this preference has all but vanished.
Additionally, we highlight the importance of this result for non-oscillation
probes of neutrinos, including neutrinoless double beta decay and cosmology.
Future experiments, including JUNO, DUNE, and T2HK will provide valuable
information and determine the mass ordering at a high confidence level.

††preprint: FERMILAB-PUB-20-330-T

Introduction. — By observing the phenomenon of neutrino oscillations, we have
determined a number of their properties. Current data allow us to understand
how neutrinos mix and that there are two non-zero mass scales. Neutrinos in
any oscillation environment are highly relativistic, so experiments are only
sensitive to differences of masses squared, $\Delta m_{ji}^{2}\equiv
m_{j}^{2}-m_{i}^{2}$, between the three neutrino mass eigenstates111Defined
where $\nu_{1}$ ($\nu_{3}$) has the largest (smallest) admixture of $\nu_{e}$.
$\nu_{i}$, with masses $m_{i}$.

Solar and reactor neutrino experiments, have determined222Barring additional
new physics in the neutrino sector Coloma and Schwetz (2016); Coloma  _et al._
(2017a, b). $\Delta m_{21}^{2}\approx+7.5\times 10^{-5}~{}{\rm eV}^{2}$
Aharmim  _et al._ (2013); Gando  _et al._ (2013). Accelerator/atmospheric
neutrinos have determined $\left\lvert\Delta m_{31}^{2}\right\rvert\approx
2.5\times 10^{-3}~{}{\rm eV}^{2}\gg\Delta m_{21}^{2}$ but, in general, are not
sensitive to the sign of $\Delta m_{31}^{2}$ – this is the neutrino mass
ordering (MO) problem – whether nature prefers $m_{1}<m_{2}<m_{3}$, the normal
ordering (NO), or $m_{3}<m_{1}<m_{2}$, the inverted ordering (IO) De Salas
_et al._ (2018).

There are two ways to determine the MO, utilizing interference or matter
effects. The first, employed by the upcoming JUNO An  _et al._ (2016)
experiment, relies on measuring neutrino oscillations where both mass-squared-
splittings are relevant. Alternatively, accelerator neutrino experiments where
effects due to $\Delta m_{31}^{2}$ are dominant and matter effects (from
neutrino interactions with rock along their journey) are relevant, are also
sensitive to the MO. A combination of measuring oscillation probabilities for
muon-neutrino disappearance $P(\nu_{\mu}\to\nu_{\mu})$ and electron-neutrino
appearance $P(\nu_{\mu}\to\nu_{e})$ (as well as the corresponding
probabilities for antineutrinos) allows for long-baseline experiments to
measure the MO. However, degeneracies exist between determining the MO, the
atmospheric octant, and the degree of CP violation in the leptonic sector
($\delta_{\rm CP}$).

The latter strategy is employed by the currently-operating Tokai to Kamioka
(T2K) Abe  _et al._ (2018a, 2020a, 2020b) and NuMI Off-axis $\nu_{e}$
Appearance (NOvA) Adamson  _et al._ (2017); Acero  _et al._ (2018, 2019)
experiments, which measure $P(\nu_{\mu}\to\nu_{e})$ and
$P(\overline{\nu}_{\mu}\to\overline{\nu}_{e})$ at long distances. Super-
Kamiokande (SK) Abe  _et al._ (2018b) also has modest sensitivity to the MO by
studying atmospheric neutrino oscillations, where matter effects are
important.

Figure 1: Bi-probability plots of oscillation probabilites for neutrinos
(x-axes) and antineutrinos (y-axes) at the $L$ and $E$ for NOvA (left panel)
and T2K (right panel) while varying $\delta_{\rm CP}$. Black (grey) crosses
indicate measurements with statistical uncertainty only for the two
experiments using their 2020 (pre-2020) results. Ellipses correspond to best-
fit points according to NOvA (blue) and T2K (red) fits under the the Normal
(solid) or Inverted (dashed) mass ordering. Dots denote probabilities for
$\delta_{\rm CP}=0$, with arrows indicating increasing $\delta_{\rm CP}$.

As of mid-2020, existing data, driven largely by these three experiments,
exhibited a strong preference for the NO over the IO:
$\Delta\chi^{2}_{\rm(NO,IO)}\equiv\chi^{2}_{\rm min,IO}-\chi^{2}_{\rm
min,NO}\approx 10$ Esteban  _et al._ (2019); Capozzi  _et al._ (2020); de
Salas  _et al._ (2020). However, T2K, NOvA, and SK have each provided
preliminary updates Dunne (2020); Himmel (2020); Nakajima (2020). We will
demonstrate that this NO preference vanished due to correlations between the
data, as well as the degeneracies between MO, octant, and $\delta_{\rm CP}$.
Additionally, we will discuss the ramifications of this result and provide
some outlook for the future.

MO sensitivity at Long-Baseline Experiments. — In long-baseline experiments
like T2K and NOvA (and the planned T2HK Abe  _et al._ (2018c) and DUNE Abi
_et al._ (2020a, b) experiments), oscillations due to $\Delta m_{21}^{2}$ have
yet to develop, so the expansion parameter $\Delta m_{21}^{2}L/4E$ is
perturbatively small. Assuming neutrinos propagate through constant-density
matter, the oscillation probability of $\nu_{\mu}\to\nu_{e}$ (with energy $E$
and distance $L$) can be approximated Nunokawa  _et al._ (2008)

$\displaystyle P_{\mu e}\equiv P(\nu_{\mu}\to\nu_{e})\approx
4s_{23}^{2}s_{13}^{2}c_{13}^{2}\frac{\sin^{2}\left(\Delta_{31}-aL\right)}{\left(\Delta_{31}-aL\right)^{2}}\Delta_{31}^{2}$
$\displaystyle+8\frac{J}{\sin\delta_{\rm
CP}}\frac{\sin\left(\Delta_{31}-aL\right)}{\left(\Delta_{31}-aL\right)}\Delta_{31}\frac{\sin{(aL)}}{(aL)}\Delta_{21}\cos{\left(\Delta_{31}+\delta_{\rm
CP}\right)}$
$\displaystyle+4s_{12}^{2}c_{12}^{2}c_{13}^{2}c_{23}^{2}\frac{\sin^{2}\left(aL\right)}{\left(aL\right)^{2}}\Delta_{21}^{2},$
(1)

where $\Delta_{j1}\equiv\Delta m_{j1}^{2}L/4E$, $s_{ij}\equiv\sin\theta_{ij}$,
$c_{ij}\equiv\cos\theta_{ij}$, and $J\equiv
s_{23}c_{23}s_{13}c_{13}^{2}s_{12}c_{12}\sin\delta_{\rm CP}$ is the Jarlskog
invariant Jarlskog (1985). Effects of propagation through matter are given by
the matter potential Wolfenstein (1978),

$a=\frac{G_{F}n_{e}}{\sqrt{2}}\approx\frac{1}{3500\
\mathrm{km}}\left(\frac{\rho}{3.0\ \mathrm{g/cm}^{3}}\right),$ (2)

where $\rho$ is the density along the path of propagation. For current/planned
$\nu_{e}$-appearance oscillation experiments,

$\displaystyle aL=\left\\{\begin{array}[]{ll}0&\text{Vacuum, any $L$ }\\\
0.065&\text{T2K/T2HK~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Abe:2018uyc}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
0.22&\text{NOvA~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Acero:2019ksn}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
0.29&\text{T2HKK~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Abe:2016ero}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
0.35&\text{DUNE~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Abi:2020evt}{\@@citephrase{(}}{\@@citephrase{)}}}}\end{array}\right.,$
(8)

while $\left\lvert\Delta_{31}\right\rvert\approx\pi/2$ so that oscillations
due to the atmospheric mass-squared splitting are maximized.

For antineutrinos, $P_{\overline{\mu}\overline{e}}\equiv
P(\overline{\nu}_{\mu}\to\overline{\nu}_{e})$ can be determined by taking Eq.
(1) and replacing $\delta_{\rm CP}\to-\delta_{\rm CP}$ as well as
$(aL)\to-(aL)$. In Fig. 1 we display how the oscillation probabilities $P_{\mu
e}$ and $P_{\overline{\mu}\overline{e}}$ vary at NOvA (left panel) and T2K
(right) baselines/energies. We assume fixed $L=810$ km (left) and $295$ km
(right), as well as $E=2.1$ GeV (left) and $0.6$ GeV (right). We fix
$\sin^{2}\theta_{12}=0.310$, $\sin^{2}\theta_{13}=0.022$, and $\Delta
m_{21}^{2}=7.53\times 10^{-5}$ eV2 Nakajima (2020); Gando  _et al._ (2013);
Adey  _et al._ (2018). Ellipses arise by varying $\delta_{\rm CP}$ for
different combinations of $\left(\sin^{2}\theta_{23},\ \Delta
m_{31}^{2}\right)$, obtained from fits to NOvA (blue ellipses) or T2K (red),
assuming the MO is normal (solid) or inverted (dashed). We discuss how these
points are obtained in the “results” section. Fig. 1 also displays measured
oscillation probabilities (with statistical uncertainty) as black (current
data Dunne (2020); Himmel (2020)) and grey (pre-2020 data Abe  _et al._
(2018a); Acero  _et al._ (2019)) crosses. Comparing older results to the
current ones, we see that the measured oscillation probabilities are trending
toward the “IO” region of this space, where
$P_{\overline{\mu}\overline{e}}>P_{\mu e}$.

We also analyze sums/differences of the neutrino and antineutrino oscillation
probabilities, $\Sigma P_{\mu e}\equiv P_{\mu
e}+P_{\overline{\mu}\overline{e}}$ and $\Delta P_{\mu e}\equiv P_{\mu
e}-P_{\overline{\mu}\overline{e}}$. Near
$\left\lvert\Delta_{31}\right\rvert\approx\pi/2$, and assuming
$(aL)\ll\Delta_{31}$,

$\displaystyle\Sigma P_{\mu e}$ $\displaystyle\to
8s_{13}^{2}c_{13}^{2}s_{23}^{2}-16s_{12}c_{12}s_{13}c_{13}^{2}s_{23}c_{23}\sin\delta_{\rm
CP}(aL)\frac{\Delta m_{21}^{2}}{\left\lvert\Delta
m_{31}^{2}\right\rvert}\mathrm{sign}\left(\Delta m_{31}^{2}\right),$
$\displaystyle\Sigma P_{\mu e}$ $\displaystyle\approx
0.17s_{23}^{2}-0.03(aL)s_{23}c_{23}\sin\delta_{\rm
CP}\mathrm{sign}\left(\Delta m_{31}^{2}\right),$ (9a) $\displaystyle\Delta
P_{\mu e}$
$\displaystyle\to\frac{32(aL)}{\pi}s_{13}^{2}c_{13}^{2}s_{23}^{2}\mathrm{sign}\left(\Delta
m_{31}^{2}\right)-8\pi s_{12}c_{12}s_{13}c_{13}^{2}s_{23}c_{23}\sin\delta_{\rm
CP}\frac{\Delta m_{21}^{2}}{\left\lvert\Delta m_{31}^{2}\right\rvert},$
$\displaystyle\Delta P_{\mu e}$ $\displaystyle\approx
0.22(aL)s_{23}^{2}\mathrm{sign}\left(\Delta
m_{31}^{2}\right)-0.05s_{23}c_{23}\sin\delta_{\rm CP},$ (9b)

Figure 2: Sums (top) and differences (bottom) of oscillation probabilities at
NOvA (x-axes) and T2K (y-axes) at fixed $L$ and $E$. Ellipses are generated by
varying $\delta_{\rm CP}$ while fixing the other oscillation parameters. Dots
denote $\Sigma P_{\mu e}$ (top) and $\Delta P_{\mu e}$ (bottom) for
$\delta_{\rm CP}=0$, with arrows indicating increasing $\delta_{\rm CP}$.
Crosses display sums/differences of oscillation probabilities assuming
statistically-independent measurements at NOvA/T2K for current (black) and
pre-2020 (grey) results.

where we have used the current best-fit measurements of $\theta_{12}$,
$\theta_{13}$, $\Delta m_{21}^{2}$, and $\left\lvert\Delta
m_{31}^{2}\right\rvert$. Analyzing Eq. (9a), we see that measurements of the
sum of oscillation probabilities allow for extracting $s_{23}^{2}$, while the
effects of CP violation and the MO have a small impact. According to Eq. (9b),
measurements of $\Delta P_{\mu e}$ can allow for extraction of the MO, octant,
and CP violation, however these are all comparable and competing effects.

We show the sums and differences of oscillation probabilities at NOvA/T2K in
Fig. 2, presenting $\Sigma P_{\mu e}$ ($\Delta P_{\mu e}$) in the top (bottom)
panel. Extracted measurements of these sums/differences are displayed as black
(current) and grey (pre-2020) crosses, assuming statistically-independent
measurements of $P_{\mu e}$ and $P_{\overline{\mu}\overline{e}}$ at each
experiment. Red/blue ellipses are generated fixing all parameters except
$\delta_{\rm CP}$ to the same combinations as in Fig. 1. In the bottom panel,
the impact of the MO is clear – NOvA requires $\Delta P_{\mu e}>0$ for NO and
$\Delta P_{\mu e}<0$ for IO. While the separation is not as powerful for T2K
(where $(aL)$ is a factor of ${\sim}3$ smaller), the NO prefers larger $\Delta
P_{\mu e}$. As with Fig. 1, we see that current data have moved towards
favoring IO for both T2K and NOvA. In what follows, we quantify these effects,
with fits to T2K/NOvA to determine their individual and joint preferences for
the MO.

Analysis. — For T2K, we consider the latest results, equivalent to
$1.97(1.63)\times 10^{21}$ protons-on-target in neutrino (antineutrino) mode
Dunne (2020). T2K observes a total of 108 (16) $\nu_{e}$
($\overline{\nu}_{e}$) like events, and 318 (137) $\nu_{\mu}$
($\overline{\nu}_{\mu}$) like events. We classify the data in the same
categories as T2K, muon-ring ($1R\mu$) and electron-ring ($1Re$) events in
neutrino and antineutrino modes, plus $\nu_{e}-CC1\pi$ events in neutrino mode
Dennis (2015). We define a log-likelihood function comparing the
expected/observed events (including normalization uncertainties). We consider
uncertainties for $1Re$ events of 4.7% (5.9%), while for $1R\mu$ we assume
3.0% (4.0%) in neutrino (antineutrino) mode. For $\nu_{e}-CC1\pi$ we use a
14.3% uncertainty.

Our analysis of NOvA is similar to T2K – we reproduce the expected spectra for
$\nu_{\mu}$ and $\overline{\nu}_{\mu}$ disappearance channels given a set of
input parameters, as well as the number of $\nu_{e}$ and $\overline{\nu}_{e}$
appearance events. We determine these spectra following the results of Ref.
Himmel (2020), which correspond to $13.5(12.5)\times 10^{20}$ proton-on-
target-equivalent in neutrino (antineutrino) mode. These spectra and event
rates enter a Poissonian log-likelihood function as with T2K – NOvA observes a
total of 82 (33) $\nu_{e}$ ($\overline{\nu}_{e}$) like events, and 211 (105)
$\nu_{\mu}$ ($\overline{\nu}_{\mu}$) like events in this dataset. Since NOvA
is statistics-limited, we do not incorporate any systematic uncertainties at
this stage of our analysis. See Appendices A and B for details of our T2K and
NOvA analyses, respectively.

In order for the long-baseline experiments to measure $\nu_{e}$ and
$\overline{\nu}_{e}$ appearance, external information is required. For
simplicity, we fix the parameters that T2K/NOvA do not measure to their
current best-fit values from other experiments: $\sin^{2}\theta_{12}=0.307$,
$\sin^{2}\theta_{13}=0.0218$, $\Delta m_{21}^{2}=7.53\times 10^{-5}$ eV2
Nakajima (2020); Gando  _et al._ (2013); Adey  _et al._ (2018). Allowing these
parameters to be free within their allowed ranges in this analysis does not
have a significant impact on the resulting interpretation of the long-baseline
results. In our conclusions, we discuss how additional reactor antineutrino
measurements can impact the MO determination. Finally, we include the
$\Delta\chi^{2}$ map from Ref. Abe  _et al._ (2018b) which we refer to as
“SK18” henceforth.

Results. — We perform three different analyses and compare their results.
First, we perform a joint analysis of T2K/NOvA/SK18. As we have observed
before, cf Fig. 1, without SK18, this fit results in a mild preference for the
IO ($\Delta\chi^{2}_{\rm(NO,IO)}=-2.6$) and a preference for $\delta_{\rm
CP}\approx-\pi/2$, maximal CP violation. When SK18 is included, this
preference changes to $\Delta\chi^{2}_{\rm(NO,IO)}=1.6$, and $\delta_{\rm
CP}\approx-3$ is favored. Fig. 3 presents the results of this fit in black,
where the top panel displays the one-dimensional $\Delta\chi^{2}$ as a
function of $\delta_{\rm CP}$ (marginalized over the other five oscillation
parameters) when we fix ourselves to be in the NO (solid black line) or IO
(dashed black line). The middle (bottom) panel presents two-dimensional
measurement contours of constant $\Delta\chi^{2}$ relative to the best-fit
point ($\Delta\chi^{2}=2.3$, dashed, and $4.61$, solid) of $\delta_{\rm CP}$
vs. $\sin^{2}\theta_{23}$, assuming NO (IO). The best-fit point,
$\sin^{2}\theta_{23}\approx 0.57$, $\delta_{\rm CP}\approx-3$, NO, is shown as
a star in the middle panel. We find that the combined results are consistent
with the hypothesis that CP is conserved ($\delta_{\rm CP}=0$ or $\pm\pi$) at
$\Delta\chi^{2}<1$.

Figure 3: Results of our fit of $\delta_{\rm CP}$ and $\sin^{2}\theta_{23}$.
Top: $\Delta\chi^{2}$ as a function of $\delta_{\rm CP}$ for fixed MO. Middle
(Bottom): $\delta_{\rm CP}$ versus $\sin^{2}\theta_{23}$ for the NO(IO).
Except for the top panel, all contours are $\Delta\chi^{2}=2.3$ (solid) and
$4.61$ (dashed)– black lines indicate a joint fit of T2K/NOvA/SK18, where blue
(red) indicate a fit to NOvA (T2K) alone. Corresponding stars indicate the
best-fit point of each fit. The text indicates the relative preference for
mass ordering by each fit.

The other two fits we perform are with only T2K or only NOvA. Results of these
two fits are shown in Fig. 3 for NOvA (blue) and T2K (red). Each fit results
in a small preference for NO over IO, again, as discussed cf. Fig. 1, with T2K
giving $\Delta\chi^{2}_{\rm(NO,IO)}=1.2$ and NOvA giving
$\Delta\chi^{2}_{\rm(NO,IO)}=0.15$. The red/blue stars in the middle panel of
Fig. 3 represent the best-fit points333In NO (IO), T2K prefers
$s_{23}^{2}=0.55$, $\Delta m_{31}^{2}=+2.56\times 10^{-3}$ eV2, $\delta_{\rm
CP}=-1.98$ ($0.55,-2.46\times 10^{-3}$ eV2, $-1.49$). NOvA prefers
$s_{23}^{2}=0.58$, $\Delta m_{31}^{2}=2.52\times 10^{-3}$ eV2, $\delta_{\rm
CP}=2.32$ ($0.57$, $-2.41\times 10^{-3}$ eV2, $-1.51$) for the NO (IO). of
these fits. The two NO best-fit regions are somewhat in tension, leading to a
joint T2K/NOvA fit preferring maximal CP violation, but IO. We show a version
of Fig. 3 without SK18 in our Appendix C.

Table 1: MO preference by an experiment or combination of experiments. The plus (minus) sign indicates preference for NO (IO). We also present the best-fit value of $\delta_{\rm CP}$ and the exclusion of CP conservation. Experiment(s) | $\Delta\chi^{2}_{\rm(NO,IO)}$ |  Best-fit $\delta_{\rm CP}/\pi$ | $\Delta\chi^{2}_{\rm CPC}$
---|---|---|---
T2K | $+1.2$ | $-0.6$ | 2.9
NOvA | $+0.15$ | $+0.7$ | 0.33
SK18/SK20 |  $+3.4$/$+3.2$ | $-0.7/{-}0.6$ | 0.35/0.81
T2K + NOvA | $-2.6$ | $-0.5$ | 2.8
T2K + SK18 | $+5.7$ | $-0.6$ | 3.0
NOvA + SK18 | $+3.6$ | $+0.9$ | 0.53
T2K+NOvA+SK18 | $+1.6$ | $-0.9$ | 0.31

Table 1 summarizes the MO preference by each experiment or combination of
experiments we consider, as well as the updated SK 2020 result Nakajima (2020)
– as these results are not yet published, we do not have a $\Delta\chi^{2}$
map for this to perform a complete joint fit. We comment on a prospective
T2K/NOvA/SK20 fit below.

Discussion & Conclusions. — The neutrino mass ordering remains one of the
largest mysteries in the Standard Model. Previously, experimental data seemed
to be preferring the normal mass ordering, $m_{1}<m_{2}<m_{3}$, corresponding
to the same ordering of the charged fermions of the Standard Model. However,
as we have shown, this evidence is waning given updated results from T2K and
NOvA, specifically when the two are combined in a joint fit. With SK18, a mild
preference for the normal ordering is obtained. However, preliminary updated
results from SK could reduce this preference – Ref. Nakajima (2020) (with
updated data) prefers NO at a lower strength than in Ref. Abe  _et al._
(2018b). As demonstrated in this work, the interplay between
$\sin^{2}\theta_{23}$, $\delta_{\rm CP}$ and the mass ordering is complex, and
a complete fit of T2K+NOvA+SK20 is surely warranted.

In this work, we focused on the experiments with recent updates, specifically
the long-baseline experiments T2K and NOvA. Reactor antineutrino experiments
such as Daya Bay and RENO, when combined with long-baseline disappearance
measurements can provide sensitivity to the MO through precise measurements of
the effective mass-squared splittings in the two oscillation environments
Nunokawa  _et al._ (2005); Minakata  _et al._ (2006). As shown in Ref. Esteban
_et al._ (2020), inclusion of reactor data pushes the preference further
toward NO at the final level of $\Delta\chi^{2}_{\rm(NO,IO)}\approx 7.1$
($2.7$ without SK18). While the resulting preference is in the direction of
NO, it is very intriguing that the different mechanisms of probing the MO
(long-baseline appearance vs. disappearance measurements) yield different MO
determinations. Given all of these effects, we must conclude that more data is
required before the neutrino mass ordering can be definitively determined.

The importance of this result cannot be understated. If neutrinos follow the
inverted ordering, this has far-reaching consequences. If, in addition,
neutrinos are Majorana fermions, there exists some minimum mass relevant for
neutrino-less double beta decay. If the inverted ordering is true and
neutrino-less double beta decay remains unobserved by upgraded experiments, we
can determine that neutrinos are Dirac fermions. Moreover, measurements of the
cosmic microwave background and the matter power spectrum allow us to infer
the sum of the neutrino masses. If neutrinos follow the inverted ordering,
their sum is at least ${\sim}100$ meV, while for normal ordering $60$ meV. The
lower limit for the inverted ordering is attainable by next-generation
experiments. Further, experiments that measure neutrino masses via kinematic
effects, such as KATRIN Aker  _et al._ (2019) and Project 8 Ashtari Esfahani
_et al._ (2017), could also be impacted by the mass ordering; the minimum
effective electron neutrino mass is about 50 meV for the inverted ordering as
opposed to 9 meV for normal ordering. The mass ordering may also play an
important role in the potential observation of relic neutrinos from the early
universe by the proposed PTOLEMY experiment Betti  _et al._ (2019).

It could be that statistical fluctuations in the data, moving in the same
direction in the bi-probability planes of Fig. 1, are the cause for this
vanishing preference for normal ordering. This highlights the importance of
two things. First, the accumulation of more data. As T2K and NOvA continue to
run, their statistical uncertainties will decrease, and thus will become more
robust against statistical fluctuations. Second, the need for the future
experiments that will definitively pin down the neutrino mass ordering.
Between JUNO’s long-baseline reactor antineutrino measurements, and DUNE and
T2HK’s long-baseline high-energy oscillation and atmospheric oscillation
measurements (which take different approaches to determine the ordering Ternes
_et al._ (2019)), we will be able to determine the neutrino mass ordering
absolutely.

Acknowledgments

KJK, PANM, SJP, and YFPG are supported by Fermi Research Alliance, LLC under
contract DE-AC02-07CH11359 with the U.S. Department of Energy. RZF is
supported by CNPq and FAPESP. This project has received support from the
European Unions Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement No 690575 and No 674896.

Note added: After the completion of this manuscript, it came to our attention
that similar considerations on the preference for inverted ordering of recent
T2K and NOvA data were made by João Coelho at the Neutrino 2020 conference
Slack channel #talks_lbl_nus_beams.

##  Appendix A T2K simulation details

As stated in the main text, the data sample is classified according to the
observed events in the Far Detector: muon-like (1R$\mu$) and electron-like
(1R$e$) ring events, together with $\nu_{e}\ \rm{CC}1\pi-$like (1R$e$+1$\pi$)
events in neutrino mode only. The expected number of events in each category
$\beta$ and bin $j$, $N_{\beta j}$ are computed as Esteban  _et al._ (2019)

$\displaystyle N_{\beta j}=N_{{\rm bkg},\beta}+\int_{E_{j}}^{E_{j+1}}dE_{\rm
rec}\,\int dE_{\nu}\,{\cal R}(E_{\rm
rec},E_{\nu})\,\varepsilon(E_{\nu})\,\phi(\nu_{\mu})\,P(\nu_{\mu}\to\nu_{\beta})\,\sigma_{\beta}(E_{\nu}),$
(10)

with $N_{{\rm bkg},\beta}$ the number of background events in the sample
$\beta$, ${\cal R}(E_{\rm rec},E_{\nu})$ the energy resolution function (taken
as Gaussian), $\varepsilon(E_{\nu})$ the detector efficiency,
$\phi(\nu_{\mu})$ the initial $\nu_{\mu}$ flux at the detector site,
$P(\nu_{\mu}\to\nu_{\beta})$ the oscillation probability and
$\sigma_{\beta}(E_{\nu})$ the correspondent cross section for the sample
$\beta$. Although eq. (10) is written for neutrino events, a similar one can
be inferred for antineutrinos. We consider the SK detector, located at 295 km
and $2.5^{\circ}$ off-axis from the neutrino source, with a fiducial volume of
22.5 kt. The initial fluxes from J-PARC and the cross sections are obtained
from Ref. Dennis (2015).

The test statistics is taken as Poissonian with priors on pull and oscillation
parameters,

$\displaystyle\chi^{2}=$ $\displaystyle\
2\sum_{\beta}\sum_{j=1}^{b_{\beta}}\left\\{(1+f_{\beta})N_{\beta j}-d_{\beta
j}+d_{\beta j}\log\left(\frac{d_{\beta j}}{(1+f_{\beta})N_{\beta
j}}\right)\right\\}+\sum_{\beta}\left(\frac{f_{\beta}}{\sigma_{\beta}}\right)^{2},$
(11)

where $d_{\beta j}$ is the T2K data, taken from (Dunne, 2020), in the sample
$\beta$ and reconstructed energy bin $j$; note that the number of bins
$b_{\beta}$ depends on the sample. The factors $f_{\beta}$ are pull parameters
for the normalization systematic uncertainties of the neutrino data sets, see
Tab. 2.

Table 2:  Uncertainties and number of bins the different samples considered in out T2K simulation. Sample | Uncertainty |  Number of bins $b_{\beta}$
---|---|---
1R$\mu$ $\nu-$mode | $3.0\%$ | 40
1R$\mu$ $\bar{\nu}-$mode | $4.0\%$ | 40
1R$e$ $\nu-$mode | $4.7\%$ | 25
1R$e$ \+ 1$\pi$ $\nu-$mode | $14.3\%$ | 25
1R$e$ $\bar{\nu}-$mode | $5.9\%$ | 25

##  Appendix B NOvA simulation details

Our simulation for NOvA follows the same strategy described above for T2K,
where we simulate event spectra as a function of reconstructed neutrino energy
(and including backgrounds) according to Eq. (10). While our T2K analysis
includes five channels (separating out the single-pion events in the $\nu_{e}$
appearance mode), our NOvA analysis only includes four – the two disappearance
channels (one in neutrino mode and one in antineutrino mode) and the two
appearance channels.

Given a set of oscillation parameters, we therefore can produce expected event
spectra for neutrino appearance and disappearance. After calculating the
expected event spectrum for (anti)neutrino disappearance, we bin our data to
match the binning in Ref. Himmel (2020). These binned data are compared
against the data extracted from Ref. Himmel (2020) using a Poissonian log-
likelihood function like the one given in Eq. 11 to determine the $\chi^{2}$
given this simulated expected event spectrum. As explained in Ref. Himmel
(2020), NOvA’s measurements are currently statistics-limited so we do not
incorporate any systematic uncertainties in our fit to NOvA data.

For the electron neutrino appearance channels in neutrino and antineutrino
modes, we perform a rate-only measurement, comparing the expected number of
events in each channel (given some set of input oscillation parameters) to the
measured event rates, 82 and 33 in neutrino and antineutrino modes,
respectively.

##  Appendix C Results of T2K/NOvA Fits without Super-Kamiokande

In Fig. 3 of our main text we presented the results of fits to NOvA and T2K’s
recently-updated data, as well as a combined fit to T2K, NOvA, and Super-
Kamiokande’s published result from Ref. Abe  _et al._ (2018b), using the
$\Delta\chi^{2}$ map from that publication. In this section, we repeat the
exercise of Fig. 3 with a joint T2K/NOvA fit without Super-Kamiokande. This is
shown in Fig. 4.

As discussed in the main text, the combination of T2K and NOvA prefer the
inverted mass ordering over the normal at the
$\Delta\chi^{2}_{\rm(NO,IO)}=-2.6$ level444Since we are displaying contours of
$\Delta\chi^{2}=2.6$ and $4.61$ in the center and bottom panels, no dashed
black contour appears in the middle panel where we are fixing the mass
ordering to be normal. This is because the combination of T2K and NOvA in our
analysis disfavors the NO at stronger than $\Delta\chi^{2}=2.3$.. Their
combination prefers maximal CP-violation, $\delta_{\rm CP}\approx-\pi/2$, and
the upper octant $s_{23}^{2}>1/2$. The marginalized one-dimensional
$\Delta\chi^{2}$ lines in the top panel of Fig. 4 allow us to determine T2K on
its own (with $\sin^{2}\theta_{13}$ fixed) can exclude a small interval of
$\delta_{\rm CP}\approx\pi/2$ at $>3\sigma$ CL. However, once NOvA is
included, the interval shrinks (note that near $\delta_{\rm CP}\approx\pi/2$,
the exclusion of the red solid line is higher than that of the black solid
line). According to our results, the combination of T2K and NOvA can only
exclude the hypothesis that CP is conserved ($\delta=0$ or $\delta=\pm\pi$) at
roughly the $\Delta\chi^{2}\approx 3$ level.

Figure 4: Results of our fit of the oscillation parameters $\delta_{\rm CP}$
and $\sin^{2}\theta_{23}$. In the top panel we show $\Delta\chi^{2}$ as a
function of $\delta_{\rm CP}$ for a fixed MO. In the middle (bottom) panel we
show $\delta_{\rm CP}$ versus $\sin^{2}\theta_{23}$ for the NO(IO). Except for
the top panel, all contours are $\Delta\chi^{2}=2.3$ (dashed) and
$4.61$(solid) – black lines indicate a joint fit of T2K/NOvA, where blue (red)
indicate a fit to NOvA (T2K) alone. The corresponding stars indicate the best-
fit point of each fit, and the text indicates the relative preference for mass
ordering by each fit
($\Delta\chi^{2}_{\rm(NO,IO)}\equiv\chi^{2}_{\rm(min,IO)}-\chi^{2}_{\rm(min,NO)}$).

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