arXiv:1711.06604v1 [math.CV] 17 Nov 2017
Division algebras of slice functions
Riccardo Ghiloni
Alessandro Perotti
Dipartimento di Matematica, Universit`a di Trento Via Sommarive 14, I-38123 Povo Trento, Italy riccardo.ghiloni@unitn.it, alessandro.perotti@unitn.it
Caterina Stoppato
Dipartimento di Matematica e Informatica “U. Dini”, Universit`a di Firenze Viale Morgagni 67/A, I-50134 Firenze, Italy
stoppato@math.unifi.it
Abstract
This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.
1
Introduction
This work addresses the study of function theory over finite-dimensional division algebras with a unified vision, thanks to the theory of slice functions introduced in [16].
As explained in [10], complex holomorphy admits a natural generalization to such algebras: the notion of slice regular function introduced in [11, 12] for the algebra of quaternions and in [13] for the algebra of octonions. The class of slice regular functions includes polynomials and convergent power series of the form
f(x) =X
n∈N
xna n,
and it has many useful properties. Quaternionic slice regular functions have been extensively studied: see [9] for a survey of the first phase of their study. Over the octonions, power series have been investigated in [17], while slice regular functions have been considered in the recent work [23]. A key tool for these studies was a quaternionic result called Representation Formula, [3, Theorem 2.27], along with the related octonionic result [17, Lemma 1].
The work [16] introduced an innovative approach. On the one hand, it provided a definition of slice regularity valid for functions with values in any alternative∗-algebraA. On the other hand,
necessarily differentiable nor continuous) for which the analog of the Representation Formula is valid. This class properly includes the class ofA-valued slice regular functions and it provides new tools to study it. For instance, the algebraic properties of slice functions, studied in [18], have been applied in [19] to the construction of Laurent series and the classification of singularities of slice regular functions.
In the present work, we turn back to the case of finite-dimensional division algebras and study slice functions in detail. In doing so, we encounter both expected and unexpected phenomena. In the second part of the work, we derive some results valid for slice regular functions over finite-dimensional division algebras, which had not been proven with the original approach to slice regularity.
In Section 2, we first overview the construction and classification of finite-dimensional division algebras. We then survey the definitions of slice function and slice regular function, along with the basic properties of these functions.
The grounds for our work thus set, we proceed in Section 3 to a detailed description of the zero sets of slice functions over finite-dimensional division algebras. Although their peculiar properties were known for quaternionic slice regular functions, [12, 6, 4, 8, 1], and for octonionic power series, [13, 17], we take a step forward and prove them for all slice functions.
Section 4 is devoted to reciprocals (multiplicative inverses) of slice functions over finite-dimensional division algebras. We present a brand new representation formula for the recipro-cal. We generalize to all such functions a formula of [22, 8, 18], which linked the values of a quaternionic slice function to those of its reciprocal. This generalization is highly nontrivial and some unexpected topological phenomena appear in the octonionic case.
In the last part of the work, we focus on slice regular functions over finite-dimensional division algebras. In Section 5, we prove the maximum modulus principle and then apply the results about reciprocals to derive a version of the minimum modulus principle. The latter principle is, in turn, applied to prove the open mapping theorem. These results subsume the separate results of [12, 7, 8, 1, 23].
Section 6 studies the possible singularities of slice regular functions. We make use of the general theory of [19] to classify them as removable singularities, poles or essential singularities, but we add a finer characterization of the essential singularities that is typical of division algebras. We then have two results proven in [22, 8] for quaternions, but new for octonions. The first one is the counterpart of the Casorati-Weierstrass theorem. The second one concerns the analogs of meromorphic functions, calledsemiregular functions. We prove that semiregular functions form an (infinite-dimensional) division algebra.
Acknowledgements. This work was partly supported by GNSAGA of INdAM and by the grant FIRB 2012 “Differential Geometry and Geometric Function Theory” of the Italian Ministry of Education (MIUR). The third author is also supported by Finanziamento Premiale FOE 2014 “Splines for accUrate NumeRics: adaptIve models for Simulation Environments” of MIUR.
2
Preliminaries
LetC,H,Odenote the∗-algebras of complex numbers, quaternions and octonions, respectively.
As explained in [5, 24, 2], they can be built from the real field R by means of the so-called
Cayley-Dickson construction:
• C=R+iR, (α+iβ)(γ+iδ) =αγ−βδ+i(αδ+βγ), (α+iβ)c=α−iβ ∀α, β, γ, δ∈R.
• H=C+jC, (α+jβ)(γ+jδ) =αγ−δβc+j(αcδ+γβ), (α+jβ)c=αc−jβ ∀α, β, γ, δ∈C.
On the one hand, this construction endows the three real vector spaces with a bilinear mul-tiplicative operation, which makes each of them an algebra. By construction, each of them is unitary, that is, it has a multiplicative neutral element 1; and R is identified with the
subal-gebra generated by 1. All alsubal-gebras and subalsubal-gebras we consider are assumed to be unitary. It is well-known that, while C is commutative and associative, H is associative but not
commu-tative. Moreover, O is not commutative nor associative but it is alternative: the associator
(x, y, z) = (xy)z−x(yz) of three elements vanishes whenever two of them coincide. Thenucleus
N(O) :={r∈O|(r, x, y) = 0∀x, y ∈A} and the center {r ∈N(O)|rx=xr ∀x∈O} of the
algebraOboth coincide withR. Alternativity implies the following properties (see [21]).
• [Moufang identities] For all elementsa, x, y of an alternative algebra,
(xax)y = x(a(xy)) (1)
y(xax) = ((yx)a)x (2)
(xy)(ax) = x(ya)x. (3)
• [Artin’s Theorem] In an alternative algebra, the subalgebra generated by any two elements is associative.
• [Power associativity] For allxin an alternative algebra, (x, x, x) = 0, so that the expression xn can be written unambiguously for alln∈N.
On the other hand, the Cayley-Dickson construction endows each of C,H,O with a ∗
-involution, i.e., a (real) linear transformationx7→xc with the following properties: (xc)c =x
and (xy)c =ycxc for everyx, y; xc =xfor everyx∈R. Thus, C,H andOare ∗-algebras. We
point out that (r+v)c=r−v for allr∈Rand all v in the Euclidean orthogonal complement
ofR. Thetrace andnorm functions, defined by the formulae
t(x) :=x+xc and n(x) :=xxc, (4)
are real-valued. In particular,Oiscompatible, i.e., its trace functiont has values in the nucleus
of the algebra. This definition has been given in [18,§1], along with the following property.
• [∗-Artin’s Theorem] In a compatible ∗-algebra, the ∗-subalgebra generated by any two
elements is associative.
Moreover, in C,Hand O, 1
2t(xy
c) coincides with the standard scalar product hx, yi and n(x)
coincides with the squared Euclidean normkxk2. In particular, these algebras are nonsingular, i.e.,n(x) =xxc = 0 impliesx= 0. IfA=C,H,O, even more is true: if we consider the central
cone ofA, namely
CA:={0} ∪ {x∈A|n(x), n(xc) are invertible elements of the center of A},
then
CA=A . (5)
The trace function t vanishes on every commutator [x, y] := xy −yx and on any associator (x, y, z) (see [18, Lemma 5.6]). It holdsn(xy) =n(x)n(y), or equivalently kxyk=kxk kyk.
Every nonzero elementxofC,HorOhas a multiplicative inverse, namelyx−1=n(x)−1xc=
xcn(x)−1. For all elements x, y:
• ifx, y6= 0 then (xy)−1=y−1x−1.
As a consequence, each of the algebras C,H,O is a division algebra: any equation ax = b or
xa=bwitha6= 0 admits a unique solution. The special caseb= 0 is equivalent to saying that there are nozero divisors. Let us recall the following well-known result:
• [Zorn’s Theorem]R,C,HandOare the only (finite-dimensional) alternative division
alge-bras.
From this point on,letAbe any of the algebrasC,H,O. Let us consider the sphere ofimaginary
units
S=SA:={x∈A|t(x) = 0, n(x) = 1}={w∈A|w2=−1}, (6)
which has, respectively, dimension 0,2 or 6. The ∗-subalgebra generated by any J ∈ S, i.e.,
CJ =h1, Ji, is ∗-isomorphic to the complex fieldC(endowed with the standard multiplication
and conjugation) through the∗-isomorphism
φJ : C→CJ, α+iβ7→α+βJ .
The union [
J∈S
CJ (7)
coincides with the entire algebraA. If, moreover,A=H,O, thenCI∩CJ=Rfor everyI, J ∈S
withI6=±J. As a consequence, every elementxofA\Rcan be written as follows: x=α+βJ,
where α∈ Ris uniquely determined by x, while β ∈Rand J ∈S are uniquely determined by
x, but only up to sign. If x ∈ R, then α = x, β = 0 and J can be chosen arbitrarily in S.
Therefore, it makes sense to define thereal part Re(x) and theimaginary part Im(x) by setting Re(x) :=t(x)/2 = (x+xc)/2 = αand Im(x) :=x−Re(x) = (x−xc)/2 = βJ. It also makes
sense to call the Euclidean normkxk=pn(x) = pa2+β2 themodulus of xand to denote it as|x|. The algebraOhas the following useful property.
• [Splitting property] For each imaginary unit J ∈ S, there exist J1, J2, J3 ∈ O such that
{1, J, J1, JJ1, J2, JJ2, J3, JJ3} is a real vector basis ofO, called a splitting basis ofO
as-sociated toJ. Moreover,J1, J2, J3 can be chosen to be imaginary units and to make the basis orthonormal.
An analogous property holds forH.
We consider on A the natural Euclidean topology and differential structure as a finite-dimensional vector space. The relative topology on each CJ with J ∈ S clearly agrees with
the topology determined by the natural identification between CJ and C. Given a subsetE of C, itscircularization ΩE is defined as the following subset ofA:
ΩE:=x∈A
∃α, β∈R,∃J ∈Ss.t. x=α+βJ, α+βi∈E .
A subset of A is termed circular if it equals ΩE for some E ⊆ C. For instance, given x =
α+βJ ∈Awe have that
Sx:=α+βS={α+βI ∈A|I∈S}
is circular, as it is the circularization of the singleton{α+iβ} ⊆C. We observe thatSx={x}
if x ∈ R. On the other hand, for x ∈ A\R, the set Sx is obtained by real translation and
conjugationz=α+iβ7→z=α−iβ, then for eachJ ∈S the mapφJ naturally embedsD into
a “slice”φJ(D) of ΩD=SJ∈SφJ(D).
The class ofA-valued functions we consider was defined in [16] by means of the complexified algebraAC=A⊗RC={x+ıy|x, y∈A}ofA, endowed with the following product:
(x+ıy)(x′+ıy′) =xx′−yy′+ı(xy′+yx′).
In addition to the complex conjugation x+ıy = x−ıy, AC is endowed with a ∗-involution
x+ıy7→(x+ıy)c:=xc+ıyc, which makes it a∗-algebra. This∗-algebra is still alternative and
compatible (see [18,§1.2]). Let us identify withCthe real subalgebraRC=R+ıRofAC, which
is the center ofAC. Then, for allJ ∈S, the previously defined mapφJ :C→CJ extends to
φJ : AC→A , x+ıy7→x+Jy .
LetD⊆Cbe preserved by complex conjugation and consider a functionF=F1+ıF2:D→AC
withA-valued componentsF1andF2. The functionF is called astem function onD ifF(z) = F(z) for everyz∈D or, equivalently, ifF1(z) =F1(z) andF2(z) =−F2(z) for everyz∈D.
Definition 2.1. A function f : ΩD →A is called a (left) slice function if there exists a stem
function F :D→AC such that the diagram
D −−−−→F AC
yφJ
yφJ
ΩD f
−−−−→ A
(8)
commutes for eachJ∈S. In this situation, we say thatf is induced byF and we writef =I(F).
If F isRC-valued, then we say that the slice functionf is slice preserving.
The term ‘slice preserving’ is justified by the following property (cf. [16, Proposition 10]): whenA=H,O, a stem functionF isRC-valued if, and only if, the slice functionf =I(F) maps
every “slice”φJ(D) intoCJ.
The algebraic structure of slice functions can be described as follows, see [18,§2].
Proposition 2.2. The stem functions D → AC form an alternative ∗-algebra over R with
pointwise addition(F+G)(z) =F(z)+G(z), multiplication(F G)(z) =F(z)G(z)and conjugation Fc(z) =F(z)c. This ∗-algebra is compatible and its center includes all stem functions D→R
C.
Let Ω := ΩD and consider the mapping
I : {stem functions onD} → {slice functions on Ω}=:S(Ω)
Besides the pointwise addition (f, g) 7→ f +g, there exist unique operations of multiplication (f, g)7→f ·g and conjugation f 7→fc on S(Ω) such that the mapping I is a ∗-algebra
isomor-phism. The∗-algebra S(Ω) is compatible and its center includes the∗-subalgebraS
R(Ω) of slice
preserving functions.
The productf ·g of two functionsf, g∈ S(Ω) is calledslice product off andg. Iff is slice preserving then (f ·g)(x) = (g·f)(x) = f(x)g(x) but these equalities do not hold in general. To computef·gwith f =I(F), g=I(G)∈ S(Ω), one needs instead to computeF G and then f ·g = I(F G). Similarly, f = fc when f is slice preserving but in general fc = I(Fc) with
f =I(F). The normal function off inS(Ω) is defined as
It coincides withf2 iff is slice preserving.
Formulae to express the operations on slice functions without computing the corresponding stem functions can be given as a consequence of the following representation formula, valid for allf ∈ S(Ω) and for allx=α+βI∈Ω (withα, β∈R, I∈S) after fixing J, K∈Swith J6=K
and after settingy:=α+βJ, z:=α+βK:
f(x) = (I−K) (J−K)−1f(y)−(I−J) (J−K)−1f(z). (9) In order to state the aforementioned formulae, it is useful to define a functionf◦
s : Ω→A, called
spherical value off, and a functionf′
s: Ω\R→A, calledspherical derivative off, by setting
The original article [16] and subsequent papers used the notationsvsf and∂sf, respectively, and
remarked that f◦
s ∈ S(Ω), fs′ ∈ S(Ω\R). As a consequence of equality (9) and of the fact that
the function Im is a slice preserving element ofS(A), for allx∈Ω\Rit holds
f(x) =fs◦(x) + (Im·fs′)(x) =fs◦(x) + Im(x)fs′(x).
Notice thatf is slice preserving if, and only if, f◦
s andfs′ are real-valued. We now come to the
We now restrict to special subclasses of the algebraS(Ω) of slice functions on Ω = ΩD. First
of all, within the∗-algebra of stem functions we can consider the∗-subalgebra of continuous stem
functionsF :D→AC. This∗-subalgebra induces, through the∗-isomorphismI, a∗-subalgebra
of S(Ω), which we denote as S0(Ω). Now suppose Ω is open: then for any J ∈ S the slice ΩJ:= Ω∩CJ=φJ(D) is open in the relative topology ofCJ; therefore,Ditself is open. We can
therefore consider the∗-subalgebras of continuously differentiable (C1) and of real analytic (Cω)
stem functionsF : D →AC. We denote as S1(Ω) andSω(Ω) the corresponding∗-subalgebras
ofS(Ω). Now, the class considered can be further refined to obtain a generalization of the class of complex holomorphic functions. This class was introduced in [11, 12] forA=Hand in [13]
for A = O, although we follow here the presentation of [16, 18]. Within the ∗-algebra of real
analytic stem functions F : D → AC, we can consider ∗-subalgebra of the holomorphic ones,
namely those such that ∂F
The corresponding∗-subalgebra ofSω(Ω) is denoted asSR(Ω) and its elements are calledslice
regular functions. Equivalently,f =I(F)∈ S1(Ω) is slice regular if
vanishes identically in Ω. The analogously defined function ∂f /∂x:=I(∂F/∂z) on Ω is called theslice derivative off.
Because we are now working with an open D, we can decompose it into a disjoint union of open subsetsDt⊆C, each of which either
1. intersects the real line R, is connected and preserved by complex conjugation; or
2. does not intersect R and has two connected components D+t, D−t, switched by complex
conjugation.
In the former case, the resulting ΩDt is called aslice domain because each intersection ΩDt∩CJ
with J ∈ S is a domain in the complex sense (more precisely, it is an open connected subset
of CJ). In case 2, we will call ΩD
t a product domain as it is homeomorphic to the Cartesian
product between the complex domainDt+and the sphereS. Thus, any mention ofS1(Ω),Sω(Ω)
and SR(Ω) will imply that Ω is a disjoint union of slice domains and product domains within the algebra A. We point out that, for quaternionic slice regular functions on a slice domain, the original reference for the construction of the algebraic structure and for the representation formula (9) is the work [4].
The relation between slice regularity and complex holomorphy can be made more explicit. The following result, called the ‘splitting lemma’, was proven in [13, 16] forA=O.
Lemma 2.3. Let {1, J, J1, JJ1, J2, JJ2, J3, JJ3} be a splitting basis for Oand letΩbe an open
circular subset of O. For f ∈ S1(Ω), let f0, . . . , f3 : ΩJ → CJ be the C1 functions such that
f|ΩJ =
P3
ℓ=0fℓJℓ, whereJ0:= 1. Thenf is slice regular if, and only if, for eachℓ∈ {0,1,2,3},
fℓ is holomorphic from ΩJ to CJ, both equipped with the complex structure associated to left
multiplication by J.
An analogous result holds forA=H, see [12, 4]. Polynomials and power series are examples
of slice regular functions, see [12, Theorem 2.1] and [13, Theorem 2.1].
Proposition 2.4. Every polynomial of the formPnm=0xmam=a0+xa1+. . . xnan with
coeffi-cientsa0, . . . , an∈Ais a slice regular function onA. Every power series of the formPn∈Nxnan
converges in a ball B(0, R) ={x∈A | kxk< R}. IfR >0, then the sum of the series is a slice regular function on B(0, R).
Actually,SR(B(0, R)) coincides with the∗-algebra of power series converging inB(0, R) with
the operations
This is a consequence of [12, Theorem 2.7] and [13, Theorem 2.12]. With the same operations, the polynomials overAform a∗-subalgebra ofSR(A).
Example 2.5. If we fixy∈A, the binomialf(x) :=x−y is a slice regular function onA. The conjugate function is fc(x) = x−yc and the normal function N(f)(x) = (x−y)·(x−yc) =
x2−x(y+yc) +yyc coincides with the slice preserving quadratic polynomial
∆y(x) :=x2−xt(y) +n(y).
Convention
Throughout the paper, all statements concerning S(Ω) and its subalgebras are valid for circular sets Ω in C,H or O. In Section 6, we set A = C,H or O. All proofs are specialized to the octonionic case for the sake of simplicity, but they stay valid when eitherCorHis substituted forO. All examples are over O, but the displayed functions can be easily restricted to the subalgebras Hand C.
3
Zeros of slice functions
In the present section, we describe the zero sets
V(f) :={x∈Ω|f(x) = 0}
of slice functions f ∈ S(Ω). In the special case of octonionic power series, similar results had been obtained in [13, 17]. For quaternionic slice regular functions, see [12, 6, 4, 8, 1].
In the general setting of slice functions, we use the theory developed in [18]. To do so, the next result will be a useful tool. Let us recall that a slice functionf is termedtameifN(f) is a slice preserving function (see Definition 2.1) and ifN(f) =N(fc).
Theorem 3.1. Every f ∈ S(Ω) is tame. As a consequence,
N(f ·g) =N(f)N(g) =N(g)N(f) =N(g·f) (13)
for allf, g∈ S(Ω).
Proof. Letf ∈ S(Ω). The functionN(f) is slice preserving because the quantities
n(fs◦(x)) + Im(x)2n(fs′(x)), t fs◦(x)fs′(x)c
appearing in Formula (11) are real numbers. Moreover, N(f) = N(fc) because the
afore-mentioned quantities are unchanged when fc is substituted for f. This is a consequence of
Formula (10) and of the equalities
n(a) =kak2=kack2=n(ac), t(abc) =ha, bi=hac, bci=t(acb)
valid for alla, b∈O. Thus,f is tame. The second assertion then follows from [18, Proposition
2.4].
We are now ready to describe the zeros of slice functionsf, gon a sphereSxand their relation
to the zeros offc, N(f), f·g.
Theorem 3.2. If f ∈ S(Ω), then for every x∈Ω the sets Sx∩V(f) andSx∩V(fc) are both
empty, both singletons, or both equal toSx. Moreover,
V(N(f)) = [
x∈V(f)
Sx= [
x∈V(fc)
Sx
Finally, for allg∈ S(Ω), [
x∈V(f·g)
Sx= [
x∈V(f)∪V(g)
Proof. Taking into account the compatibility of O, its nonsingularity and equality (5), when
Theorem 3.1, the third assertion follows. Our final remark is that, sinceN(f) is slice preserving, V(N(f)) is circular, whence equal toSSx⊆V(N(f))Sx. The second assertion follows.
The general picture is much more manifold than the previous example tells. The next result describes the ‘camshaft effect’ (so called in [17], which studied the special case of octonionic power series).
We point out that in case4.(b) the pointwcan be equivalently computed by means of each of the formulae appearing in2. and3.
Proof. Taking into account the compatibility ofOand equality (5), whenA=Owe can apply [18,
Theorem 5.5] to allf, g∈ S(Ω). Moreover, the inclusionsSx∩V(f·g)⊆ {w}appearing in cases
2.,3. and4.(b)of [18, Theorem 5.5] become equalities whenA=Oby virtue of Theorem 3.2.
The zero sets of slice regular functions can be further characterized, as follows. For J ∈S,
we will use the notationsC+
J ={α+βJ ∈CJ|β >0}and Ω
+
J := Ω∩C
+
Theorem 3.5. Assume thatΩ is a slice domain or a product domain and letf ∈ SR(Ω).
• If f 6≡0then the intersection V(f)∩C+
J is closed and discrete inΩJ for allJ ∈Swith at
most one exception J0, for which it holds f| Ω+J
0
≡0.
• If, moreover, N(f)6≡0 thenV(f)is a union of isolated points or isolated spheresSx.
Proof. By [18, Theorem 4.11], iff 6≡0 then for eachJ ∈SeitherV(f)∩C+
J is closed and discrete
in ΩJ orf|Ω+
J
≡0. Moreover, by the same theorem and by the lack of zero divisors inO, if there
exists J0 ∈ S such that f|
Ω+J 0
≡ 0 then such a J0 is unique unless f′
s ≡ 0. But f| Ω+J
0
≡0 and
f′
s≡0 would implyf|ΩJ
0 ≡0, whence the contradictionf ≡0.
IfN(f) =N(fc)6≡0 then, taking into account equality (5), we can apply [18, Corollary 4.17]
to conclude thatV(f) is a union of isolated points or isolated spheresSx.
An example withf 6≡0 andf|C+
J0
≡0≡N(f) has been provided in [16, Remark 12]:
Example 3.6. FixJ0∈S. Letf(x) = 1 + Im(x)
|Im(x)|J0for each x∈O\R. Thenf is slice regular
inO\RandV(f) =C+
J0. Moreover N(f)≡0.
Understanding whenN(f)≡0 impliesf ≡0 is the same as characterizing the nonsingularity ofSR(Ω), which can be done as follows.
Proposition 3.7. The ∗-algebra SR(Ω) is nonsingular if, and only, Ω is a union of slice
do-mains.
The previous proposition follows directly from [18, Proposition 4.14], taking into account that
O is nonsingular. Similarly, the next result derives from [18, Proposition 5.18], [18, Corollary
5.19] and Theorem 3.1.
Proposition 3.8. Assume thatΩis a slice domain or a product domain. Takef ∈ SR(Ω)with N(f)6≡0. Then, forg∈ S0(Ω),f·g≡0 org·f ≡0impliesg≡0. In particular, ifΩis a slice domain, then no element of SR(Ω) can be a zero divisor in S0(Ω).
4
Reciprocals of slice functions
This section treats the multiplicative inverses f−• of slice functions. We will make use of the
general theory of [18] but also prove two new formulas for f−•: a representation formula and
a formula that links the values of f−• to those of f. The representation formula forf−• is a
completely new result. The second formula was only known for the quaternionic case, see [22, 8, 18].
Our first statement follows immediately from [18, Proposition 2.4], from formula (11) and from Theorem 3.1.
Proposition 4.1. Let f ∈ S(Ω). If Ω′ := Ω\V(N(f)) is not empty, then f admits a
multi-plicative inverse f−• inS(Ω′), namely
f−•(x) = (N(f)−•·fc)(x) = (N(f)(x))−1fc(x).
If x ∈ Ω′∩(R∪V(f′
s)) then f−•(x) = f(x)−1. Furthermore, if Ω′ is open then f−• is slice
We point out that here and in the rest of the paper the restrictionf|Ω′ is denoted again asf
andf−•stands for f |Ω′
−•
. Similarly, the productf·gof two slice functionsf, gwhose domains of definition intersect in a smaller domain ˜Ω6= ∅ should be read as f|Ω˜ ·g|Ω˜. The consistency
of this notation is guaranteed by the fact that slice multiplication is induced by the pointwise multiplication of the corresponding stem functions.
We remark that if Ω = ΩD is a slice domain andf ∈ SR(Ω) hasN(f)6≡0, thenV(N(f)) is
the circularization of a closed and discrete subset ofD.
Example 4.2. For fixedy∈Oandf(x) :=x−y, we have
forf•n(x) when they are unambiguous. In such a case,xwill necessarily stand for a variable.
Now let us see how f−• can be represented in terms of f◦
We point out that the definition is well-posed because O is a compatible ∗-algebra. Notice
and
Φ(a1, a2)a2−Φ(a2, a1)a1=n(a1)a
c
1a2+ac2a1n(a2)−n(a2)ac2a1−ac1a2n(a1) (n(a1)−n(a2))2+ (t(ac
2a1))2 = 0.
Thus,g·f|α+βS ≡1. Since Proposition 4.1 guarantees the existence off
−•∈ S(Ω′), we conclude
that
f−•(α+βJ) =g(α+βJ) = Φ(a1, a2)−JΦ(a2, a1) for allJ ∈S, which is our first statement.
The second statement follows, for the casex∈Ω′∩R, from the fact that
1 = (f·f−•)(x) =f(x)f−•(x) =f◦
s(x)f−•(x).
For allx∈(Ω′\R)∩V(f′
s), it follows from the fact thata2= 0 implies, for allJ∈S, the equality
f−•(α+βJ) = Φ(a1,0)−JΦ(0, a1) =a−1
1 along with the equalityf(α+βJ) =a1=fs◦(α+βJ).
By the same argument we derive the inclusionV(f′
s)∩Ω′ ⊆V((f−•)′s). Sincef = (f−•)−•,
the inclusionV((f−•)′
s)⊆V(fs′) also holds, whence the third statement.
We will now study the relation between the valuesf−•(x) and the values f(x)−1.
Theorem 4.5. Letf ∈ S(Ω)and suppose thatΩ′:= Ω\V(N(f))=6 ∅. For allx∈Ω′∩(R∪V(f′
s))
it holdsf−•(x) =f(x)−1=f(y)−1 for each y ∈S
x. If Ω′′ := Ω′\(R∪V(fs′))6=∅, then for all
x∈Ω′′
f−•(x) =f(Tf(x))−1 (15)
where
Tf(x) := (fc(x)−1((xfc(x))fs′(x)))fs′(x)−1. (16)
Tf is a bijective self-map of Ω′′. For all y∈Ω′′, the restriction (Tf)|Sy is a conformal
transfor-mation of the sphereSy. Iff|
Ω′′ ∈ S
0(Ω′′), thenT
f is a homeomorphism.
If Ω′ is open andf
|Ω′ ∈ Sω(Ω
′)⊇ SR(Ω′)thenf′
sextends toΩ′ in real analytic fashion. Let
us denote the zero set of the extension as V and set Ω := Ωb ′\V. Then formula (16) defines a
real analytic diffeomorphism Tf ofΩb onto itself, fulfilling equality (15)for all x∈Ω.b
Finally, in all cases described Tf−• is the inverse map toTf.
Proof. The first statement for Ω′∩(R∪V(f′
s)) is a direct consequence of Theorem 4.4.
Let us prove the second statement concerning Ω′′. Forα, β∈Rsuch thatα+βS⊆Ω′′, we
know thatf(α+βI) =a1+Ia2 for allI∈S, witha1:=f◦
s(α+βi) anda2:=βfs′(α+βi), and
we know thata26= 0. Atx=α+βI we have that
Tf(x) =α+βJ, J = (fc(x)−1((Ifc(x))a2))a−21. (17) Now,J ∈S. Indeed,n(J) =n(I) = 1 because the norm functionn is multiplicative. Moreover,
since the trace functiont vanishes on all associators and commutators,
t(J) =t((fc(x)−1Ifc(x))(a2a−1
2 )) =t(fc(x)−1Ifc(x)) =t(I) = 0. Consequently,
f(Tf(x))−1= (a1+Ja2)−1
= a1+fc(x)−1((Ifc(x))a2)−1 = (fc(x)a1+ (Ifc(x))a2)−1
This quantity coincides withf−•(x) = (N(f)(x))−1fc(x) if, and only if,
fc(x)a1+ (Ifc(x))a2=N(f)(x).
The last equality is equivalent to each of the following equalities:
(ac
1+Iac2)a1+ (Iac1−ac2)a2=n(a1)−n(a2) +It(ac2a1), (Iac
2)a1+ (Iac1)a2=I(ac2a1) +I(ac1a2), (I, ac
2, a1) =−(I, ac1, a2),
−(I, a2, a1) = (I, a1, a2),
where we have taken into account formulae (10) and (11) and the fact that t(a2), t(a1) are elements of the nucleus of O. The last equality is true by the alternating property ofO. This
proves the second statement concerning Ω′′.
Now let us fixα+βS⊆Ω′′and prove that (Tf)|
α+βS is a conformal transformation ofα+βS.
By formula (17),Tf(α+βS)⊆α+βS. According to Theorem 4.4, equality (15) can be rewritten
forx=α+βI andTf(x) =α+βJ as
Φ(a1, a2)−IΦ(a2, a1) = (a1+Ja2)−1 whence
J =−a1a−21+ (Φ(a1, a2)−IΦ(a2, a1))−1a−21.
All affine transformations ofOare conformal (see [24,§4.6, p.205]) and the mapρ(w) =w−1=
|w|−2wc is conformal onO\ {0}, as it is the composition between the reflectionw7→wc and the
inversion in the unit sphere ofR8 centered at 0. Thus, (T
f)|α+βS is a conformal transformation
ofα+βS, as desired.
We conclude that Tf is a bijective self-map of Ω′′ because Ω′′ is a disjoint union of spheres
Sy (for appropriatey∈Ω′′), each mapped bijectively into itself byTf.
Now let us prove that Tf−• is the inverse map toTf : Ω′′ →Ω′′. We have, for f−• ∈ S(Ω′),
thatV(N(f−•)) =V(N(f)−•) =∅. Moreover,
Ω′\(R∪V((f−•)′s))) = Ω′′
by Theorem 4.4. Thus,Tf−• is a bijective self-map of Ω′′ mappingSy into itself for ally ∈Ω′′.
By applying formula (15) twice, we get that for allx∈Ω′′
f(x) = (f−•)−•(x) =f−•(Tf−•(x))−1= (f(Tf(Tf−•(x)))−1)−1=f(Tf(Tf−•(x))).
Since for each y∈Ω′′ the compositionT
f ◦Tf−• mapsSy into itself and f|Sy is an affine
trans-formation ofSy into another spherea1+Sa2, we conclude thatTf◦Tf−•(x) =xfor allx∈Ω′′.
To conclude the proof, let us consider the case when some regularity is assumed forf and let us apply [16, Proposition 7].
We first deal with the case when f ∈ S0(Ω′′). Then fc, f′
s : Ω′′ → O\ {0} are
continu-ous, whence Tf is continuous in Ω′′. Because f−• is also an element of S0(Ω′′), the inverse
transformationTf−• is continuous, too, andTf is a homeomorphism.
Secondly, we treat the case when Ω′ is open and f ∈ Sω(Ω′). In this case,fc: Ω′ →O\ {0}
is real analytic and f′
s extends to a real analytic function Ω′ → O. If V is its zero set and
b
Ω := Ω′\V then Tf extends to a real analytic map on Ω by the same formula (16). For allb
x∈Ωb \Ω′′, we observe that xbelongs to the nucleusR of Oso that T
f(x) =x. This implies
both that equality (15) is still fulfilled (thanks to the first statement) and thatTf Ωb
=Ω. Theb inverse map ofTf :Ωb →Ω is the analogous real analytic extension ofb Tf−• to Ω. In particular,b
The study conducted for quaternionic slice regular functions in [22, Theorem 5.4] and in [8, Proposition 5.2] is consistent with the previous theorem:
Remark 4.6. Formula (16)reduces to
Tf(x) =fc(x)−1xfc(x) (18)
whenever (x, fc(x), f′
s(x)) = 0. If this associator vanishes for all x∈ Ω′ and fs′ 6≡0, then the
previous formula extends Tf to a bijective self-map of Ω′ (a homeomorphism if f ∈ S0(Ω′), a
real analytic diffeomorphism ifΩ′ is open andf ∈ Sω(Ω′)) with inverse
Tf−1(x) =Tf−•(x) =Tfc(x) =f(x)−1xf(x).
On the other hand, in the octonionic case equality (18) does not always hold true:
Example 4.7. Consider the octonionic polynomial f(x) = ℓ+ 2xi. By direct computation, f′
s≡2i andfc(x) =−ℓ−2xi; in particular, fc(j) = 2k−ℓ. Thus,
Tf(j) =−((2k−ℓ)−1((j(2k−ℓ))i))i=
−3j+ 4ℓi 5
while
fc(j)−1jfc(j) = (2k−ℓ)−1j(2k−ℓ) =−j . We point out that
f(fc(j)−1jfc(j))−1=f(−j)−1= (ℓ+ 2k)−16= ℓ−2k
3 =f
−•(j),
where we have taken into account thatf−•(x) =N(f)(x)−1fc(x) =−(1 + 4x2)−1(ℓ+ 2xi). The previous example shows that [23, Formula (5.2)] is only true under additional assump-tions, such as those of Remark 4.6.
Remark 4.8. Formula (16)reduces to Tf(x) =xwhenever
• fc(x)∈R; or
• xbelongs to a slice CJ that is preserved byf.
The thesis follows by direct computation in both cases. In the second case, we take into account the fact that x, fc(x) andf′
s(x) all belong to the commutative subalgebraCJ.
In general,Tf does not always admit a natural extension to Ω′. Let us begin with a general
remark and then provide some examples.
Remark 4.9. Let f ∈ S(Ω), set Ω′ := Ω\V(N(f)) and let y ∈Ω′∩V(f′
s). If the set Ω′′ :=
Ω′\(R∪V(f′
s))6=∅includes a subsetU (whose closure includesy) such thatlimU∋x→yfc(x) =a
andlimU∋x→y f
′
s(x)
|f′
s(x)| =u, then
lim
U∋x→yTf(x) = (a
−1((ya)u))uc.
We are now ready to provide an example where Tf admits an extension to Ω′, though not
through formula (18), and an example where it does not. In both examples, we will use the Leibniz rule (12) for spherical derivatives and the fact that for ∆(x) =x2+ 1 we have ∆′
s(α+βI) = 2α,
∆◦
Example 4.10. Consider the octonionic polynomialf(x) =−i+ (x2+ 1)j. By direct computa-tion,
f′
s(α+βI) = ∆′s(α+βI)j = 2αj .
The zero set of the extension of f′
s to Ois ImO. Thus, Tf extends to a real analytic
transfor-ifRe(x)<0. Thus, the transformationTf can be analytically extended to Ω′ by setting
Tf(x) =−(fc(x)−1((xfc(x))j))j .
The previous examples notwithstanding, Theorem 4.5 has the following useful consequence.
Corollary 4.12. Let f ∈ S(Ω) and set Ω′ := Ω\V(N(f)). If C is a circular nonempty subset
of Ω′ then
f−•(C) ={f(x)−1|x∈C}.
5
Openness of slice regular functions
In this section, we will state and prove the counterparts of the maximum modulus principle, the minimum modulus principle and the open mapping theorem for slice regular functions. Our statements subsume those proven in [12, 7, 8, 1, 23]. In the quaternionic case, a completely different approach will be adopted in [15].
Let us start with the first of these results, proven in [12, 7, 8, 23] for slice domains and in [1] for quaternionic product domains.
Theorem 5.1 (Maximum Modulus Principle). Let f ∈ SR(Ω) and suppose |f| has a local maximum point x0∈Ω.
2. IfΩis a product domain and x0∈C+
J thenf is constant in Ω
+
J.
Proof. Suppose Ω to be either a product domain or a slice domain. That is, Ω = ΩD where
either: D intersects the real line R, is connected and preserved by complex conjugation; orD
does not intersectRand has two connected components switched by complex conjugation.
Ifx0∈CJ, consider an orthonormal splitting basis
{1, J, J1, JJ1, J2, JJ2, J3, JJ3}
forOand apply Lemma 2.3: there are holomorphic functionsfℓ: ΩJ →CJ such that
f|ΩJ =
3
X
ℓ=0 fℓJℓ,
withJ0:= 1. Let us define a mapF = (F0, F1, F2, F3) :D→C4by letting
Fℓ:=φ−J1◦fℓ◦φJ,
whereφ−J1 denotes the inverse of the bijection φJ :D→ΩJ, α+iβ7→α+Jβ. The Euclidean
norm kF(z)k equals |f(φJ(z))|, whence kFk has a local maximum point z0 := φ−J1(x0) ∈ D.
SinceFis holomorphic, it follows from the maximum modulus principle for holomorphic complex maps [20, Theorem 2.8.3] thatF is constant in the connected component ofD that includesz0. As a consequence,f is constant in the connected component of ΩJ that includesx0.
If Ω is a product domain and x0∈C+
J thenf is constant in Ω
+
J. If, on the other hand, Ω is
a slice domain thenf is constant in ΩJ, whence in Ω.
In the case of a product domain, a function that is constant on a half-slice Ω+J may well have a local maximum point.
Example 5.2. Let f ∈ SR(O\R)be defined by the formula
g(x) = 3i+x·f(x) = 3i+xf(x), f(x) = 1 + Im(x)
|Im(x)|i
(using the function of Example 3.6). In particular, g|C+
i
≡3i and|g||C+
i
≡3. We can see that i
is a local maximum point for|g| as follows. First, we observe that
|g(x)|2−32=|x|2|f(x)|2+ 6hi, xf(x)i.
If x∈C+
J then f(x) = 1 +Ji,|f(x)|2= 2−2hi, Jiand
hi, xf(x)i=hi, xi+hi, xJii=hi,Im(x)i+h1, xJi=hi,Im(x)i − |Im(x)|
=|Im(x)|(hi, Ji −1) =−1/2|Im(x)| |f(x)|2.
Thus,
|g(x)|2−32= (|x|2−3|Im(x)|)|f(x)|2
If we consider product domainU :={x∈O:|x|2<3|Im(x)|}, which includesi, then
|g(i)|= 3 = max
U |g|.
We point out that the same is true if we replacei with any x0∈U+
i , while for V :=O\U and
Before turning towards the minimum modulus principle, we prove a technical lemma (cf. [14, Proposition 6.13] for the quaternionic case).
Lemma 5.3. Let f ∈ S(Ω). Choose y=α+Jβ∈Ω(with α, β∈R, β >0, J∈S) and let
v:=fs◦(y)fs′(y)c.
1. Ifv∈Rthen|f||S
y is constant.
2. Suppose v 6∈ R and set I := Im(v)
|Im(v)|. Then the function |f||Sy attains its maximum at
α+βIand its minimum atα−βI. In particular, the maximum and minimum of|f||Sy are
attained at points belonging to the subalgebraAf,y generated byfs◦(y)andfs′(y). Moreover,
|f||Sy has no other local extremum.
Thus, iff(y) = 0 then either f|Sy ≡0 ory is the unique local minimum point of|f||Sy.
Proof. Forx∈Sy it holds
f(x) =fs◦(x) + Im(x)fs′(x) =fs◦(y) + Im(x)fs′(y),
whence
|f(x)|2=|f◦
s(y)|2+|fs′(y)|2+ 2hfs◦(y),Im(x)fs′(y)i=|fs◦(y)|2+|fs′(y)|2+ 2hv,Im(x)i.
If Im(v) = 0 then hv,Im(x)i= 0 and |f(x)|2 is constant in S
y. Otherwise, |f(x)|2 is maximal
(respectively, minimal) when Im(x) is a rescaling of Im(v) with a positive (respectively, negative) scale factor. Moreover, it does not admit any other local extremum.
We are now ready for the minimum modulus principle. In the quaternionic case, separate results had been proven in [12, 7, 8, 1]. In the octonionic case, [23] considered only the case of a slice regular function whose modulus has a local minimum point inR. Forf ∈ SR(Ω), after
restrictingf to Ω′ := Ω\V(N(f)), we will deal with the points of Ω′′ := Ω′\(R∪V(f′
s)) by
means of the transformation Tf : Ω′′ →Ω′′ defined in Theorem 4.5. The points in Rand the
interior points ofV(f′
s) will be even easier to deal with, while any point of the boundary∂ V(fs′)
(defined as the closure minus the interior of the set, as usual) will require an extra assumption.
Theorem 5.4. Let f ∈ SR(Ω). Suppose x0 ∈Ωto be a local minimum point for |f|, but not a zero off. In case x0∈∂ V(fs′), take one of the following additional assumptions:
(a) there exists a circular neighborhoodC of x0 such that|f(x0)|= minC|f|; or
(b) there exist w0 ∈Sx0 and an open neighborhood H of w0 in Ω such that Tf continuously
extends toH and the extension mapsw0 tox0.
IfΩis a slice domain, thenf is constant. IfΩis a product domain then there existsJ ∈Ssuch
that f|
Ω+J ≡f(x0).
Proof. Sincef(x0)6= 0, Lemma 5.3 tells us that f (whence N(f)) has no zeros inSx
0. In other
words,Sx
0 is included in the domain Ω
′:= Ω\V(N(f)) of the reciprocalf−•. LetU be an open
neighborhood ofx0 in Ω′ such that
|f(x0)|= min
We consider the set
K:= (U ∩R)∪ {y∈Ω′\R| f◦
s(y)fs′(y)c∈R, Sy∩U 6=∅},
which includesU∩RandU∩V(fs′). Thus,U\K is included in Ω′′:= Ω′\(R∪V(fs′)) and we
can setW :=Tf−1(U\K).
Claim 1. If x06∈K, then the equality
|f−•(y0)|=|f(x0)|−1= max
W∪K|f
−•|
holds fory0=Tf−1(x0). Ifx0∈K then it holds for all y0∈Sx0.
Proof. We apply Theorem 4.5, Corollary 4.12 and Lemma 5.3 repeatedly. We first observe that
sup
Ifx0∈RthenU includes a circular open neighborhood ofx0, which is also included inW ∪K.
If x0 is an interior point of V(f′
As a consequence, we can apply the Maximum Modulus Principle 5.1 to f−• at some point
y0∈Sx
0. If Ω (whence Ω
′) is a slice domain, we conclude thatf−• is constant in Ω′. Thus,f is
constant in Ω′, whence in Ω. If Ω is a product domain, we reason as follows.
• The function f−• is constant in the half-slice Ω′+
I through y0. Moreover, the point y0
(whence I) can be chosen so that the constant is f(x0)−1. With this choice, every y ∈
the subalgebraAf,xgenerated byfs◦(x) andfs′(x), which is associative by Artin’s theorem.
• IfS is not empty then, sinceS is an open subset of Ω+J, it follows thatf ≡f(x0) on Ω+J.
• If S is empty then WI+ is empty. Because WI+∪KI+ is a neighborhood of y0 in Ω+I , it follows that KI+ is a neighborhood of y0 in Ω+I. As a consequence, the circular setK is a neighborhood ofSy0 =Sx0 in Ω′. Since|f| ≡ |f(x0)| in K, the point x0 is also a local
maximum point for|f|. By Theorem 5.1,f is constant on the half-slice containingx0.
Remark 4.6 allows us to draw the following consequence, which applies to all slice preserving regular functions and to all quaternionic slice regular functions.
Corollary 5.5(Associative Minimum Modulus Principle). Letf ∈ SR(Ω) and assume that the associators(x, fc(x), f′
s(x))vanish for all x∈Ω\R. Suppose|f| admits a local minimum point
x0 ∈Ω, which is not a zero of f. If Ω is a slice domain, thenf is constant. If Ωis a product domain then there existsJ ∈Ssuch that f|
Ω+J ≡f(x0).
In the octonionic setting, whenx0is a boundary point ofV(fs′) but neither of the assumptions
(a)and(b)of Theorem 5.4 holds, it may well happen that noy0∈Sx
0 is an interior point of the
setW ∪K considered in the proof.
Example 5.6. Consider again the octonionic polynomial f(x) =−i+ (x2+ 1)(j+xℓ)of Ex-ample 4.11. We already saw that the zero set of f′
s(α+βI) = 2αj+ (3α2−β2 + 1)ℓ is S.
Moreover,
fs◦(α+βI) = −i+ ∆◦s(α+βI)(j+αℓ)−β2∆′s(α+βI)ℓ
=−i+ (α2−β2+ 1)j+ (α3−3αβ2+α)ℓ ,
whence the conditionf◦
s(α+βI)fs′(α+βI)c∈Ris only satisfied at S.
For each I∈S, we saw in Example 4.11 that both(i((Ii)j))j and(i((Ii)ℓ))ℓ are limit points
of Tf atI. Now letU :=B(ℓ,1/2) andW :=Tf−1(U\S). We will prove by contradiction that,
for allI∈S, the set W∪S is not a neighborhood of I.
If W ∪S were a neighborhood ofI, we would have (i((Ii)j))j,(i((Ii)ℓ))ℓ∈U, whence
(i((Ii)j))j=ℓ+γ, (i((Ii)ℓ))ℓ=ℓ+δ
for someγ, δ∈B(0,1/2). By direct computation, this would imply
I=ℓ−((i(γj))j)i, I=−ℓ+ ((i(δℓ))ℓ)i ,
whence the contradictionI∈B(ℓ,1/2)∩B(−ℓ,1/2) =∅.
Despite such pathological phenomena, Theorem 5.4 allows to establish the next result.
Theorem 5.7(Open Mapping Theorem). Let f ∈ SR(Ω). Assume either: Ωis a slice domain and f is not constant; or Ω is a product domain and f is not constant on any half-slice Ω+J. Then
f|Ω\V(fs′)
is an open map. Moreover, the image through f of any circular open subset U ⊆Ωis open. In particular, f(Ω) is open.
Proof. LetU be an open subset of Ω\V(f′
s) or a circular open subset of Ω. Let us prove that
• Iff′
s(x0)6= 0, the pointx0must be an isolated zero for the functiong(x) :=f(x)−y0inU.
Thus, there exists an closed Euclidean ballK:=B(x0, r)⊆U such thatg never vanishes in K\ {x0}.
• Now suppose that f′
s(x0) = 0 but U is circular. For an appropriate R >0 we have that
K:={x∈O|dist(x,Sx0)≤R} ⊆U and that gnever vanishes inK\Sx0.
We claim that ε := 13min∂K|g| is the desired radius. Indeed, for all y ∈ B(y0, ε) and for all
x∈∂Kthe inequality 3ε≤ |g(x)|implies
|f(x0)−y|=|y0−y|< ε <2ε≤ |g(x)| − |y0−y| ≤ |f(x)−y|.
Thus,|f(x)−y|admits a minimum (whence a zero by Theorem 5.4) at an interior point ofK. As a consequence,y∈f(K)⊆f(U), as desired.
The quaternionic case had been proven in [12, 7, 8, 1]. In the octonionic case, the second statement has been recently proven in [23], while the first statement is proven here for the first time.
We point out that, in the quaternionic and octonionic cases, restricting f to Ω\V(f′
s) is
essential in order to have an open map.
Example 5.8. The slice regular polynomialf(x) =x2hasf′
s(α+βJ) = 2αfor allα, β∈Rwith
β6= 0andJ ∈S. Thus,fs′ (extended toO) hasImOas its zero set. Consider the imaginary unit
i∈V(f′
s) and notice that it has distance 1 from CJ for all J ∈S orthogonal to it, e.g., J =k.
Thus, the Euclidean ballB(i,1) does not intersect Ck. Becausef is slice preserving, f(B(i,1))
does not intersectCk\R. As a consequence,f(i) =−1 is not an interior point off(B(i,1))and
f(B(i,1)) is not open in O.
6
Singularities of slice regular functions
In this section, we first recall from [19] the construction of Laurent-type expansions and the related classification of singularities as removable, essential or as poles. We then state a charac-terization of each type of singularity and prove an analog of the Casorati-Weierstrass theorem for essential singularities. Finally, we study the algebra of semiregular functions, namely func-tions without essential singularities, proving that it is a division algebra when the domain is a slice domain. These results are new in the octonionic case. Over the quaternions, part of the characterization of essential singularities is new, while the other results had been proven in [22, 8].
Two distinct Laurent-type expansions have been presented in [19] for slice regular functions on an alternative∗-algebra. Throughout this section, we continue to focus on an algebra A=C,H
orO. In this situation, the sets of convergence of these expansions are balls, or shells,
Σ(y, R) :={x∈A|σ(x, y)< R},
Σ(y, R1, R2) :={x∈A|τ(x, y)> R1, σ(x, y)< R2}, U(y, r) :={x∈A|u(x, y)< r},
with respect to the distanceσand the pseudodistances τ,u defined as follows onA:
σ(x, y) :=
|x−y|if x, y lie in the sameCJ
q
(Re(x)−Re(y))2+ (|Im(x)|+|Im(y)|)2otherwise , (19)
τ(x, y) :=
|x−y|if x, y lie in the sameCJ
q
(Re(x)−Re(y))2+ (|Im(x)| − |Im(y)|)2otherwise , (20)
u(x, y) := q|∆y(x)|. (21)
The expansions are based on functions such as those mentioned in Examples 3.3 and 4.2. The first result is [19, Theorem 4.9]:
Theorem 6.1. Consider a slice regular function f ∈ SR(Ω). Suppose thaty∈A andR1, R2∈
[0,+∞] are such that R1 < R2 and Σ(y, R1, R2) ⊆Ω. Then there exists a (unique) sequence
{an}n∈Zin Asuch that
f(x) =X
n∈Z
(x−y)•n·a
n (22)
inΣ(y, R1, R2). If, moreover,Σ(y, R2)⊆Ω, then for alln <0we have an = 0and formula(22)
holds inΣ(y, R2).
The second result is a consequence of [19, Remark 7.4]:
Theorem 6.2. Let f ∈ SR(Ω), lety ∈CJ ⊆A and let r1, r2∈[0,+∞] withr1< r2 such that
U(y, r1, r2)⊆Ω. Then
f(x) =X
k∈Z
∆ky(x)(xuk+vk) (23)
for allx∈U(y, r1, r2), where
uk= (2πJ)−1R∂UJ(y,r)dζ∆y(ζ)−k−1f(ζ), (24)
vk= (2πJ)−1
R
∂UJ(y,r)dζ(ζ−2 Re(y)) ∆y(ζ)
−k−1f(ζ). (25)
If, moreover, U(y, r2)⊆Ω, then for all k <0 we haveuk = 0 =vk and formula (23) holds in
U(y, r2).
The previous results allow to adopt the following terminology.
Definition 6.3. Consider a slice regular function SR(Ω). A point y is a singularity for f if there exists R > 0 such that Σ(y,0, R) ⊆Ω, so that Theorem 6.1 and Theorem 6.2 hold with inner radii of convergence R1= 0 =r1 and positive outer radii of convergence R2, r2.
In the notations of Theorem 6.1, the pointyis said to be a poleforf if there exists anm≥0 such thatan= 0for alln <−m; the minimum suchmis called the orderof the pole and denoted
asordf(y). Ify is not a pole, then it is called an essential singularityforf andordf(y) := +∞.
In the notations of Theorem 6.2, the spherical order of f atSy is the smallest even natural
number2k0 such thatuk = 0 =vk for all k <−k0. If no suchk0 exists, then we setordf(Sy) :=
+∞.
Singularities can be characterized as follows.
Theorem 6.4. Let Ωe be a circular open set, let y ∈Ωe\Rand set Ω := Ωe\Sy. Iff ∈ SR(Ω)
then one of the following assertions holds:
1. Every point ofSyis a removable singularity for f, i.e.,f extends to a slice regular function
on Ω. It holdse ordf(Sy) = 0andordf(w) = 0for allw∈Sy.
2. Every point of Sy is a non removable pole for f. There exists k ∈ N\ {0} such that the
function defined on Ωby the expression
x7→∆k
y(x)f(x)
extends to a slice regular function g ∈ SR(Ω)e that has at most one zero in Sy. It holds
ordf(Sy) = 2k. Moreover, ordf(w) =k andlimΩ∋x→w|f(x)|= +∞for all win Sy except
the possible zero ofg, which must have order less than k.
3. Every point w∈Sy, except at most one, is an essential singularity, i.e., ordf(w) = +∞.
It holds ordf(Sy) = +∞. For all neighborhoodsU of Sy inΩe and for all k∈N,
sup
x∈U\Sy
|∆k
y(x)f(x)|= +∞.
In particular, to check which is the case it suffices to check whetherordf(y),ordf(yc)are both 0,
both finite (but not both0) or not both finite; or, equivalently, (ifJ ∈SAis such thaty∈CJ\R)
whether the function defined onΩJ by the expression
z7→(z−y)k(z−yc)kf(z)
is bounded near y andyc for k= 0, for some finitek or for nok∈N.
Proof. We can apply [19, Theorem 9.4] tof. Moreover, according to Theorem 3.2, the function gappearing in case2can have at most one zero. Finally, from the same theorem and from [19, Lemma 10.7] it follows that in case3there can be at most one point inSy that is not an essential
singularity forf.
A similar characterization is available for real singularities, see [19, Theorem 9.5]. It is completely analogous to the complex case.
We can now consider the analogs of meromorphic functions.
Definition 6.5. A functionf is (slice) semiregularin a (nonempty) circular open setΩe if there exists a circular open subsetΩof Ωe such thatf ∈ SR(Ω) and such that every point of Ωe\Ωis a pole (or a removable singularity) forf.
The algebra of semiregular functions can be studied as follows.
Theorem 6.6. Let Ωbe a a circular open set. The setSEM(Ω) of semiregular functions onΩ is an alternative ∗-algebra with respect to+,·,c.
• If Ωis a slice domain, SEM(Ω) is a division algebra.
Proof. The first statement is derived from [19, Theorem 11.3]. When Ω is a slice domain, [19, Corollary 11.7] tells us that the nonzero tame elements of the algebra form a multiplicative Moufang loop. But in our setting all slice functionsfare tame by Theorem 3.1, whence the second statement follows. Finally, if Ω is a product domain then the algebra is singular by Proposition 3.7 (see also Example 3.6). However, every f with N(f)6≡ 0 admits a multiplicative inverse (still semiregular in Ω) by [19, Theorem 11.6], if we take into account again Theorem 3.1.
We are now ready to prove our last result: an analog of the Casorati-Weierstrass theorem for essential singularities.
Theorem 6.7. Let Ωe be a circular open set. Let y ∈Ωe and set Ω :=Ωe\Sy. Suppose y to be
an essential singularity for f ∈ SR(Ω). Then, for each neighborhood U of Sy in Ω, the imagee
f(U\Sy)is dense in A.
Proof. We will prove that, if there exist a neighborhoodU of Sy and a Euclidean ball B(v, r)
withr >0 such that
f(U\Sy)∩B(v, r) =∅,
then y is not an essential singularity for f. We assume, without loss of generality, U to be circular.
The previous equality implies that the function g:=f−vmapsU\Sy into the complement
ofB(0, r). Now considerg−•, which is semiregular in Ω. By Corollary 4.12,
g−•(U\S
y) ={g(x)−1| x∈U \Sy} ⊆B(0,1/r),
whenceg−• is bounded nearS
y. By Theorem 6.4,g−• extends to a regular function onΩ. Usinge
Theorem 6.6 twice, we conclude that g and f are semiregular in Ω, so thate y cannot be an essential singularity forf.
References
[1] A. Altavilla. Some properties for quaternionic slice regular functions on domains without real points. Complex Var. Elliptic Equ., 60(1):59–77, 2015.
[2] J. C. Baez. The octonions. Bull. Amer. Math. Soc. (N.S.), 39(2):145–205, 2002.
[3] F. Colombo, G. Gentili, and I. Sabadini. A Cauchy kernel for slice regular functions. Ann. Global Anal. Geom., 37(4):361–378, 2010.
[4] F. Colombo, G. Gentili, I. Sabadini, and D. Struppa. Extension results for slice regular functions of a quaternionic variable. Adv. Math., 222(5):1793–1808, 2009.
[5] H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Pres-tel, and R. Remmert. Numbers, volume 123 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1990. With an introduction by K. Lamotke, Translated from the second German edition by H. L. S. Orde, Translation edited and with a preface by J. H. Ewing, Readings in Mathematics.
[6] G. Gentili and C. Stoppato. Zeros of regular functions and polynomials of a quaternionic variable. Michigan Math. J., 56(3):655–667, 2008.
[8] G. Gentili and C. Stoppato. The zero sets of slice regular functions and the open mapping theorem. In I. Sabadini and F. Sommen, editors,Hypercomplex analysis and applications, Trends in Mathematics, pages 95–107. Birkh¨auser Verlag, Basel, 2011.
[9] G. Gentili, C. Stoppato, and D. C. Struppa. Regular functions of a quaternionic variable. Springer Monographs in Mathematics. Springer, Heidelberg, 2013.
[10] G. Gentili, C. Stoppato, D. C. Struppa, and F. Vlacci. Recent developments for regular functions of a hypercomplex variable. In I. Sabadini, M. V. Shapiro, and F. Sommen, editors, Hypercomplex analysis, Trends Math., pages 165–185. Birkh¨auser Verlag, Basel, 2009.
[11] G. Gentili and D. C. Struppa. A new approach to Cullen-regular functions of a quaternionic variable. C. R. Math. Acad. Sci. Paris, 342(10):741–744, 2006.
[12] G. Gentili and D. C. Struppa. A new theory of regular functions of a quaternionic variable. Adv. Math., 216(1):279–301, 2007.
[13] G. Gentili and D. C. Struppa. Regular functions on the space of Cayley numbers. Rocky Mountain J. Math., 40(1):225–241, 2010.
[14] R. Ghiloni, V. Moretti, and A. Perotti. Continuous slice functional calculus in quaternionic Hilbert spaces. Rev. Math. Phys., 25(4):1350006, 83, 2013.
[15] R. Ghiloni and A. Perotti. On a class of orientation-preserving maps ofR4. Under construc-tion.
[16] R. Ghiloni and A. Perotti. Slice regular functions on real alternative algebras. Adv. Math., 226(2):1662–1691, 2011.
[17] R. Ghiloni and A. Perotti. Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect. Ann. Mat. Pura Appl. (4), 190(3):539–551, 2011.
[18] R. Ghiloni, A. Perotti, and C. Stoppato. The algebra of slice functions.Trans. Amer. Math. Soc., 369(7):4725–4762, 2017.
[19] R. Ghiloni, A. Perotti, and C. Stoppato. Singularities of slice regular functions over real alternative∗-algebras. Adv. Math., 305:1085–1130, 2017.
[20] M. Klimek. Pluripotential theory, volume 6 of London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications.
[21] R. D. Schafer. An introduction to nonassociative algebras. Pure and Applied Mathematics, Vol. 22. Academic Press, New York, 1966.
[22] C. Stoppato. Poles of regular quaternionic functions. Complex Var. Elliptic Equ., 54(11):1001–1018, 2009.
[23] X. Wang. On geometric aspects of quaternionic and octonionic slice regular functions. J. Geom. Anal., 27(4):2817–2871, 2017.
