#  Order preserving Quotient Lifting Properties

T. S. S. R .K. Rao Adjunct Professor, CARAMS, MAHE, Manipal, India
<EMAIL_ADDRESS>
###### Abstract.

Several properties of subspaces $J\subset X$ in a pair $(X,J)$ with quotient
lifting properties of different types are derived, giving emphasis to order-
preserving properties for order unit spaces and spaces of affine continuous
functions on Choquet simplexes.

###### 2000 Mathematics Subject Classification:

Primary 46B40, 47L05, 46G10, 46B25
Keywords and phrases. Quotient lifting properties; Compact and order
preserving Quotient lifting properties; Choquet simplexes, Banach lattices

##  1\. Introduction

The Quotient Lifting Property ($QLP$) for pairs of Banach spaces deals with
the existence of norm preserving lifts of operators.

######  Definition 1.

( [1]) Let $X$ be a real Banach space and $J$ be a closed subspace of $X$. The
pair $(X,J)$ has the QLP if and only if for every Banach space $Y$ and every
bounded operator $S:Y\to X/J$ there exists a bounded operator $T$ from $Y$ to
$X$ lifting $S$ while preserving the norm, i.e. $\|T\|=\|S\|$ and $\pi\circ
T=S$, where $\pi:X\rightarrow X/J$ is the quotient map.

If $(X,J)$ has the QLP then $J$ is the kernel of a contractive projection and
hence a complemented subspace of $X$, see [1].

Motivated by this, we introduce a related and weaker property involving the
lifting of non-negative bounded linear operators to have non-negative, norm-
preserving lifting in the class of order unit spaces (see [2] and [6], Chapter
1) and compact operators to compact operators with the same norm. We also give
conditions under which this property is preserved by subspaces and quotient
spaces.

Let $K$ be a compact convex set in a locally convex topological vector space
and let $A(K)$ denote the space of real-valued affine continuous functions on
$K$, equipped with the supremum norm and point-wise order. For a closed split
face $F\subset K$, if $J=\\{a\in A(K):a(F)=0\\}$, we show that for the pair
$(A(K),J)$, the order preserving, linear quotient lifting property is
equivalent to $F$ being the range of an affine continuous map on $K$, which is
the identity on $F$. We give examples of subspaces $J$ of $A(K)$, for a
Choquet simplex $K$, so that $(A(K),J)$ has the compact extension property ,
using vector-valued Choquet theory. Our approach follows the popular route of
converting the lifting problem to a vector-valued extension problem.

The general question about how lifting properties behave with respect to
injective tensor products, among various categories of Banach spaces, has not
received much attention. Unlike the QLP, for the compact quotient lifting
property (CQLP), the subspace may not be complemented, making the passage to
tensor products more difficult.

Let $\Omega$ be a compact Hausdorff space and let $C(\Omega)$ denote the space
of continuous functions on $\Omega$. We conclude the paper with an application
of a remarkable result of Aviles and Troyanski relating the existence of
uniformly rotund in every direction (URED) renorming of $C(\Omega)$ to that of
a strictly positive measure supported on $\Omega$, to lifting lattice
homomorphism preserving operators to $C(\Omega)$.

##  2\. Various forms of lifting properties

The results of this section show an interdependence between geometric
properties of a Banach spaces and the lifting properties. We also give results
showing how these properties behave with respect to order ideals or lattice
ideals, in specific function spaces.

Let $J\subset X$ be a closed subspace, we say that $(X,J)$ has the compact
lifting property (CQLP), if for all Banach spaces $Z$ and compact linear
operators, $T:Z\rightarrow X/J$, there is a compact linear operator
$S:Z\rightarrow X$ such that $\pi\circ S=T$ and $\|T\|=\|S\|$.

Clearly if $(X,J)$ has the QLP, since the lifting is via composition with the
associated projection $P$, QLP implies the CQLP. More precisely, we recall
from [1] that $P$ is the projection defined by the composition of
$\pi:X\rightarrow X/J$ followed by $\tilde{Id}$, which is the norm-preserving
lift of the identity map, $I:X/J\rightarrow X/J$. Also if $J$ is of finite
codimension, then the CQLP implies the QLP.

######  Example 2.

Let $\Omega$ be a compact Hausdorff space and let $C(\Omega,X)$ be the space
of $X$-valued continuous functions, equipped with the supremum norm. We recall
the identification of the space of compact operators
${\mathcal{K}}(X,C(\Omega))$ as the space $C(\Omega,X^{\ast})$ via the
embedding $T\rightarrow T^{\ast}\circ\delta$ from $\Omega$ into $X^{\ast}$ (
$\delta:\Omega\rightarrow C(\Omega)^{\ast}$ is the evaluation map) is an onto
isometry. For a closed set $E\subset\Omega$, if $J=\\{f\in
C(\Omega):f(E)=0\\}$, note that $C(\Omega)/J$ can be identified via the
‘restriction’ map with $C(E)$, and hence $\mathcal{K}(X,C(\Omega)/J)$ is
identified with $C(E,X^{\ast})$, an application of Dugundji’s extension
theorem, implies that $(C(\Omega),J)$ has the CQLP.

Given an infinite discrete set $\Gamma$, using the identifications of
$\ell^{\infty}(\Gamma)$ with $C(\beta(\Gamma))$ and $c_{0}(\Gamma)$ with
$\\{f\in C(\beta(\Gamma)):f(\beta(\Gamma)-\Gamma)=0\\}$, we have that
$(\ell^{\infty}(\Gamma),c_{0}(\Gamma))$ has the CQLP, but not the QLP as
$c_{0}(\Gamma)$ is not even a complemented subspace of
$\ell^{\infty}(\Gamma)$.

In the next proposition we give an application of the CQLP.

######  Proposition 3.

If $(X,J)$ has the CQLP, let $\\{\pi(x_{n})\\}_{n\geq 1}$ be a relatively
compact sequence in $X/J$. Then there is a relatively compact sequence
$\\{z_{n}\\}_{n\geq 1}\subset X$ having the same bound as
$\\{\pi(x_{n})\\}_{n\geq 1}$ such that $\pi(z_{n})=\pi(x_{n})$ for all $n$.

###### Proof.

It is easy to see that the canonical bounded linear map $T:\ell^{1}\rightarrow
X/J$ defined by $T(e_{i})=\pi(x_{i})$ (where $e_{i}$ is the canonical basis in
$\ell^{1}$) is a compact operator. Now by hypothesis, we have
$S:\ell^{1}\rightarrow X$ a compact operator with $\pi\circ S=T$,
$\|S\|=\|T\|$. It is easy to see that $\\{z_{i}=S(e_{i})\\}_{i\geq 1}$ is the
required sequence in $X$. ∎

Our next two results deal with the stability of the property CQLP. In the
following proposition we use the canonical embedding of a space in its bidual.
In particular, $(X/J)^{\ast\ast}=X^{\ast\ast}/J^{\bot\bot}$ and the quotient
map, $\pi:X^{\ast\ast}\rightarrow X^{\ast\ast}/J^{\bot\bot}$ is the biadjoint
of the corresponding quotient map from $X\rightarrow X/J$, still denoted by
$\pi$. If $J\subset X$ is a reflexive space, then $J^{\bot\bot}=J$. We recall
that $J$ is said to be factor reflexive space, if $X/J$ is a reflexive space.

######  Proposition 4.

  1. (1)

Let $J\subset Y\subset X$. Suppose $(X,J)$ has the CQLP. Then $(Y,J)$ has the
CQLP.

  2. (2)

If $J$ is factor reflexive and $(X,J)$ has the CQLP, then
$(X^{\ast\ast},J^{\bot\bot})$ has the CQLP.

  3. (3)

If $J$ is reflexive, then $(X^{\ast\ast},J)$ has the CQLP implies $(X,J)$ has
the CQLP.

###### Proof.

Let $T:Z\rightarrow Y/J\subset X/J$ be a compact operator. By hypothesis,
there is a compact operator $S:Z\rightarrow X$ such that $\pi\circ S=T$ and
$\|S\|=\|T\|$. For any $z\in Z$, $\pi(S(z))=T(z)=\pi(y)$, for some $y\in Y$.
Hence $S(z)-y\in J$, so that $S(z)\in Y$. The conclusion follows.

Now suppose $J$ is factor reflexive and $(X,J)$ has the CQLP. Let
$T:Z\rightarrow X^{\ast\ast}/J^{\bot\bot}=X/J$ be a compact operator. By
hypothesis, there is a compact operator, $S:Z\rightarrow X$ such that
$\pi\circ S=T$ and $\|S\|=\|T\|$. The conclusion in (2) follows in view of our
remark about quotient maps.

When $J$ is reflexive, in view of the inclusions $J\subset X\subset
X^{\ast\ast}$, the conclusion in (3) follows from (1). ∎

In the following proposition, we use the notation $\pi$ for several quotient
spaces, with appropriate interpretation.

######  Proposition 5.

Let $Z\subset Y\subset X$ be Banach spaces. Suppose $Y$ is of finite
codimension and $(X,Y)$ has the CQLP. Then $(X/Z,Y/Z)$ has the CQLP.

###### Proof.

Since $I:X/Y\rightarrow X/Y$ is a compact operator, by hypothesis, there is a
compact operator $I^{\prime}:X/Y\rightarrow X$ such that $\pi\circ
I^{\prime}=I$ and $\|I^{\prime}\|=1$. Now $P:X\rightarrow X$ defined by
$P(x)=I^{\prime}(\pi(x))$ is a contractive projection with $ker(P)=Y$. Define
$P^{\prime}:X/Z\rightarrow X/Z$ by $P^{\prime}(\pi(x))=\pi(P(x))$. If
$\pi(x_{1})=\pi(x_{2})$, then $x_{1}-x_{2}\in Z\subset Y$, so
$P(x_{1})=P(x_{2})$. Thus $P^{\prime}$ is well defined. It is routine to check
that it is a contractive projection. If $y\in Y$, then
$P^{\prime}(\pi(y))=\pi(0)$, so $Y/Z\subset ker(P^{\prime})$. Conversely if
$P^{\prime}(\pi(x))=0$, then $\pi(x)=\pi(x-P(x))$, as $(x-P(x))\in Y$, we get
$\pi(x)\in Y/Z$. Hence $ker(P^{\prime})=Y/Z$. Now it follows from our remarks
at the beginning of Section 2, that $(X/Z,Y/Z)$ has the CQLP. ∎

Let $X$ be an order unit space and let $J\subset X$ be a proper order ideal.
Now for the order unit space $X/J$, we can consider the pair $(X,J)$ having
the order preserving quotient lifting property (OQLP), if for any order unit
space $Z$ and for any linear, bounded, order unit preserving operator
$T:Z\rightarrow X/J$, there is a linear bounded, order unit preserving
$S:Z\rightarrow X$ such that $\pi\circ S=T$ and $\|T\|=\|S\|$.

When the pair $(X,J)$ has an appropriate lattice structure, a similar lifting
property can be defined. We will be implicitly using this without giving a
formal definition. A version of Propositions 4 and 5 can be contemplated, when
$X$ is also an ordered ideal in its bidual or $X/Y$ is an order unit space.

In what follows we will be using the notations and terminology of convexity
theory and order unit spaces from Chapters I and II of Alfsen’s monograph. See
also Section 18 of [6]. We illustrate situations where apart from quotient
lifting, one can also achieve ‘lifts’ that preserve the order structure.

Let $K$ be a compact convex set in a locally convex topological vector space.
Let $F\subset K$ be a closed, extreme convex set (face) and let
$\phi:K\rightarrow F$ be an affine continuous map which is the identity on
$F$. It is easy to see that $P:A(K)\rightarrow A(K)$ defined by
$P(a)=a\circ\phi$ is a positive, contractive linear projection such that
$P(1)=1$, $ker(P)=\\{a\in A(K):a(F)=0\\}$ and it is an order ideal in $A(K)$.
It is also easy to see that the canonical map, $\Phi:A(K)/ker(P)\rightarrow
A(K)$ defined by $\Phi(\pi(a))=P(a)$ is an order-preserving linear
contraction. Now for any order unit space $E$, with order unit $e$, if
$T:E\rightarrow A(K)/ker(P)$ is a linear, continuous, order-preserving map
with $T(e)=\pi(1)$, then it is easy to see that $T^{\prime}=\Phi\circ T$ is an
order-preserving, norm-preserving quotient lifting.

In the following theorem we preserve the notation from above.

######  Theorem 6.

Let $K$ be a compact convex set and let $F\subset K$ be a closed split face.
Suppose for all order unit spaces $E$ and for all non-negative, bounded linear
operators $T:E\rightarrow A(K)/J$, there is a non-negative bounded linear
lifting $T^{\prime}:E\rightarrow A(K)$ which is also norm preserving. There
exists an affine continuous map $\phi:K\rightarrow F$ such that $\phi$ is the
identity on $F$. Hence $(A(K),J)$ has the QLP.

###### Proof.

Since $F$ is a closed split face, we use the identification of $A(K)/J$ with
$A(F)$ via the map, $\pi(a)\rightarrow a|F$ both as an isometry and order-
preserving map. Now the identity map $I:A(K)/J\rightarrow A(K)/J$ is a linear
order-preserving map and hence by hypothesis, admits a norm-preserving, order-
preserving lifting $I^{\prime}:A(F)\rightarrow A(K)$.

We next recall that $K$ can be identified with the state space $\\{\Lambda\in
A(K)^{\ast}:\|\Lambda\|=\Lambda(1)=1\\}$, equipped with the weak∗-topology,
via the affine homeomorphism $k\rightarrow\delta(k)$ where $\delta(k)$ is the
evaluation map. Now since $A(K)/J=A(F)$ is an order unit space, applying the
hypothesis to $I:A(F)\rightarrow A(K)/J$, let $I^{\prime}:A(F)\rightarrow
A(K)$ be the order unit , norm preserving quotient lift. It is easy to see
that using implicitly the canonical embeddings noted above,
$(I^{\prime})^{\ast}|K\rightarrow F$ is the required continuous, affine map.
The quotient lifting property implies that this is the identity on $F$. ∎

We next prove a vector-valued affine continuous function version of Example 2.
This requires some knowledge of Choquet simplexes from Alfsen’s monograph and
a vector-valued integral representation analogue of Singer’s theorem (see
[4]), due to Saab [9]. See also [4] for general theory of vector measures and
tensor product theory. For a Choquet simplex $K$, we first establish the
canonical isometry between $A(K,X^{\ast})$, the injective tensor product space
$A(K)\otimes_{\epsilon}X^{\ast}$ and ${\mathcal{K}}(X,A(K))$.

Let $K$ be a compact Choquet simplex and let $F\subset K$ be a metrizable
closed face. Since $K$ is a simplex, $F$ is a split face and also a Choquet
simplex (see Chapters II and III of [2]). Note that the set of extreme points
$\partial_{e}F=\partial_{e}K\cap F$. Since $F$ is metrizable, $\partial_{e}F$
is a $G_{\delta}$-subset of $F$ and as $F$ is closed, $\partial_{e}F$, is a
Borel set in $K$.

Since $K$ is a simplex, $A(K)$ has the metric approximation property (as
$A(K)^{\ast}$ is isometric to a $L^{1}(\mu)$ space, see [6] Section 18) ,
consequently for any Banach space $X$, ${\mathcal{K}}(X,A(K))$ can be
canonically identified with the injective tensor product space,
$A(K)\otimes_{\epsilon}X^{\ast}$.

Let $\delta:K\rightarrow A(K)^{\ast}$ be the canonical evaluation map, which
is an affine homeomorphism onto $\delta(K)$ when the latter is equipped with
the weak∗-topology. Now as in the case of vector-valued continuous functions,
for $T\in{\mathcal{K}}(X,A(K))$, $T^{\ast}\circ\delta:K\rightarrow X^{\ast}\in
A(K,X^{\ast})$ and conversely for any $a\in A(K,X^{\ast})$, $T:X\rightarrow
A(K)$ defined by $T(x)=x\circ a$ for $x\in X$ is a compact operator. It is
easy to see that this association is an onto isometry between
${\mathcal{K}}(X,A(K))$ and $A(K,X^{\ast})$. In particular we have identified
$A(K)\otimes_{\epsilon}X^{\ast}$ and $A(K,X^{\ast})$.

In what follows these identifications will be used implicitly. For a Banach
space $X$ we recall that $X^{\ast}_{1}$ denotes the closed unit ball with the
weak∗-topology. We recall that for $k\in\partial_{e}K$ and
$e^{\ast\ast}\in\partial_{e}X^{\ast\ast}_{1}$, $\delta(k)e^{\ast\ast}$ denotes
the extreme functional or boundary measure (in the sense in [9]),
$(\delta(k)e^{\ast\ast})(a)=e^{\ast\ast}(a(k))$ for $a\in A(K,X^{\ast})$.

######  Theorem 7.

Let $K$ be a Choquet simplex and let $F\subset K$ be a closed metrizable face.
Let $J=\\{a\in A(K):a(F)=0\\}$. Then $(A(K),J)$ has the CQLP.

###### Proof.

Let $X$ be any Banach space. We first note that since $A(K)/J=A(F)$ and $F$ is
a simplex, ${\mathcal{K}}(X,A(F))$ has the canonical identifications as
$A(F,X^{\ast})$ and $A(F)\otimes_{\epsilon}X^{\ast}$. Our theorem will be
proved once we show that any $a\in A(F,X^{\ast})$ has a norm-preserving
extension in $A(K,X^{\ast})$.

To do this we show that if $J^{\prime}=\\{a\in A(K,X^{\ast}):a(F)=0\\}$, then
the map $\Phi:A(K,X^{\ast})/J^{\prime}\rightarrow A(F,X^{\ast})$ defined by
$\Phi(\pi(a))=a|F$ is a surjective isometry and for any $b\in A(F,X^{\ast})$,
if $\Phi(\pi(a))=b$, then there is a $c\in J^{\prime}$ such that
$\|a-c\|=d(a,J)=\|a|F\|$ (i.e., $J^{\prime}$ is a proximinal subspace of
$A(K,X^{\ast})$). Thus $a-c$ is a desired norm-preserving extension.

It is easy to see that $\Phi$ is a one-one, linear contraction. Fix $\pi(a)$.
Let $\Lambda\in\partial_{e}(J^{\prime})^{\bot}_{1}$ be such that
$\Lambda(a)=d(a,J^{\prime})$. We next use the metrizability of $F$ and vector-
valued Choquet theorem of Saab, to identify $J^{\prime}$ as the space of
$X^{\ast\ast}$-valued boundary measures supported on $\partial_{e}F$. Once
this is done, $\Lambda=\delta(k)x^{\ast\ast}$ for some $k\in F$ and
$x^{\ast\ast}\in\partial_{e}X^{\ast\ast}_{1}$. Thus
$\Lambda(a)=d(a,J^{\prime})=x^{\ast\ast}(a(k))\leq\|a(k)\|\leq\|a|F\|$.
Therefore $\Phi$ is an into isometry.

Let $\tau\in(J^{\prime})^{\bot}$ be a boundary measure. Fix $x^{\ast}\in
X^{\ast}$, for any $a\in A(K)$, $a(F)=0$, since $ax^{\ast}\in J$, for the
scalar boundary measure $x^{\ast}\circ\tau$,
$\int_{K}ad(x^{\ast}\circ\tau)=0$. Now by the scalar theory, by the
metrizability of $F$, $x^{\ast}\circ\tau$ is supported on $\partial_{e}F$.
Therefore, $\tau$ is supported on $F$. The other inclusion is easy to see.

Now let $\lambda$ be a boundary measure in $A(F,X^{\ast})^{\ast}_{1}$
vanishing on the range of $\Phi$. Since $F$ is metrizable, we again have,
$\lambda$ is supported on $\partial_{e}F$. Now extend $\lambda$ to a regular
$X^{\ast\ast}$-valued Borel measure on $K$. Since $\partial_{e}F$ is a Borel
set, we see that $\lambda$ is a boundary measure in
$A(K,X^{\ast})^{\ast}_{1}$. Also for any $a\in A(K)$, $x^{\ast}\in X$,
$\int_{\partial_{e}F}ad(x^{\ast}\circ\lambda)=0$. Since $K$ is a simplex, we
again have $\lambda=0$. Thus $\Phi$ is onto.

Let $M(F^{c},X^{\ast\ast})$ denote the space of $X^{\ast\ast}$-valued boundary
measures on the Borel $\sigma$-field, supported by the open convex set
$F^{c}$. As in the scalar-valued case, by the integral representation theorem,
it is easy to see that
$A(K,X^{\ast})^{\ast}=(J^{\prime})^{\bot}\bigoplus_{1}M(F^{c},X^{\ast\ast})$
is an $\ell^{1}$-direct sum. Therefore $J^{\prime}$ is a $M$-ideal in the
sense defined in Chapter I of [5] and hence is a proximinal subspace of
$A(K,X)$ by Proposition II.1.1 of [5]. (See also [8] for a simpler proof of
this proposition). This completes the proof.

∎

Our next corollary is an application of the proof of the above theorem to
$A(K,X)$, to describe the state space (always considered with the
weak∗-topology) of certain vectors in $A(K,X)$.

######  Corollary 8.

Let $K$, $F$ be as in the above theorem and let $J^{\prime}=\\{a\in
A(K,X):a(F)=0\\}$. Let $0\neq a\in A(K,E)$ be such that
$d(a,J^{\prime})<\|a\|$. Then the state space, $S_{a}=\\{\Lambda\in
A(K,X)^{\ast}_{1}:\Lambda(a)=\|a\|\\}=\overline{CO}\\{\delta(k)e^{\ast}:(\delta(k)e^{\ast})(a)=e^{\ast}(a(k))=\|a\|~{},~{}k\in\partial_{e}F^{c}~{},e^{\ast}\in\partial_{e}X^{\ast}_{1}\\}$.
If $\tau\in\partial_{e}S_{a}$, then $\tau=\delta(k)e^{\ast}$ for some
$e^{\ast}\in\partial_{e}E^{\ast}_{1}$ and an accumulation point $k$ of
$F^{c}$.

###### Proof.

Clearly the set on the R. H. S, when non-empty, is contained in $S_{a}$. Since
$S_{a}$ is a face, if $\lambda\in\partial_{e}S_{a}$, then
$\lambda\in\partial_{e}A(K,X^{\ast})_{1}$. In view of the
$\ell^{1}$-decomposition, similar to the one, given at the end of the above
Theorem, applied now to $A(K,X)^{\ast}$, as $d(a,J)<\|a\|$, we get
$\lambda\in\partial_{e}M(F^{c},X^{\ast})_{1}$. As before, it is easy to see
that $\lambda=\delta(k)e^{\ast}$ for some $k\notin F$ and
$e^{\ast}\in\partial_{e}X^{\ast}_{1}$ (in particular, the set in the R. H. S
in the formula is non-empty). Thus by an application of the Krein-Milman
theorem, the first conclusion follows.

If $\tau\in\partial_{e}S_{a}$, then by Milman’s converse of the Krein-Milman
theorem, and using the compactness of $K$ and $X^{\ast}_{1}$, there are nets
$\\{k_{\alpha}\\}_{\alpha\in\Delta}\subset F^{c}$ and
$\\{e^{\ast}_{\alpha}\\}_{\alpha\in\Delta}\subset\partial_{e}X^{\ast}_{1}$,
$\delta(k_{\alpha})e^{\ast}_{\alpha}\rightarrow\tau$ in the weak∗-topology,
which we may and do assume are also such that $k_{\alpha}\rightarrow k$ and
$e^{\ast}_{\alpha}\rightarrow e^{\ast}$, for some $k\in K$ and $e^{\ast}\in
X^{\ast}_{1}$.

Thus $\tau=\delta(k)e^{\ast}$. As $\|\tau\|=1$, we have, $\|e^{\ast}\|=1$.
Since $\tau\in\partial_{e}A(K,X)^{\ast}_{1}$, it is easy to see that
$e^{\ast}\in\partial_{e}X^{\ast}_{1}$. ∎

We do not know how to prove compact and order-preserving lifting theorems in
the category of order unit spaces or lattices $X$, and $A(K,X^{\ast})$ and for
subspaces of the type $J$ as above.

A particularly interesting situation occurs by taking $X=A(K_{2})$ for a
Choquet simplex $K_{2}$. In the following theorem we again get a vector-
valued, this time non-negative extension theorem. We only present an outline
proof of this result, omitting the technical details.

######  Theorem 9.

Let $K_{1},K_{2}$ be simplexes . Let $F\subset K_{1}$ be a closed face. Let
$f\in A(F,A(K_{2}))$ be a non-negative, affine continuous function. $f$ has a
non-negative, norm-preserving extension in $A(K_{1},A(K_{2}))$.

###### Proof.

It can be shown using the analysis carried out above and the tensor product
theory of simplexes developed by I. Namioka and R. R. Phelps, in [7], that
$A(K_{1},A(K_{2}))$ is isometrically and in order preserving way identified
with $A(K^{\prime})$ for a unique (upto affine homeomorphisms) simplex
$K^{\prime}$( formally called the tensor product of $K_{1},K_{2}$). Now if $F$
is a closed face of $K_{1}$ then there is a closed face $G$ of $K^{\prime}$,
which corresponds to the tensor product of the simplexes $F,K_{2}$. Thus the
vector-valued extension theorem is reduced to extending a non-negative affine
continuous function on $G$ to a non-negative affine continuous function on
$K^{\prime}$ with the same norm, which in the scalar case was done in Chapter
3 of [2]. ∎

We next give another application of lifting lattice homomorphisms to a
geometric notion.

Given a compact set $\Omega$, a positive regular Borel measure $\mu$ is called
strictly positive, if $\mu(U)>0$, for any non-empty open set $U\subset\Omega$.
For example, if $\Omega$ is also separable, with a dense set
$\\{\omega_{n}\\}_{n\geq 1}$, then
$\mu=\sum_{1}^{\infty}\frac{1}{2^{n}}\delta(\omega_{n}))$ is a strictly
positive regular measure. As before, we denote by $\delta(w)$ the point
evaluation functional at $w$ or the Dirac measure concentrated in $w$.

We also recall the definition of uniform rotundity in every direction (URED).

######  Definition 10.

A norm on a space $X$ is said to be URED if whenever $\\{x_{n}\\}_{n\geq 1}$
and $\\{y_{n}\\}_{n\geq 1}$ are two sequences of unit vectors satisfying the
the following two conditions:

  * •

$\lim_{n}\|\frac{x_{n}+y_{n}}{2}\|=1;$

  * •

there exists $z\in X$ and scalar $r_{n}$ such that $x_{n}-y_{n}=r_{n}z,$ for
all $n\in\mathbb{N},$

then $lim_{n}\|x_{n}-y_{n}\|=0.$

If the second hypotheses on the definition is removed then we have the normal
uniform rotundity. The main theorem from [3] implies that $C(\Omega)$
admitting an equivalent URED norm is equivalent to $\Omega$ to supporting a
strictly positive measure.

######  Theorem 11.

Let $\Omega$ be a compact space such that $C(\Omega)$ admits an equivalent
URED norm. Let $E\subset\Omega$ be a closed set and let $J=\\{f\in
C(\Omega):f(E)=0\\}$. Suppose for every Banach lattice $X$, every bounded
linear lattice homomorphism, $T:X\rightarrow C(\Omega)/J$ has a norm-
preserving linear lifting $T^{\prime}$, which is also a lattice homomorphism.
Then $C(\Omega)/J$ has an equivalent URED norm.

###### Proof.

As before we use the canonical identification of the space $C(\Omega)/J$ as
$C(E)$. Hence by hypothesis, the identity map $I:C(E)\rightarrow
C(\Omega)/J=C(E)$ admits a linear norm-preserving lifting
$I^{\prime}:C(E)\rightarrow C(\Omega)$ which is also a lattice homomorphism.
Now by the structure of scalar-valued lattice homomorphisms on continuous
function spaces, if $\phi:\Omega\rightarrow C(\Omega)^{\ast}_{1}$ is the
canonical embedding, then $(I^{\prime})^{\ast}\circ\phi:\Omega\rightarrow E$
is a continuous map which is the identity on $E$.

Now by the main result of [3], the existence of an equivalent URED norm
implies that there is a strictly positive measure $\mu$ supported on $\Omega$.
Now $\nu=\mu\circ\phi^{-1}$ is a strictly positive measure supported on $E$.
Therefore by the theorem from [3] again $C(E)=C(\Omega)/J$ admits an
equivalent URED norm. ∎

######  Remark 12.

As before the lattice lifting assumption implies the existence of a linear
lattice, contractive projection $P:C(\Omega)\rightarrow C(\Omega)$ such that
$ker(P)=J$.

The author thanks Professors F. Botelho and R. J. Fleming for providing the
initial impetus for this work. Thanks are also due to the Editorial team for
the prompt handling of this article during the pandemic. The author’s first
article on Choquet simplexes appeared in 1982 and over four decades they
continue to provide nice illustrations for Banach space theoretic concepts.

## References

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# Detecting anthropogenic cloud perturbations with deep learning

Duncan Watson-Parris  Samuel Sutherland  Matthew Christensen  Anthony Caterini
Dino Sejdinovic  Philip Stier

###### Abstract

One of the most pressing questions in climate science is that of the effect of
anthropogenic111A term commonly used in climate science to mean caused by
human activity. aerosol on the Earth’s energy balance. Aerosols provide the
‘seeds’ on which cloud droplets form, and changes in the amount of aerosol
available to a cloud can change its brightness and other physical properties
such as optical thickness and spatial extent. Clouds play a critical role in
moderating global temperatures and small perturbations can lead to significant
amounts of cooling or warming. Uncertainty in this effect is so large it is
not currently known if it is negligible, or provides a large enough cooling to
largely negate present-day warming by CO2. This work uses deep convolutional
neural networks to look for two particular perturbations in clouds due to
anthropogenic aerosol and assess their properties and prevalence, providing
valuable insights into their climatic effects.

Deep Learning, ICML, Climate

##  1 Introduction

The planetary energy balance (between incoming and outgoing radiation), and
hence global temperature, is very sensitive to the properties and distribution
of clouds in the atmosphere. In turn, the properties of clouds depend on the
availability of condensation nuclei in the form of aerosol (Lohmann et al.,
2016). An increase in aerosol particles, due to human activity for example,
generally enhances the number of small cloud droplets for a given amount of
liquid water. Smaller droplets are more reflective and so increase the albedo
of the cloud (Twomey, 1974). These effects are difficult to observe though
because they are small compared to the other main drivers of cloud formation,
such as moisture convergence and the availability of condensable water. These
changes also affect the availability of aerosol, and clouds themselves change,
remove, and even produce aerosol, creating complex feedbacks which are hard to
disentangle.

One approach to tackle this problem is by looking for strong local aerosol
perturbations in otherwise clean environments, ideally with slowly varying
cloud properties. Ship tracks –- tracks of cloud made brighter than their
surroundings by ship emissions, provide ideal cases (Christensen & Stephens,
2011). Finding and analysing these tracks is, however, a laborious and time-
consuming process, and hence no global long-term databases of these important
phenomena exist.

Another, even less well understood, process by which changes in aerosol amount
can change cloud properties is the so called lifetime effect. Cloud droplets
will start to precipitate once they reach a certain critical diameter, usually
around 14$\mu$m. All other things being equal, a smaller distribution of cloud
droplets can delay the onset of precipitation in a cloud, potentially
enhancing its lifetime and fractional area (Albrecht, 1989). This acts to
increase the albedo of the cloud further, potentially creating a large cooling
effect (relative to the unperturbed cloud) (Boucher et al., 2013).

Large continuous decks of stratocumulus (Sc) clouds occur in the cold
upwelling regions of the major oceans and play a crucial role in the global
energy balance because of their size and persistence. However, an important
phenomena occurs where a large region of the cloud deck dissipates through the
onset of precipitation and leaves open regions, so called Pockets of Open
Cells (POCs) (Stevens et al., 2005). It has been hypothesized that these
occurrences could be affected by anthropogenic activity, specifically through
aerosol perturbations delaying or even inhibiting their onset and causing a
large cooling effect through the mechanisms described above (Rosenfeld et al.,
2006). Hence, POCs could have important implications for climate change. Again
however, besides from a handfull of small-scale case studies, no database of
POC occurrence exists.

Here we present a method of automatically detecting ship tracks and POCs in
140 TB of high resolution satellite imagery using deep convolutional neural
networks. This model is then run over many years of satellite data producing a
unique database of observations that allows detailed studies of the
probability of their occurrence in different regions and physical conditions.
These implications will feed back into regional and global assessments of
their effect on global cloud forcing.

The following section will outline the data used and methodology developed for
the detection of these phenomena, before moving on to a presentation of the
main findings so far in Section 3. We conclude with a discussion of the
implications of our work and an outlook in Section 4.

##  2 Methodology

###  2.1 Data

All of the data used in the methods outlined below were obtained from the
Moderate Resolution Imaging Spectrometer (MODIS) instruments on the NASA Aqua
and Terra (MODIS Science Team, ) satellites. The Level 1B data sets were used
which provide calibrated and geolocated radiances at-aperture for all 36 MODIS
spectral bands at 1km resolution. Due to the different scales and properties
of the two different phenomena, however, the data were prepared independently.
The models used are also slightly different in each case, though in the future
we hope to combine them into a single model.

For the ship track detection dataset, the original imager reflectances from
Channels 1, 20 and 32 (corresponding to wavelengths of 645nm, 3.75$\mu$m and
12$\mu$m respectively) were combined into a three-channel false-color
composite. This composite was designed to provide information in the visible
(towards the middle of the solar spectrum), the near-infrared (which provides
information about the cloud droplet size) and the infra-red (which allows
discrimination of cloud liquid and ice). The original 1350x2030 pixel images
were split into 12 440x440 pixel padded images to reduce the size of image
used in training, while maintaining the full 1km resolution. The training data
was provided in the form of 4,500 hand-logged tracks (Segrin et al., 2007;
Toll et al., 2017; Christensen & Stephens, 2012) which were interpolated and
converted into pixel masks for use in training the model. An example image and
the corresponding hand-logged data is shown in Fig. 1b.

For the POC detection dataset, the micro-physical properties of the cloud were
less important and a true-color RGB composite was used from channels 1, 4 and
3 (corresponding to wavelengths of 645nm, 555nm and 469nm respectively). We
use SatPy (Raspaud et al., 2018) to prepare the composite images from the raw
HDF4 datafiles. Due to the relatively large size of the features and to speed
up training the images were rescaled from 1350x2030 pixels to 648x1296 and
then split in two to create images 648x648 in size, which were then further
rescaled to 224x224 to match ResNet-152. The training dataset was created by
hand-logging 1029 images which contained 715 POCs (as shown in Fig. 1a).

###  2.2 Model

Both of the models used are based on ResUnet architecture which has shown good
performance in remote observation settings (Zhang et al., 2017).

The ship-track detection model currently uses the ResUnet model with a binary
cross-entropy loss function and train it over 100 epochs. We use the Adam
optimization (Kingma & Ba, 2014) with a step-down in the learning rate from
1e-5 to a minimum of 5e-7 in fractional steps of 0.2 when the loss plateaus.
While this model shows some qualitative skill, as shown in Fig. 1b, at the
time of writing training and development was still ongoing.

(a) (b)

Figure 1: From left to right: An example input image, the hand logged
validation mask, and the inferred mask from the model for both the POC model
(a) and the ship track model (b). Note that not all the ship tracks are
included in the hand logged mask due to human error and/or the selection
criteria used.

The POC detection model uses a modified ResNet-152 whose dense layers have
been removed and replaced by three up-sampling blocks based on the second half
of the ResUnet model used for the ship-track detection. The ResNet-152 portion
of the model had been pre-trained on ImageNet, since this gives strong texture
recognition, which is important in this case as the different cloud structures
give very different textures. The upsampling blocks were trained using the
Adam optimization algorithm and step-down learning rate from 0.001 with
fractional steps of 0.2. The loss function used was the DICE coefficient, as
it is robust to class imbalance and gave very good results in testing. The
final masks were refined using a reduced ResUnet model that was trained in the
same way as the ResNet-152 model. The model obtains a precision score of 0.73
on the test set, but visually performs extremely well. 8,491 POCs were found
in the 25,582 files on which the model was run, creating the largest dataset
of this phenomenon to date.

##  3 POC Results and Analysis

By applying the POC detection model to all of the MODIS images which intersect
the main Sc regions off the coast of California, Peru and Namibia we are able
to build a large database of the temporal and spatial distribution of POCs and
their properties. Here we breifly summarise their main characteristics.

Figure 2: The density of POCs in each of the main Stratocumulus (Sc) regions
is shown normalised by the number of images used for inference and the
climatological fraction of Sc.

Figure 2 shows the density of POCs in each of the main Sc regions, normalized
by the number of images used and the climatological fraction of Sc (Schiffer &
Rossow, 1983). Both the Californian and Namibian POCs show highest densities
towards the edge of the cloud decks, whereas the Peruvian POCs are more evenly
spread across the deck. The Peruvian Sc region also shows the highest number
of POCs overall. The temporal distribution of POCs shows a strong seasonal
cycle and peaks in the local winter, although POCs in the Californian Sc deck
also have a peak in northern hemisphere summer (not shown).

Figure 3: The mean cloud optical thickness as a function of distance from the
POC boundary. Shading indicates the standard deviation in the mean over each
distance bin. Each color represents a region.

A number of retrievals are performed on the raw MODIS spectral radiances to
determine cloud properties at each pixel. By applying the inferred POC masks
to the retrieved MODIS cloud properties (MODIS Atmosphere Science Team, 2015)
we are able to build statistics about the POCs and their surrounding
environment. Using OpenCV (Bradski, 2000) to extract regions of fixed distance
from each POC we can plot the average properties as a function of distance
from all of the detected POCs. Figure 3 shows the average retrieved cloud
optical thickness for all of the POCs binned as a function of distance from
the POC boundary. The increase in optical thickness inside the POC is clearly
evident and due to the reduction in liquid water content through
precipitation. The similarity between the three regions is striking and
implies a common mechanism driving POC formation.

By combining the spatio-temporal distribution of POCs with their average
optical depth and an assumed cloud droplet asymmetry parameter we are able to
calculate the change in albedo due to POCs (Stephens, 1994). Calculating a
global mean value and multiplying by the incident solar flux, we find that
POCs make a very small change to the amount of energy reflected by the Sc,
only 0.02 Wm-2. Even if anthropogenic activity suppressed POC formation
completely the climate effect would be negligible compared to a present day
CO2 forcing of 1.7 Wm-2.

##  4 Conclusions and discussion

Automatically detecting, and even classifying, clouds in satellite data is
easy and done routinely, however the detection of _perturbations_ in those
clouds is more challenging. In this work we have demonstrated the first
application of deep convolutional neural networks in the detection of ship-
tracks and POCs. The detection of ship tracks provides positive cases of the
direct perturbation of clouds by anthropogenic aerosol, while the detection of
POCs provides clues about the adjustments of clouds to these perturbations.

By running inference over all images from the MODIS satellite intersecting the
main Sc regions we have created a database of the properties of 1000s of POCs
and their spatial and temporal characteristics. This has enabled a global,
long term analysis of these phenomena, which will provide the keys to
unlocking new understanding of the conditions under which aerosols influence
cloud physics and ultimately the climate.

While the ship track model is currently still under development, early results
provide encouraging evidence that the model has some skill and could be
improved further using some of the techniques developed for POC detection.

Looking forward, we hope to develop a single model which is able to detect
both these, and other cloud perturbations in order to produce a global, open,
database spanning more than a decade of satellite observations. This will
prove invaluable in our efforts to better constrain the effects of aerosol on
clouds and their overall contribution to anthropogenic climate change.

## Acknowledgements

We gratefully acknowledge the support of Amazon Web Services through an AWS
Machine Learning Research Award. We also acknowledge the support of NVIDIA
Corporation with the donation of a Titan Xp GPU used for this research. DWP
and PS acknowledge funding from the Natural Environment Research Council
project NE/L01355X/1 (CLARIFY). PS and MC acknowledge funding from the
European Research Council project RECAP under the European Union’s Horizon
2020 research and innovation programme with grant agreement 724602.

### Software and Data

The MODIS data used were the 021km and 06 products available through
https://modis.gsfc.nasa.gov/data/. The models were built using Keras (Chollet
et al., 2015) with the Tensorflow engine (Abadi et al., 2015). The models and
training data will be made freely available on publication.

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# Mirages and Large TeV Halo-Pulsar Offsets from Cosmic Ray Propagation

Yiwei Bao1,2 Gwenael Giacinti2,3<EMAIL_ADDRESS>Ruo-Yu Liu1,4
<EMAIL_ADDRESS>Hai-Ming Zhang1 Yang Chen1,4<EMAIL_ADDRESS>1Department of
Astronomy, Nanjing University, 163 Xianlin Avenue, Nanjing 210023, China
2Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 201210,
China
3School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China
4Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University,
Ministry of Education, Nanjing, China

###### Abstract

The study of extended $\gamma$-ray sources usually assumes symmetric diffusion
of cosmic rays. However, recent observations of multiple sources near single
pulsars and significant offsets between TeV halo centroids and their parent
pulsars suggest that this assumption is overly simplistic. In this Letter, we
demonstrate that asymmetric propagation of cosmic rays near their accelerators
may create multiple TeV sources instead of a single symmetric source. This
mechanism also explains the large offsets between TeV halo centroids and their
pulsars. We demonstrate that several perplexing detected sources can be
naturally explained without invoking additional invisible accelerators.

††preprint: arXiv:0000.00000

Introduction— HAWC [1] and LHAASO [2] have detected almost a hundred gamma-ray
sources at TeV energies and beyond. Many of these sources are extended and
likely due to inverse Compton (IC) from relativistic electrons. One prominent
and intriguing example of such leptonic sources is the “TeV halo”. TeV halos
are extended gamma-ray emitting regions around middle-aged pulsars. They are
due to electrons with energies up to about a PeV [3, 4, 5], and with typical
spatial extensions of the order of $\approx 50$ pc. Within these regions, the
diffusion coefficient of cosmic rays (CRs) is suppressed by a factor of
$10^{2}$–$10^{3}$ compared with the value derived from the Boron-to-Carbon
ratio [6]. Studying TeV halos not only provides insights into these unique
celestial phenomena but also offers valuable information about the turbulent
magnetic fields in the interstellar medium around these middle-aged pulsars
[7].

The LHAASO catalog identifies dozens of TeV sources that are associated with
middle-aged pulsars, and like in HESS and HAWC observations [1], a large
fraction of these sources exhibit significant offsets between their centroids
and the positions of the pulsars. These offsets, however, can neither be
explained by the interaction of the reverse shock with the pulsar wind nebula
(PWN) [8], nor by the proper motion of the pulsar when $>10$ TeV $\gamma$-rays
are considered [9]. Moreover, some of these extended sources are not even
circularly symmetric. Notable examples include 1LHAASO J0206+4302u and 1LHAASO
J0212+4254u. These discoveries challenge the traditional cosmic-ray diffusion
model, which predicts that the accelerator should always be located at the
center of the observed emission and that the radial profile of the source
should be a symmetric Gaussian —or an elliptical Gaussian if anisotropic
diffusion along a large-scale magnetic field is considered.

In reality, cosmic-ray propagation is expected to be very filamentary on
scales smaller than the coherence length of the turbulence, because these
particles follow their local magnetic field lines [10]. In contrast, standard
diffusion models approximate their propagation as isotropic or anisotropic
diffusion, with a homogeneous diffusion coefficient or tensor around their
sources. This approximation is valid only if the diffusion length scale is
much larger than the coherence length $L_{\rm c}$ of the magnetic field
($\approx 1/5$ of the outer scale of the turbulence for a Kolmogorov
spectrum), with the specific influence of given magnetic field structures
averaging out thanks to the particle traversing many different field
structures whose directions are random. As a result, in such models, the
extended gamma-ray source will always be centered at the particle accelerator.
In practice, this phenomenological approximation is often sufficient, because
the coherence length in spiral arms —where most pulsars are located— may be as
low as 2 pc [11], and the lifetime of low-energy electrons is sufficiently
long to allow them to cross several coherence lengths. However, challenges
have now started to arise as more sources have been discovered above 20 TeV in
the inter-arm regions (where the coherence length may be much larger [11]) and
off-plane regions (where the average coherence length could reach around 200
pc [12]). For such sources, the diffusion approximation is expected to break
down because their electrons can only propagate within a coherence length.
Hence, it is necessary to treat electron propagation in a more realistically
way to explain their extended gamma-ray emissions. As shown below, if the
magnetic field is bent away from or towards the observer, the electrons
accumulate along the line of sight, forming apparent, off-centered gamma-ray
sources, whose directions on the sky may be far from that of the true parent
electron source. Due to their misleading nature, we call them “mirage
sources”, although the underlying physics is different from that of optical
mirages. For this new type of gamma-ray sources, no astrophysical accelerators
lie at their centroids, which is dramatically at odds with any naive or
standard expectation. The electrons responsible for the detected sources have,
in fact, been accelerated in a nearby accelerator, which may even be located
outside the detected gamma-ray source.

In this Letter, instead of adopting the traditional diffusion approximation to
investigate electron propagation around their sources, we calculate from first
principles the motion of individual electrons in synthetic three-dimensional
turbulence. The latter approach can give vastly different results, and it is
more realistic because the (effective) cosmic-ray diffusion tensor strongly
varies in space on length scales that are smaller than the coherence length.
Our method is more fundamental and more flexible than using the diffusion
approximation. Indeed, it can both resolve the filamentary propagation of
electrons on small scales, and recovers the predictions from the diffusion
approximation on large scales. By considering the short lifetime of $\gtrsim$
100 TeV electrons, we find that the real distribution of these electrons
around their sources can be very filamentary even in strong turbulent magnetic
fields. Also, we find that both the presence of large offsets from the pulsars
and the presence of mirage sources are two interlinked phenomena arising from
this filamentary propagation. Our results suggest that in the inter-arm region
and in the Galactic halo where the coherence length is large, a large pulsar
halo could be identified as multiple distinct smaller halos. Moreover, when
the primary identified halo exhibits low luminosity, these ancillary mirage
halos could emerge as the sole identified entities, leading to considerable
offsets between the detected gamma-ray sources and the actual astrophysical
source of the electrons.

Methods— We use test particle simulations, where we integrate the trajectories
of $>100$ TeV electrons in a turbulent magnetic field, as in e.g. Refs. [13,
14, 15]. For the trajectory integrator, we use the original Boris pusher [16].
These electrons display an inverse Compton (IC) spectrum mainly off the cosmic
microwave background (CMB), peaking around 20 TeV. For the magnetic field, we
use the combination of a regular component $\bm{B}_{\rm r}$ and a turbulent
component $\bm{B}_{\rm t}$ which can be expressed by a superposition of
$N_{\rm m}$ plane waves[13]. Hereafter, we adopt $N_{\rm m}=600$, over 6
orders of magnitude in wavelength. The electrons are injected at a point
source in the simulation, assumed to be a pulsar, and with a power-law
spectrum ${\rm d}N/{\rm d}\gamma\propto\gamma^{-\alpha}$, where $N$ denotes
the electron number and $\alpha\approx 2$ the power-law index. The spin-down
luminosity of the pulsar is represented as $L(t)=L_{0}/[1+(t/\tau_{0})^{2}]$,
with $L_{0}$ the initial spin-down luminosity and $\tau_{0}$ the initial spin-
down timescale. To mimic the continuous injection of electrons, we inject
$\approx 10^{4}$ numerical particles continuously over time with energy
$\gamma(t_{\rm inj})=\int^{t_{\rm inj}}_{t_{\rm age}}\left[\dot{\gamma}_{\rm
sync}(\gamma)+\dot{\gamma}_{\rm IC}(\gamma)\right]+{\rm
100\,TeV}/{m_{e}c^{2}}$. This ensures that they are all cooled down to 100 TeV
at the time $t_{\rm age}$, where $t_{\rm inj}$ is the injection time of the
particle and $t_{\rm age}$ the current age of pulsar. The value of $t_{\rm
inj}$ is set such that every numerical particle represents roughly the same
number of real electrons. We only study here sources located within a few kpc
from the Sun, due to their better detectability. Therefore, for the seed
photon energy density, we use the interstellar radiation field averaged over a
2 kpc region surrounding the Sun.

For the simulation data analysis, we assume that the TeV halos generated by
these pulsars are observed by LHAASO KM2A [17]. We consider the observations
to be at 20 TeV, and we set here the reference luminosity of the pulsar halo
to $L_{\rm ref}=6\times 10^{32}\,$erg/s. This $\gamma$-ray luminosity is about
6 times that of the observed luminosity of Geminga’s halo in the range of
$8-40$ TeV [6], but still reasonable. A wider range of luminosities is
explored in [18] (paper II). For fainter sources, the lower luminosity can be
compensated by an increase in observation time. Assuming all arriving photons
to be at 20 TeV, the resulting annual counts is $8.1\times 10^{5}(250\,{\rm
pc}\,d^{-1})^{2}$ (with $d$ being the distance to the pulsar). Our background
is set to be consistent with that in the Crab Nebula region, which results in
1 event per hour within a 1∘ cone [5]. We set the observation time to 3 yr,
which is roughly the LHAASO effective operation time, and we assume that the
source is visible for 6 hours per day.

For smoothing, we apply a point spread function (PSF) of 0.3∘ (LHAASO KM2A
resolution at 20 TeV [17]). The fraction of total counts in each bin is
computed by summing the Gaussian of 10240 numerical particles in each
observation bin (of 0.1∘). We employ the standard “on-off method” for
analyzing these observational counts as is used by the LHAASO Collaboration.

Multiple sub-halos and large TeV halo-pulsar offsets— In our simulations, we
frequently detect multiple halo-like sources forming near a single
accelerator. The underlying reason for having such multiple sources is that
when the electrons diffuse along magnetic field lines, these magnetic field
lines bend on scales comparable with the coherence length. When these bent
field lines align roughly with our line of sight, a projection effect occurs,
causing the electrons to accumulate along the line of sight and to produce a
mirage source.

The offsets observed in the data between TeV halo centroids and pulsars
locations can also be attributed to this phenomenon of mirage sources. More
precisely, if the primary halo that is closer to the pulsar happens to be
dimmer than its adjacent halos, the main halo may become elusive, and might
even drown in the background noise. As a result, the only detectable halo
could be the brighter adjacent one, resulting in a surprizingly large apparent
offset between the detected TeV halo and the location of its parent pulsar.

In paper II, we investigate how mirage source properties are influenced by the
regular-to-turbulent magnetic field strength ratio. Provided that this ratio
$B_{\rm r}/B_{\rm t}$ is $\lesssim 1$ (we explore up to 3, higher ratio is not
realistic in the Galaxy), it does not significantly impact the properties of
mirage sources. Therefore, in the following, we will not further discuss the
impact of $B_{\rm r}/B_{\rm t}$.

In Figure 1, the particles are injected at the center of each panel at (0,0),
in two different realisations of the magnetic turbulence with $L_{\rm c}=40$
pc, $B_{\rm r}=0$ and $B_{\rm t}=3\,\mu$G. The distance of the pulsar is set
to 250 pc. The colors represent the gamma-ray emission from these sources, and
the noise is not displayed. One can see that a source forms near the injection
site in each panel (orange dot for its centroid). Along the filamentary
magnetic field, we also detect two other sources with similar brightness
(black and green dots). Therefore, a single large halo can be very filamentary
and can potentially be identified as triple Gaussian sources lying along the
filamentary structure.

In Figure 2, two configurations with very large offsets are displayed. As can
be seen, the main sources near the parent pulsars could have half of the
surface brightness of the secondary sources. With noises taken into
consideration, we may only see the secondary sources, with the main sources
being drowned in the noise. In these two panels, we put the sources at a
distance of $1\,$kpc (at larger distance the offsets would be even more
obvious, because the brightness ratio of two sources are the same, and the
main sources are more easily buried by the noise), and the parameters are set
to $L_{\rm c}=200$ pc (the mean Galactic value according to [12]), $B_{\rm
r}=0$ and $B_{\rm t}=0.5\,\mu$G (a reasonable turbulent field strength, see
table of paper II).

Applications— Despite their complicated evolution in their early ages, a large
fraction of pulsar wind nebulae (PWNe) will ultimately evolve into a bow shock
pulsar wind nebula (BSPWNe) in the interstellar medium (ISM) due to their high
kick velocity ($100$–$500$ km s-1 [19]). This transition is a result of the
constant proper motion velocity of the pulsar over time and the significantly
decreasing expansion velocity of the supernova remnants. Consequently, these
pulsars will eventually leave their parent SNRs and move supersonically in the
ISM. In Figure 3, we show three configurations which can reproduce the sources
1LHAASO J0206+4302u and 1LHAASO J0212+4254u. The offset from 1LHAASO
J0206+4302u to the millisecond pulsar J0218+4232 can be attributed to the
BSPWN whose tail can extend for tens of pc before it gets destroyed by neutral
mass loading [20, 21]. The real separation of the two sources is $\approx 70$
pc, and the resulting required $L_{\rm c}$ would be $\sim 280$ pc, which is
reasonable at high Galactic latitude. In the first panel, $L_{\rm c}=200$ pc,
and the injection is isotropic. In the other panels, $L_{\rm c}=400$ pc, and
all electrons have an initial positive y velocity to mimic the particle
injection at the tail of the BSPWN. The phenomenon is more obvious with
narrower injection. The total $\gamma$-ray luminosity is roughly 10 times that
of Geminga. Parameters are set to $B_{\rm r}=0.7\,\mu$G and $B_{\rm
t}=0.7\,\mu$G (see paper II for the turbulent and regular field strength at
the location of PSR J0218+4232). With the observed $\gamma$-ray index $\approx
2.5$, we use $\alpha=1.76$, and the distance is set to be $d=3$ kpc (distance
of the milli-second pulsar). Meanwhile, Faraday rotation measurements have
revealed a large-scale magnetic field parallel to the Galactic plane [22],
which is consistent with our study here because 1LHAASO J0206+4302u and
1LHAASO J0212+4254u lie almost parallel to the Galactic disk.

The electron density inside the BSPWN tail is extremely low, and therefore the
tail can hardly be seen. The injection of the electrons could happen at the
end of the BSPWN tail, as in the traditional picture, or everywhere around the
BSPWN [23]. The latter explanation can hardly explain the offset, and we then
adopt the traditional scenario. The combined flux of all the sub-sources
perceived by KM2A, is $0.54\times 10^{-16}$ cm-2 s-1 TeV-1 [2].

Meanwhile, using this model, the offsets observed in many TeV halos can be
naturally accounted for, such as J2005+3415 and J2005+3050 (see paper II for
detailed calculations).

Discussion— Even though mirage sources look similar to real sources, it is
possible to identify them as mirages. The most intrinsic feature of a mirage
source is that there is always a connecting structure in gamma-rays (probably
slightly dimmer than the sources in TeV) connecting the pulsar and the mirage
source. Although with current observations the connecting structure may be
drowned in the noise, it might be possible to detect it in the future with a
larger observation time, or with experiments with better sensitivities.

It is also possible to distinguish mirage halos thanks to observations in
other wavelengths. The magnetic flux tube in the halo is oriented nearly along
the line of sight, resulting in the synchrotron emission from the electrons
being mainly emitted perpendicular to the line of sight. Therefore, mirage
sources are expected to be very faint in X-rays, compared to their
$\gamma$-ray luminosity. On the other hand, the thin structure connecting the
primary source to the mirage sources should be brighter in X-rays and easier
to detect, also because of the orientation of the magnetic fields. In
contrast, if the gamma-ray sources are contributed by invisible radio-quiet
pulsars, there should be no structures connecting the pulsars and the offseted
sources, and there will be a small X-ray PWN surrounding the very vicinity of
the pulsar, just like the Geminga PWN [24]. Therefore, future X-ray telescopes
with low sensitivities could help confirm whether the source is a mirage, or
is contributed by a radio-quiet invisible pulsar.

When the source is bright, the asymmetry increases the number of detected
sources, and when the source is dim, the asymmetry decreases the number of
detected sources because it is harder to identify a source with a standard
Gaussian if the source is faint and far away ($\sim 3$ kpc) from us. This
helps account for the relatively few detected TeV halos compared to the
numerous detected pulsars. However, more sources will become identifiable as
observation time increases.

Our simulations use synthetic turbulent magnetic fields based on Ref.[13], and
assume isotropic turbulence with a Kolmogorov power spectrum. However, the
properties of the interstellar turbulence may be different, see e.g. [25, 26].
Although probably not accurate at gyration scales, our turbulence does capture
well the more isotropic long-wavelength modes in the turbulence, which are
those driving the filamentary propagation of cosmic-rays on $L_{\rm c}$ scales
[10], as well as the formation of mirage sources and large offsets. Our method
can handle a wide range of turbulence length scales within reasonable
computing times. More realistic simulations, like particle-in-cell or 3D
magnetohydrodynamic simulations, only cover limited spatial scales,
restricting the range of electron energies that can be studied with them.

Conclusions and perspectives— The assumption of isotropic cosmic-ray
diffusion, which is traditionally used in the study of TeV halos around
pulsars, is getting challenged by recent observations of multiple sources near
single pulsars and significant offsets between TeV halo centroids and their
parent pulsars. In this Letter, we introduced the novel concept of mirage
sources. We demonstrated, by propagating individual electrons from first
principles, that the intrinsically asymmetric propagation of electrons around
pulsars can create multiple detectable TeV sources instead of a single
symmetric TeV halo. This mechanism also explains large offsets between TeV
halo centroids and their parent pulsars.

We showed that many perplexing gamma-ray sources can be interpreted through
asymmetric propagation very naturally. In particular, one asymmetric halo
generated by a pulsar may be identified as multiple sources. Our model
explains observed halo-like TeV $\gamma$-ray emissions with significant
offsets from nearby pulsars. It suggests that these halos may be attributed to
those pulsars despite their significant offsets, rather than by invisible
radio-quiet pulsars closer to the TeV halo centers. Our results are also
consistent with the large scale magnetic field found parallel to the Galactic
disk [22]. Our model predicts that new TeV halos discovered at high Galactic
latitude, if any, will likely show a filamentary structure and maybe also
mirage sources/large offsets.

Figure 1: Cases in which a single halo could potentially be misinterpreted as
triple halos. The color representation indicates the number of counts in each
grid point (smoothed with Gaussian PSF). The noise has been removed. The best-
fit points are the best-fit center of Gaussians, and the particles are
injected at (0,0).

Figure 2: Cases in which a large offsets can arise from the same mechanism.
The color representation indicates the number of counts in each grid point.
The noise has been removed.

Figure 3: Three configuration which can reproduce the 1LHAASO J0206+4302u and
1LHAASO J0212+4254u, with the color represent in the number of counts in each
grid point. The noise has been removed.

Acknowledgements- We thank Chunkai Yu, Hao Zhou and Giovanni Morlino for
discussion. This work is supported by NSFC under grants No. 12393853,
12350610239, 12393852, 12173018, 12121003 and U2031105.

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