The principal extension of Riemann surfaces to a Banach algebra

algebra

Citation for published version (APA):

Ackermans, S. T. M. (1968). The principal extension of Riemann surfaces to a Banach algebra. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 68-WSK-04). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1968

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The Principal Extension of Riemahn Surfaces to a Banach Algebra by S.T.M. Ackenuans T.H.-Report 6S-WSK-04 November 1968

theory in Banach algebras given by De Bruijn

[3].

The analytic functions

in this theory are funct ions defined in a Banach algebra wi.th identity

element. which have the property that in a neighbourhood of every point of the domain of definition they coincide with the principal extension of a locally analytic complex valued function defined on an open sub-set of the complex numbers. The notion "principal extension of a lo-cally analytic functiorr' is introduced by Dunford; for its definition

and properties we refer to [6J ch.

5;

or to

[5]

sec.

7.3

for the case

of operator algebras.

The main purpose of De Bruijn t s program is to give a manifold theory belonging to this kind of function theory. Every ordinary Riemann

sur-face has in a certain sense an extension to the Banach algebra. Such

80-called principal extensions are introduced by De Bruijn

[3],

and in

the present paper they are discussed in some detail. The general notion of an analytic manifold related to a Banach algebra presents some dif-ficulties. MOst of these difficulties arise from unpleasant properties of one-to-one analytio mappings in Banach a.lgebras. In order to avoid these difficulties we propose a definition of analytio manifold whioh differs with De Bruijnts definition.

Our notion of a manifold -called strong Banaoh manifold- is based on a very restrioted olass of functions. which we introduce in this paper calling them strongly schlicht functions. As a consequence we obtain fewer manifolds than De Bruijn. But our manifolds, unlike De Bruijn's, have the nice property that they always can be embedded in principal

There are six sections in this paper. Sec.1 and sec. 2 contain preliminaries on Riemann surfaces and Banach algebra function theory respectively. In sec. 3 we introduce the class of the strongly schlicht functions. De Bruijn's construction of the principal extension of a Riemann surface to the Banach algebra is described in sec. 4, whereas sec.

5

contains the definition of strong Banach manifolds and the proofs of the facts that principal extensions of Riemann surfaces are

strong Banach manifolds and that every strong Banach manifold can be embedded in the principal extension of a Riemann surface. Several topo-logical and spectral properties of principal extensions of Riemann surfaces are studied in sec. 6. This section concludes with a dis-cussion of the principal extensions of locally analytic functions on Riemann surfaces. The operational calculus is extended by this

dis-cussion to principal extensions of Riemann surfaces.

Besides N.G. de Bruijn, W. Peremans also excercised a great influence on the present work. The author is very indebted to both of them for help and many encouraging discussions; he realises that a major part of the ideas involved in this paper is theirs. Several unpublished manuscripts by De Bruijn and Peremans were at the author's disposition in the preparation of this paper.

As it may be convenient to the reader we now give a short survey of the notations used in this paper. All definitions, of course, are post-poned. The letters I and J _{always denote index sets; i, i', i 1, i 2, •••••}

j, j', j1' j2' •••••• are indices. Throughout this paper A denotes a Banach algebra with identity element ej the algebra of the complex num-bers is denoted by

r ,

its identity, of course, Qy 1. It is the charac-teristic feature of our notation that corresponding entities related to the complex numbers and to the Banach algebra, whenever possible, are denoted by corresponding Greek and Roman types respectively.

This feature is clearly seen in the following list of letters used in this paper and their meaning.

a:,~,y, _•••

,I: ,

a,b,c, _••• , z,

.

..

Q,Q' ,Q1 ,Qi'

·

..

0,0' ,0_{1 'ai'}

·

..

lP,'P', _{• ••} F,G, _••• lh.1₁,.1' , ••• D,D₁,D', _••• n,p, _{• ••} p,r, •••

L:,EI,Ei ,E1' •••

S,Sl'Si,S1' •••

analytic functions in A,

the spaces of Riemann surfaces,

the spaces of strong Banach ma.nifolds, elements of

r,

elements of A, subsets of

r,

subsets of A, analytic functions in

r,

points on 6,.1 1, ••• points on D,D 1, subsets of _{6,Ll1 ' •••} subsets of D,D 1, ••• mappings Ll -

r,

6₁

- r

,

_•••

r -

_{6 1, •••} mappings

r -

6, • ••

·

..

<P,q>i,q>1' c¥,q,i,!jI1' T'l'111' ••• f,f i ,f1, ••• g, gi' g1'··· h,h 1, ••• 8,8 1, ••• T,T 1, ••• mappings 6 - 6₁• mappings D _ A, D 1 _ A, ••• mappings A _ D, A - D₁, ••• mappings D - D 1• ana~tic structures on 6,6₁, •••

strong~ analytic structures on D,D

1, ••• the algebras of the local~ analytic functions on Q,E.

Principal extensions are denoted by the symbol M ( ); so M(a) is a subset of A, M(~) a function in A, etc.

For composition of functions and mappings we use the symbol

0,

most~ without reference to the domain of definition of the composed function or mapping. So, if we speak of the function FoG, it is

im-plicit~ understood that this function is defined only for z such that G(z) is in the domain of F.

If notations are introduced by means of formulas containing an

equality sign we place: on the side of the new notation. So, by Q:= ~(~)

we indicate that the set ~(~) is denoted by Q.

1. Riemann Surfaces

For the sake of reference in the following sections we give in full

the definition of an abstract Riemann surface (cf. [1] p. 32). We also

state some properties used in other sections.

Let t. be a Hausdorff topological space.

Definition 1.1

A coordinate system (~,~) on t. is a nonempty open subset ~ of ~ and a

homeomorphism ~ of ~ onto an open subset of

r.

Definition 1.2

A coordinate system (~,~) is compatible with (~1'~1) if ~

n

~1

=

~ or

if

~1

0

~-1

is an analytic mapping of

~(~

n

~1)

onto

~1(~

n

~1).

Definition

1.3

An analytic structure on ~ is a family

e

=

{(~.,~.); i E

r}

of

coor-- 1 1

(1.3.1) U_i E I ~i

=

~.

(1.3.2) For all i E I, j E I, (~."p.) is compatible with (~.,(l).:).

1 1 J oJ

(1.3.3) If a coordinate system (~,,) is compatible with (~i"i) for all i E I, then (~,~) E

e.

Definition 1.4

A

Riemann surface

(A,e)

is a Hausdorff space

A

with an analytic struc-ture 8 on it.

First we must remark that this definition differs from the usual one (see

[2]

p. 114, or [9] p.6) as our Riemann surfaces need not be oon-nected.

Condition (1.3.3) in the definition of a Riemann surface is superfluous and usually omitted, (we state it explioitly as we will have to do so in the Banach algebra analogue of this definition). This fact is accounted for by the following remark.

If 8₁ is a family of coordinate systems on A , satisfying conditions (1.3.1) and (1.3.2),then there exists a unique analytic structure e on ~ with 8

1 c 8.

This 8 is the set of all coordinate systems compatible with ever,y e1e-ment of

e

₁•

I:efinition 1.5

If (A,8) is a Riemann surface, 8

=

{(~.".); i E I} , and if ~t is a

r

-1 1

-1 valued function on ~ c A , then ~t is called ana1Ytic if , t 0 ' i is

analytic on ~i (~

n

~i) for ever,y i E I for which L:

n

L:

Definition 1.6

The Riemann surfaces (6,e) and (e" 9

1) where 9 == {(L:i , !Pi); i E: I} , A_

= {(

Ll .,!p.);j E:

J}

are analytic equivalent if there erlsts a

homeo-I J J

morphism 1) of 6 onto

~

such that <Pj 0 1) 0 <Pi-1 is an analytic function

for every i E: I and j E: J for which it is defined. Such 1) is called an analytic homeomorphism.

We shall reformulate this definition in an equivalent form, which is, however, more sui ta.ble for the generalization to Banach algebras which is our purpose.

Let 1) : I:. -

'i

be a homeomorphism, let (Ll

i ,<Jli)E: 9 then 1)(~,Cli) :

=

:= (1)(i:i)' lJl.i 0 TJ-1) is a coordinate system in 6

1; let

TJCs)

:=

:= {!J(Lli'lIli); i E: I} • The following theorem gives the desired refor-mulation.

Theorem 1.7

The Riemann surfaces (6, 9) and (~,8

1) are analytic equivalent iff there exists a homeomorphism TJ of 6 onto 6

1 such that

ri-

e) =

e

1•

Definition 1.8

Let (I:.,e) be a Riemann surface; let

For pairs

9== {(L:.,cp.); iE: I}, Q. :=m.(Ll.)

1 1 ~ T1 1

(Q.,a.)

with

a.

E:

Q.

we define a relation ~

b.Y

1 1 1 1

This relation is an equivalence relation in the set

The sets

«Q.,a.);

a

i E:

Q.}

are called charts of the Riemann surface;

1 1 1

the totality of all charts, together, with the above equivalence rela-tion is called the atlas of the Riemann surface. This atlas has the following properties.

(1.8.2) I f (O.,a.)... (O.,et.) for some a. and ex. then there exists a

1. 1. J J 1. J

uniquely determined open set 0ij with et

i EOij C 0i and an

analytic homeomorphism\I> .. of 0 .. into O. such that (Q.,e),..,

l.J 1.J J 1.

,.." (O.,¢?.(c» for all CEO .. and such that this 0 .. , moreover,

J l.J 1.J l.J

has the property that there is no

C.

E O.

\0 ..

with (o.,e.) ,..,

1. 1. l.J 1. 1.

,.., (Q.,e.) for some

C.

EO .• To prove this it suffices to remark

J J J J

thatQ

iJ. :=cp.(E. nIl.)and<1? .. :_cp. ocp.-1 onO1. 1. -J l.J J 1. iJ. meet there-quirements. It follows easily that <P ••(Q .. ) ... Q.. and

\I>:-~

=

¢?J'1.'.

l.J 1.J J1. 1.J

(1.8.3) I f (QJ..,O:i) ~ (Q.,et.) [~denotes the negation of,..,],then there

J J

exist open sets Q! and Qt with ex. E O! c Q., a. E Q.! c QJ"

1. J 1.. 1. 1. J J

such that for all e. E Q!, C. E Q! we have

1. J. J J

(0.

,C.)

~ (O.,t: .).

J. J. J J

Since CPi-1(a.) "cp.-1(ex.) and

~

is Hausdorff, there are open sets

1. J J

E! and E! with cp:-1 (a.) E E!, cp:1(a.)E E!, E!

n

E! ...

¢.

J. J J. 1. J. J J J J. J

NowO! := cp.(E.

n

E!); O! := cp.(E.

n

E!) meet the requirements.

J. 1. 1. 1. J J J J

Let

[(Q.,a.)]

denote the equivalence class of

(Q.,a.);

let

J. 1. 1. 1.

6

1 :- {[(O.,a.)] ;J. 1. i E I, a. EO.} ; letJ. J. <jI.J. : 0 .... 6J. 1 be

de-fined by ~.(a.)

=

[(O.,a.)J , then {<\I. (Oi); i E I} is a basis of

1. 1. J. J. 1.

a topology in

~1.

Let El₁ := {(<\Ii(Oi)'

~~1);

i E I} , then

(~1,e1)

is a Riemann surface.

MJreover (~,El) is analytic equivalent to (6

1,El1). In order to indicate a mapping TJ : I::. ... 1::.

1 meeting the requirement of

defini-tioJl1.6 we remark that if for 1tE:!::.,1tE E.n E., then (O.,tIl·(n» ,..,_J.

J J. TJ.

>V (O.,cp.(n) t hence <\I. (I'll.(n» =: ,I ••(I'll'(n».

Thus, TJ(n):= <II.(CR. ( ~) if nE: E. defines a. mapping 6. -6.

1•

~ ~ ~

The verification that (6.

1,81) is a Riemann surface, and that TJ

is an analytio homeomorphism is st::raightforward.. We remark that

-1 -1

~j 0 ~i

=

~j 0 ~i • ~ij.

(1.8.5) If (6.,8) and (6.

1,81) are analytio equivalent then the atlasses of (6.,8) and (6.

1,81) are identical. In the present paper ar~lytic

equivalent Riemann surfaces are considered to be equal.

(1.8.6) Let

{Q

j ; j E

J}

be a family of open sets in r and let there exist an equivalence relation in

{(Q.,a.);

j E: J,

a.

E

Q.}

which

. J J J J

satisfies

(1.8.1),

(1.8.2) and

(1.8.3).

We set 6.'

:-:=

{C(Q.,a.)] ,

j E: J,

a.

E

Q.}

and define the mappings

J J J J

~! : Q. - 6' by~! (a.)

=

[(Q.,a.)] •

J J J J J J

Now {~!(Q!); j E: J, Q! c Q., Q! open} is a basis for a Hausdorff

J J J J J

topology in 6.'. MJreovere':= {(~!(Q.), ~1:-1)} satisfies

(1.3.1i)

J J J

and (1.3.2). So there is an analytic structure en on 6.' with 8n ~

e

f , and {Q.; j E

J}

together with the equivalence relation

J

in {(Q .,a.); j E: J, a. E: Q.} is part of the atlas of (6.1_,8").

J J J J

The following theorems are needed, we give them without details. Theorem

1.9

If (6,e) is a Riemann surface and n E 6., then there exists a complex function cp which is analytic in

6\{n}

and which has a pole of positive order in n.

A function <p is said to have a pole in n, if for every i for which 1t E: E_i,

cp 0

,7

1 has a pole in cp.(n). To prove theorem 1.9 we take MOon the

::l.. ~ T

components of 6. which do not contain n The component of n with the coordinate systems defined in i t is an ordinary Riemann surface.

The existence of a ~ on this connected Riemann surface is a well known result from the theory of Riemann surfaces. For closed surfaces it

fol-lows from the Riemann-Roch theorem, the. order of the pole being .to; g+1,

where g is the genus

([2]

p.;29). For open Riemann surfaoes see [9]

theorem 40.3.3 p. 199.

Theorem 1.10 (Riemannts mapping theorem).

I f (t.,e) is a Riemann surface and t!. is a simply connected topological

space, then (t!.,e) is analytic equivalent with either the unit disk, or

the complex plane, or the Riemann sphere.

By unit disk, complex plane and Riemann sphere is meant the Riemann

surfaces consisting of these spaces and as coordinates all one-to-one

analytic functions on them. For a proof of theorem 1.10 see

[8]

p.196-214.

2. Function theory in a Banach algebra.

2.1

Throughout this paper A denotes a Banach algebra. with identity

element e. For the definition and properties of Banach algebras we refer

to [10J. The most important concept for our theory is that of the

spec-trum. I f a E A, the spectrum a(a) is the set of all complex numbers X

such that a - Xe is singular, it is a non-void, compact subset of

r.

I f Q is a subset of

r,

M(Q) denotes the subset of A consisting of all

elements a with a(a)cQ. M(Q) is called the principal extension of Q.

Several properties of principal extensions of complex sets are discussed

If 0 is an open set, <I> a locally analytic function on 0, then the

principal extension of <I> (denoted M(<I») is a funotion on M(O) whioh can be defined by

where the integration is taken along a system of oontours in 0 which has winding number 1 at every point of (f(a). For all details of this definition (which is due to Dunford) see [6] seo. 5.2. or [5]seo.7.3; an alternative definition is given in (3). If <I>(~) is a rational function, <I>(C) =

cr(C-t

_{1 ) •••}

{C-C

_k

)(C-C

_k+1)-1 •••

(C-em)-1 ,

then M(<I» (z)

=

a(z - t1e) ••• (z-tke) (Z-Ck+1e)-1 ••• (z - tme)-1; M(<I» is defined on M(r\{Ck+1 t •••

,C

_m}).

Let Zo be the algebra of all local~ analytic functions on OJ M(ZO) := := {M(<I»; <I> E

Zg},

then M(ZO) is an algebra of continuous functions and the principal extension is an isomorphism of Zo onto M(ZO).

Zo is closed under local~ uniform convergence and so is M(ZO) and we have

rrM(<I>n) (z) - M(<I» (z)n ~ 0 r<I>n(C) - <I>(t)1 ~ 0

locally uniform~ on M(O) iff

loca1~ uniform~on o.

For proofs of the above facts as well as of the following

two

theorems, both due to Dunford, we refer to

[5]

or [6] lac. cit.

Theorem 2.2 (Spectral mapping theorem). If <I> is local~analytic on 0, a E M(o),then

(f (M(<I» (a» = <I> «f (a) )

where <I>«f(a»:= {CJ?(a); aE (f(a)} •

Theorem

2.3

(Composite function theorem)

As in

[3) ,

principal extensions and restrictions of principal extensions

are called strongly analytic functions. We mention some more properties of principal extensions.

(2.4) If ill E ZQ' a E M(Q), b regular, then M(ip) (bab-1) =

=

b(M(ip) (a» b-1•

(2.5) I f ip E ZQ' '1' E ZQ' a E M(Q),then (M(ip) (a» (M(W) (a»

=

= (M(W) (a» (M(ill) (a».

(2.6) We use the notation a(O) := U_zE c/(z). I f 0 is open, a(O) is open.

The proof of this faot is immediate. If 0 is open in A, cxE:a(O) then

a Ea(e.) for some exEO. As 0 is open, a + Xe EO for small

Ixr.

But as

(a + X) E a (a + Ae), (a + ~JEa(O) for small values of

I

AI.

Definition

2.1

Let 0 be a set in A, let F be defined on O. F is anal~tic on 0 if each

point aE 0 has an open neighborhood 0 such that there exists a locally

a

analytic function q> on a( 0a) with

F(z) = Id(q» (z) for zE (0

n

M(a(O_a

»).

It is this notion of analyticity that ccours in De Bruijn's funotion

theory (see

[3] ).

The set of all funotions analytic on 0 is denoted by ZO; Zo is an

algebra. For several properties of analytic functions we refer to

[3] •

The most important result on analytic functions is the identity theorem 2.8 which follows from the G-differentiability of these

Theorem 2.8 (Identity theorem).

If a is a oonneoted open set in A, if F E ZO' GE Zo and if F(z) - G(z) on some non-empty open set 0

0 c 0, then F(z) - G(z) on O.

;. StronglY schlicht funotions

Analytic funotions in A which are one-to-one do generally not share the nice properties of one-to-one analytic functions in

r,

(see

[3] ).

Therefore the theory of analytic manifolds over a Banach algebra rill be based on a class of one-to-one analytic functions whioh have several addi tional properties. This is the class of the strongly schlicht

functions discussed in this section.

Definition

3.1

A function F defined on an open set 0c A is called strongly schlicht, if it is strongly analytic on 0, whereas there exists a strongly analytic funotion G on F(O), such that G(F(z))

=

Z for all z E O. From this definition it follows that FCO) is open, see theorem

3.4

below. As it also follows that F(G(z)

=

z for all z E F(O) this implies that G is strongly schlicht.

Theorem

,.2

I f Q c r and g;, is schlicht (i.e. one-to-one and locally analytic) on Q, then M(g;,) is strongly schlicht on M(g). MCg;,) is a homeomorphism of M(g;,) onto M(g;,(g».

Proof

Let F := M(q;), G := M(W). By theorem 2.3 we have F(G(z»

=

z for all

z E: M(g); G(F(z»

=

z for all z E: M(iP(Q». As F and Gboth are con-tinuous and one-to-one, both are homeomorphisms.

Theorem

3.3

If F is strongly schlicht on 0,then there exists a schlicht function ~

on 0(0) such that F(z) ::: M(q; Hz) for all z E: O.

Proof

As F is strongly analytic on

a

there exists a locally analytic function ~

on aCO) such that F(z)

=

M(cf? Hz) for all zEd. Let Gbe the inverse of

F, then there exists a locally analytic function Win a neighborhood of

a(F(d»

=

cf?(a(d» such that G(z)

=

M(W)(z) for all Z E: F(O).

As G(F(z»

=

Z for z E:

a

theorem

2.,

yields W(~(,=»

=

I;; for CE: a(d)J

henoe, iP is schlicht.

Theorem

3.4

If F is strongly schlicht on 0, then F is a homeomorphism of d onto

F(d); and F(d) is open. Proof

Let cf? be the schlicht function on aCO) with F(z)

=

M(cf?)(z) for all

Z E: O. Let F₁ := M(cf?) then F₁ is a continuation of F to

M(a(O».

F

1 is

a homeomorphism by theorem 3.2, hence, 80 is F and F(O) ::: F

1(d) is open.

Rema~_

The manifold theory in [3J is based on a class of functions called

schlicht functions which are defined in the following way.

A function F defined on an open set

a

is called schlicht if it is

analytic on

a

and if there exists a function G, analytic on F(O) such

Obvious~, the requirements for a function to be stronglY schlicht are much stronger than to be schlicht. In the following example we show that even a function which is strongly analytic and schlicht according to the above definition need not be stronglY schlicht.

Example

3.6

Let A :=

r

x

r,

i.e. A "" {(t

1

,C

2); t1 E

r,

C

2 E

r};

ITCC1

,C

2)ff

=

=

max

(!c

₁

1,rc

₂t);

A(C

₁

,C

₂₎ ==

(AC

_{1 ,AC2) ;} (~1'~2) +

(C

₁

+t

_{2 )} ==

== (t₁+C1,t₂

+t

_{2 );} (t₁,t₂)(C₁,C_{2 )} == (t_1t1';2Ca!fO«C1,C₂

»c

{C1'C_{2 } •}

Let O:e {(C₁,C_{2), Re} t_{1 <0, Re} t₂

>O,r~1r~

Ic

₂

1h

F(z) := z2:a

(t~,t~),

so F is

strong~

analytic. F(O) ""

{(~1,t2);

rt

₁

f

f

lt₂1 , t₁ nor t

₂

~eal

and'"

° }.

0(0)

==

{C;

Re

~

O} •

~(C)

==

C

2 is not schlicht on

0(0),

hence F is not

strong~ schlicht.

Nevertheless F is schlicht, since the function G(z) == (-

-..R

1

,-..R

2), where

J,.

z ""

(C

1

,C

2) and

r

denotes the principal branch of the function

C

2

, is analytic on F(O) (since F(O) does not contain scalar points) and

G(F(z» == z (z EO).

We conclude this section with some useful results on strongly schlicht functions, the first of which should be compared with the previous example.

Theorem

3.7

Let 0 be an open set in A with

0(0)

e c O. If F is strongly analytic on 0 and schlicht on 0, then F is strongly schlicht on O.

Proof

The function F is the principal extension of a locally analytic function ~

theorem 3.1. If a₁ € a(O) a₂ E 0(0), a₁

f

a₂ then a₁e and a₂e are

different points in O. Hence,

F(a1e)

f

_F(a2e),

but since F(a₁e )

=

~(a1)e,

F(a2e )

=

~(a2)e. we also have ~(a1) ~ ~(a2)'

Theorem 3.8

If 0 is open inA;o(O)e cO; F strongly schlicht on 0, then o(F(O»ecF(O). Proof

Let ~ be the schlicht function on 0(0) with M(~)(z) :::: F(z) for z € O. Then o(F(O»

=

~(0(0». If a E o(F(O», then a

=

~(~) for some ~ € 0(0). So ~ eE o(O)e c a and F(~e) :::: ~(~)e =

ae

E F(a).

Theorem

3.9

If F is strongly schlicht on

a

and G is strongly schlicht on F(O), then G0 F is strongly schlicht on

a.

The proof is straightforward.

4.

The principal extension of (6,8)

In this section we describe De Bruijn's construction of the principal extension of a Riemann surfaoe. Let (6,8) be a Riemann surface,

8

=

{(~1'<Pih i E I }; Q. := <p.(~.). J. J. J.

Since the atlas of (6,8) (see definition 1.8) is a complete description of (6,8) in terms of

r,

it is suitable for principal extension to A. Let the relation~, the sets Q

ij (i € I, j € I) and the funotions ~ij

be as in definition 1.8.

Let F .. := M:(~.. ) defined on M(Q .. ). We now define a relation (again

J.J lJ lJ

denoted "') for pairs (M(Q. ),a.) with o(a.. )cQ .•

Definition

4.1

Let a. € M(O.) a. E: M(O .). We say that (M(O.), a.) ,...., (M(O .), a.) iff

:1. :1. J J :1.:1. J J

a. € M(O .. ) and F ..(a.)

=

a .•

1 1J 1J:1. J

Theorem 4.2

The relation

(4.1)

is an equivalence relation. Proof

As <P •• is the identity on 0., F .. is the identity on M(O.) (i € I).

:1.:1. :1. :1.:1. :1.

Hence the relation is reflexive. I f (M(O.), 80

1) '" (M(O .), a.) then a(a.) ::

:1. . J J J

=~.. (a(a.» by theorem 2.2. So a. E: M(O .. ) and F ..(a.) =(F .. 0 F ..)(a.).

:1.J :1. J J1 J:1. J J1 1J 1

By theorem 2.3 F .. 0 F .. is the principal extension of il2. 0 iP.j which

J:1. :1.J J:1. :1.

is the identity on O ..• So F ..(a.) = a .• This proves the relation is

1J J1 J :1.

symmetric. The transitivity of the relation is proved by another

application of the composite function theorem.

I t should be remarked that the functions F .. are strongly schlicht,

1J

hence they are homeomorphisms of M(O .. ) onto M( 0..).

1J J:1.

We denote the equivalence class containing (M(Oi)' a.) by [(M(o..),a.)].

1 :l.:1.

Let

The mappings gi : M(Oi)'"

M(t.)

(i € I) are defined by gi(a

i ) := := [(M(o..), 8..)

]Ca.

E: 111(0-1

».

:1. :1. :1. ...

Elements of

M(t.)

are denoted b,y the letters p, q, •••

Theorem

4.4

The family of sets {gi(Oi); i E: It 0i c M(Qi)' Gi open} is a basis of a topology in

M(t.).

Proof

Let pE (g.(O!)

n

g.(O!», then there exist 8.. and 8.. with a.E O! cM(Q.);

1 1 J J 1 J 1 1 1

a.E O! cM(Q.) and (M(g.), a.) - (M(g.), a.) Ep.

J J J 1 1 J J

Let ot! := 0i'nF.i(O!nM(Q ..

»,

thenpEg.(otnc(g.(o!)ngj(Ol».

1 J J J1 1 1 1 1 J

From now on M(6) is considered as a topological space with this topology.

Theorem

4.5

M(~) is a Hausdorff space.

Proof

Let pEM(6), qEM(l'.), p ~ q, (M(Q.), a)Ep, (M(Q.),b)Eq.

1 J

Now (M(Q.),a)

f

(M(Q.),b) and we will show that there exist open sets 0

1

1 J

and O! with aEO

i' c M(g.), b EO! c M(g.) and (M(Q.),y),j, (M(Q.),z) for

J 1 J J 1 J

all yEO! and all Z EO!. By considering neighborhoodbases of a and b we

1 . J

see that it suffices to prove that from a E M(Q .. )(n =: 1,2, ••• )

n 1J

lim a == a, lim F .. (a ) == b i t follows that F .. (a) exists;

11:""00 n l1:-"X' 1J n 1J

the continuity of F.. then implies that F .. (a) "" b. So we have to prove

1J 1J

that form lim a == a, a(a)c Q.. , a(a)c Q.it follows that o(a)c Q4J"

rr.oo n n 1J 1 ...

or that a - A.e is regular for A.E Q.\Q ..•

3. J.J

Let A.E Q.\Q .. , then ~.(A.) == [(Q.,A.)] == : 1t is a point of the Riemann

J. J.J 1 J.

surfaoe

(61

,e

₁₎

which we consider equal to (l'.,e) (see

1.8.4).

By theorem 1.9 there is an analytic function<p on 6

1

\{1t}

which has a

pole in 1t of order (k+1). Now <p 0 tV. is analytic in Q.\{A} and has a

3. 1

pole in A.; <p 0 ~. is analytic in Q .• Let b := F .. (a ), G.:== M(<p 0 tV.)

J J n 1J n J J

on M(Q .), then lim G.(F .. (a

»

== lim G.(b ) == G.(b).

J frIGO J J.J n frIGO J n J

Let G. := M(<p 0 ~.) on M(Q.\{A.} ) J M(Q .. ), then Gi(z) = G.(F .. (z» for

J. 1 3. 1J J 1J

Z E M(Q .. ); so G.(a ) =: G.(b ). Hence, lim G.(a) == G.(b).

1J 1 n J n ~ 1 n J

(C E Q.). J.

There exists a function 'l' analytic on Q. and ayE

r, y

~ 0, such that

3.

Let G := M(W) on M(Q.),then

1.

(s. _,\,e)k G. (s. ) == y (a. _ Ae) -1 + G(a. );

n 1. n n n

so

lim y-1 (s. _ Ae)k G.(s. ) _ y-1 G(s. ) ==

~ n 1. n n

== y-1(s. _ Ae)k G.(b) _ y-1 G(a.) == lim (a _ Ae)-1 == : c.

J n...oo n

We have c(s. - Ae) == (s. - Ae)c == e and this completes the proof.

Theorem

4.6

The mappings gi : M(Qi) - M(6) are homeomorphisms. For all i E I and

. h -1

J E I for whic g. 0 g. is defined, it is the principal extension on

J 1.

M(Q .. ) of <po 0 <p,:,1 •

1.J J 1.

The proof is immediate.

Definition

4.7

M(e) :== {(g.(M(Q.», g,:,1); i E I}. (M(6), M(e» is called the principal 1. 1. 1.

extension of (6,e).

We often use the notations and properties: M(Qi) == : 0.; g.(O.)== : S.;

1. 1 1. J.

-1 _ • f. -1 -1 ( )

g. - . . • F..: == g. 0 g. == f. 0 f. == Mq).. , where q). . is the

1 1. 1.J J 1. J 1. lJ l.J

. . - 1 - 1

schllcht functlon <po 0 <po ==~. 0 ~. on Q..•

J 1 J 1. lJ

Theorem

4.8

If (6,61) and (6₁

,e

₁) are analytic equivalent then the principal extensions are identical.

5.

Strong Banach Manifolds

In this section we give the definition of an analytic manifold belonging to A. Although this definition is independent of that of a Riemann surface, it is strongly supported by the analytic properties of

r

and their relation to A.

The main results in this section are the theorems expressing that the principal extension of ever,y Riemann surface is a strong Banach manifold and that ever,y strong Banach manifold is embeddable in the principal extension of a Riemann surface.

Let D be a Hausdorff space.

Definition

5.1

A (Banach) coordinate system on D is a pair (S,f) of which S is a non-empty open subset of D and f a homeomorphism of S onto an open subset f(S) of A.

Definition

5.2

A strongly ana~tio structure on D is a family T = {(S.,f.); i E I}

J. J.

of coordinate systems on Dt satisfying the following three conditions

I f S.

n

J. strongly

s.

f

¢

then

M(a(f.

(S.

n

S.»)

=

f.(S.

n

Sj) J J. J. J J . J . schlicht function on f.(S.

n

S.). J. J. J -1 and f. 0 f. is a J J..

I f a Banach coordinate system (Stf) satisfies the conditions (i) and (ii) below, then it is in T

(i) M(a(f(S»)::: f(S).

(ii) If (S

n

Si)

f~,

then M(o(fi(S

n

Si») ::: fi(S

n

Si) and f 0

f~1

is

a strong~ schlicht function on fi(S

n

Si).

I f one takes i

=

j in (5.2.2) it follows that for every coordinate system (S,f) in a strong~ ana~tic structure M(a(f(S»)

=

f(S).

Remark (5.2.5)

If (S,f) is a coordinate system and if M(a(f.(S

n

S.»)

=

f.(S

n

Si) and

1. l. l.

f 0

f~1

is

strong~

schlicht on it, then also f(S

n

S.) ::: M(a(f(S

n

S.»)

1. l.

l-and fl.' 0 £-1 is

strong~

schlicht. For, if f.(S

n

S.) ::: M(Q') and

1. l.

f 0

f~1

::: M(if?), then f(S

n

5.) ::: (f 0 f,:,,1) (M(Q'» ::: lVf(if?(Q'».

l. l. 1.

(theorem 3.2). Definition

5.3

A strong Banaoh manifold is a pair (D,T) where D is a Hausdorff spaoe a.nd T is a strongly a.nalytic struoture on D.

I f A :::

r,

then the definitions of Riemann surface and strong Banaoh manifold coincide.

The following theorem uses the notations of the previous section. Theorem 5.4

I f (6,8) is a Riemann surface, then (M(6), M(a» is a strong Banach manifold.

Proof

In view of theorem 4-5 we on~ have to prove that M(a) is a strongly analytic struoture on M(ll). I t is trivial that M(e) Sf~,t;isfies (5.2.1).

I f in

Mel)

we have S. n s.

:J

0,

then f. (S. n S.) ""

M(Q .. )

and f. 0 f,:,,1 ""

~ J ~ 1 J l J J 1

""

M(~

.. ) ""

M(~.

0

~,:,,1).

Hence,

(5.2.2)

is also satisfied; we turn our

1J J ] '

attention to

(5.2.3).

Let (S,f) be a coordinate system satisfying the conditions (i) and (ii) from

(5.2.3).

Let f(S) :: :

M(Q).

For indioes i such that S

n s.

~~,

let f(S

n

S.) :: :

M(Q!)

and let f. 0 f-1

= :

]. 1 ] , ] .

= :

M(~i) on

M(Ql).

We construot a mapping ~ of

Q

into ~1 (see

1.8.4).

For aCQ let f-1(ae) ( S.; then we set ljI(O:) :==

~.

(eIl!(a».

]. ]. ].

We first show that the value of ~(a) is independellt of i. Let also

r..

1(ae) € S.,

r.

0

r-1

= :

M(~!)

on

M(Q!)

:= f(S

n

S.). Then

g.(~!(o:)e)

==

J J J J J ]. ].

""

g.(~!(o:)e)

==

r-

1(ae). So (f. 0 g.)(eIl!(a)e)

=

~!(a)e.

But as f. 0 g.

=

J J J 1 ] , J J ] .

""

M(ljI~1

0

~.)

this yields

J 1

(<Ji~1

0 <Ji. 0 <P!) (0:)

=

eIl!(o:).

J ]. 1 J

Moreover we shall show that $ is one-to-one on Q. Let ~(a)

""

<Ji(~),

then

ljI.(<P!(a»

""

$.(<P!(~»

for some i and some j ( I, or

(~:1o

<Ji.)(<Pi'(o:» ""

]. ]. J J J ].

=

W!(~). Taking principal extensions and using theorem

2.3

we find

J

(r.

0 g.)(<P!(a)e) "" <P!(f3)e, hence

r-

1(ae) ::

f-1(~e)

and a::

~.

J ]. 1 J

We further show that

(ljI(Q),~-1)

(8,. Recall that 8₁

=

{T}(z.;.,~.);(~.,~.)(

8} 1 1 ]. 1

where T} is the analytic homeomorphism construoted in

(1.8.4).

Let

~(Q)n

T}(6.)

.;.~;

now cp. 0 T}-\ fjI maps ljI-1(<Ji(Q)

n

TJCz.;.» onto

]. 1 ].

(~.

0

'11-1

_)(~(Q)

_n

r(,r..

»,

and on the first set we have

~

= ljJ. 0

<P!,

]. . 1 1 1

-1 .

80 ~i 0 T} 0 ~= <Pi is sohllcht there. Since 8

1 is an analytio structure, axiom 1.3.3 yields (ljJ(O),</i-1) (8

1•

This, however, means that the open set Q occurs in the atlas of (~,e) and (Q,a) "'" (Oi,a

1.) iff (cp.]. 0 T)-1 0 ljJ) (0:)

=

ct ••1 Let Q = : Q.,1 , then

If g :"'" g., is the mapping M(Q) _ M(~) defined by g(a) := [(M(Q),a)] , ].

We complete the proof that (S,f)€ M(e) by observing that g(M(Q»

=

Sand that g-1

=

f. Suppose a E M(Q) and f-1

(&)

E S., then f-1(a)

=

[(M(Q.),

~ ~

(f. 0 f-1)(a)] • Since a is in the domain of definition of f. 0 f-1, a(a)

~ ~

is in Q! and (f. 0 f-1)(a) is the principal extension of

~i'

evaluated in

~ ::L

a; this means (M(Q),a)

~

(M(Q.),

(f. 0 f-1)(a» so gee) m f-

1

(a).

~ J.

Definition

5.5

Let (D,T) and (D₁,T₁) be strong Banach manifolds. The manifold (D,T) is called analytic equivalent to (D

1,T1) if there exists a homeomorphism h

of D onto D₁ such that T

1

=

{(h(S), f 0 h-1

); (S,f)E T }= : h(T).

If A

=

r,

then in view of theorem 1.1 this definition is the same as for Riemann surfaces.

Example

If A is either the open set Q c

r

or

r

itself and

e

=

{(E,~); E open, E c A, ~ schlicht },

then the principal extension (M(6),M(e» of the Riemann surface (Ate) is analytic equivalent with the strong Banach manifold (M(6),T) where

The principal extension of the Riemann sphere, (which is the one-point compactification of

r

with all schlicht parameters on it) plays an important role in the theory. For matrix algebras a detailed discussion of the properties of this generalised Riemann sphere is given in

[4] •

Since the open set of a coordinate system from a strongly analytic

structure is homeomorphic to the principal extension of an open set in

r,

there may be no arbitrarily small coordinate systems. So it is in

general not true tr.lB.t the strongly analytic structure on (D,T) induces a. strongly a.nalytic structure on every open subset of J.

In this respect strong Banach manifolds may differ from ordinary Riemann surfaces. The next theorem describes the situation. Theorem 5.6

If (D,T) is a strong Banach manifold, T

=

{(S.tf.);i E I }, D₁ is an

1. 1.

open subset of D; T₁ = {(So_1.

n

D₁, f._1. I_I S._1.

n

D₁); (S.,f.)_{1. 1.}

E

T } (f

I

S denotes the restriction of f to S), then (D

1,T1) is a strong Banach manifold iff T₁ c T.

Proof

Let Si

n

_D1

= :

Si1' f_i r Si

n

D₁ := : f

i1• I f T1 c T, then it is obvious that T₁ satisfies (5.2.1) and (5.2.2). As to (5.2.3), let (S,f) be a coordinate system in D₁ satisfying M(a(f(S») := f(S) for which

M(a(fi1 (S

n

Si1»)

=

_{f i1 (S}

n

Si1) and f 0

f~~

is strongly schlicht on

this set for every i with S

n

Si1

I

¢.

But then the same holds for T instead of T1, while S

n

Si := S

n

Si1' Therefore (S,f) E T, and since

S

n

D₁ := S, we have (S,f) E T

1, so T1 is a strongly analytic struoture. On the other side, if (D₁,T₁) is a strong Banaoh manifold, then for

every coordinate system from T₁: (D₁

n

S, f [ D₁

n

S) where (S,f) E T, we have D₁

n

S

n

Si

=

(D₁

n s) n

(D₁

n

Si) and (fl D₁

n

S) 0

f~1

:=

=

_{(f[ D1}

n

S) 0 (fil D1

n

Si)-1 for ever,y (Si,fi)ET. So, since T

1 and T are strongly analytic structures, application of (S.2.2)and (5.2.3) yields that T

1 c T. Remark

In the situation of theorem 5.6 we have T₁ = {(S,f), (S,f) E T, S c D₁} ,

but a subset of T having this form is in general not a strongly analytic str'J.cture on D₁; it may even be empty, whereas D₁ is not empty •

Definition

5.7

A strong Banach manifold (D,T) is said to be embeddable into the strong Banach manifold (D',TI) if there exists a mapping h : D - D' such that

h is a homeomorphism of D onto h(D) and h(T) := {(h(S), f 0

h-1 );

(S,f)ET} is a strongly analytic structure on h(D), sa.tisfying heT) cT'.

We shall now prepare the main result of this section, namely that ever,y strong Banach manifold is embeddable i. the principal extension of a Riemann surface.

Definition 5.8

Let (D,T) be a strong Banach manifold, where T

=

{(SI,f

il );iEI}.Let d.: •

~ ~

: • f?(S!). The atlas of (D,T) is the system of sets 0i together with

J. 1.

the equivalenoe relation for pairs (O.,a.) where a. E d. defined b,y

1. 1. 1. 1.

(O.,a.) '" (O.,a.) iff f~ 0 f1-1(a.)

1. 1. J J J J. 1.

(see 1.8).

=

a .•

J

It should be verified that this, in fact, is an eqUivalence relation. Moreover this relation has the following properties.

(5.8.1 )

If (O.,a) '" (O.,b), then a

=

b.

1. 1.

If (0. ,a.) ,.." (O.,a.) for some a. and a., then there exists a uniquely

1 1. J J 1. J

determined open set O.. with a. E 0 .. c O. and a strongly schlicht

1.J 1. 1.J 1.

function F .. of 0 .. into O. such that (0. ,z) ,..., (O.,F .. (z» for all

lJ lJ J 1. J l.J

Z E 0., and such that this O.. , moreover, has the property that there

1.J 1.J

is no z. E 0.\ 0 .. with (O.,z.),..., (O.,z.) for some z. EO,.

1. 1. 1.J 1. 1. J J J J

( ) f ,-1 'ly

In fact, 0 ..

=

f! 51

n

5~ and F" == f! 0 • on 0 ..• It easl.

l.J 1. 1. J 1.J J 1. 1.J

follows that F .. (0. ,) == 0 .. and that

F7~

== F ...

Sf and

J

the

If (Oi,ai)..f (OJ,a

j ), then there exist open sets

0;'

and

OJ

with

a. E O! c 0., a. E O! cO. such that for all z. E O!, Zj E O! we have

J. ]. ]. J J J ] . ] . J

(Oi'Z')

J

(O.,z.). Since f!-1(a.)

~

f!-1(a.) and D is Hausdorff,

]. J J ] . ] . J J

there are open sets Sil' and Sf with

n-

1(a.) E S'!I',

f~-1

(a.) €

]. J ] . ] . ]. J J

S*J.'

n

SJf ==

¢.

Now Oil == f!(Sil'

n

SJ); O! : == f!(S~

n

S!) have

].]. ]. J J J J

reqUired properties.

From the atlas of (D,T) we shall now construct a Riemann surface.

Let O. :== a(O.) for all i E If 0 .. :== a(O .. ) for all i E I, j E I;

] . ] . ].J J.J

let ~.. be the schlicht function on 0 i ' with F ..

=

M(~.. ).

].J J ]'J ].J

Definition

5.9

In the set {(Oi,ai); i € I, ai E 0i} the relation", is defined by

(0 . ,ai) ... (0j ,a .) iff W••(a.) == IX ••

]. J ] . J ] . J

This relation is an equivalenoe relation; in terms of Aand (D,T) it can be formulated in the following way :

(O.,ai)"" (0 .,a.) iff f!-1(a.e) == f!-1(a. e ).

]. J J ] . ] . J J

We define A :- {[(Oi,ai)] ; i E I,ai Eoil. (For the equiva.lence class to which (O,a) (a EO) belongs we use the notation [(O,a)].) We use small Greek types to denote elements of A. The mappings tPi : 0i - A

are defined by

Obviously, we have

-1 <Pi :=

cPi •

Theorem 5.10

The family of sets

{,I..

_~].(O!);_]. i E I, O!_{] . ] . ] .}cO., O! open1_J topology in A •

Proof

If 1tE("'i(O!)

n ....

(O!» then (O.,a.)E1t, (O.,a.)Enfor somea_i and CX

J,;

1 J J 1 1 J J

and ex. EO!; a. EO!; <.P •.(a.) = a.. I f0 '.' := O!

n

<.P.. (O!

no .. ),

then

1 1 J J 1J 1 J 1 1 J1 J J1

From now on we consider ~ to be a topological space with this topology. Theorem 5.11

Under the foregoing assumptions, A is a Hausdorff space. Proof

Let1tE~, P EA, 1t~p, (O.,a.)En, (Q.,a.)Ep; then<.p .. (a.) ~ a. and

1 1 J J 1J 1 J

(O.,a.e)

~

(O.,a.e).

Elf

(5.8.3)

there exist

0'

and Ot with a.e E O! cO.,

1 1 J J 1 J 1 1 1

aJ.e E at c O. such that for all z. EO!, z.E: Ot we have (O.,z.) ~ (O.,z.).

J J 1 1 J J 1 1 J J

I f O'! := O! 11 re,

or!

=

ot n

re then a(OI~) and a(O'j') are open and for all

1 1 J J 1

1:. Ea(O'!), t.Ea(O'!) we have (o.,e.) ~ (O.,t.). So the open sets

1 1 J J 1 1 J J

.... (cr(O'!» =: E! and .... (cr(OI!» ;:;l;! satisfy :nEl;!, pE E!, l;!

n

l;!

=

13.

1 1 1 J J J 1 J 1 J

Theorem 5.12

The set e₁ ;:; {(~i(Oi)'~i); i E I} is a family of coordinate systems on A satisfying (1.3.1) and (1.3.2).

Proof

The facts that the ~. (0.) are open in A, that the lp. are homeomorphisms, 1 1 1

and that U. E I ~.(O.)

=

A follow immediately from the definition of A. 1 1 1

(1.3.2) is satisfied since <po 0 <p,:,,1 ;:; <.P •. is schlicht on

~.(~.(Q.)

n

J 1 l J 1 1 1

n

~j(Oj». Theorem 5.12 and the remark concerning definition 1.4 yield the following result.

Theorem 5.13

There is a unique analytic structure 8 on A for which 8

1c 8 and (A,e) is a Riemann surface.

We consider the strong Banach manifold (M(A),M(a» (see section

4).

This is the manifold in which (D,T) will be embedded.

LetEi := ~i(Oi) for i E II a

=

{(Ei'~i)1 i E I UJ} where I

n

J ... ~;

OJ.' :- ~.(E.)_{J. J.} for i E J, then

For further notations see section

4.

Since points, sets, and parameters of (D,T) are now marked with dashes, no confusion with corresponding quantities in (M(A),M(a» can occur.

Definition 5.14

For each point pI E D we define

h(pl) := [(M(Oi),fj(PI»] if pI E S! (i E I).

If p' E S!

n

S!, then (M(O.),f!(pl»

~

(M(Q.), f!(p'» since f! 0 f!-1

J. J J. J. J J J ~

is the principal extension of tI? • and (f! 0 f !-1 ) (f! (p.»

=

f! (p'),

J.J J J. J. J

so h(p I) is independent of i and h is a mapping D -;. M(A). Theorem

5.15

The mapping h is an embedding of (D,T) in (M(A),M(a». Proof

The proof consists of three parts. First, we have to show that h is a homeomorphism of D onto h(D). Second, we must prove that h(T) :=

:= {(h(SI),f' 0 h-1); (S',f l ) E T } is a

strong~

analytic structure

on h(D). The third property of h which should be verified is the obvious fact that h(T) c M(a).

1) The mapping h is one-to-one; for, if h(p')

=

h(qt), p'ES!, qlESj, then (:M(Q.), n(pl»,..., (M(O.),f!(ql», hence

J. J. J J

(f! 0 f!-1 )(f!(pI»

=

f! (q I )

J J. J. J

i)

Moreover h is a homeomorphism. For, if SI is a neighborhood of pI, pIES!, then also S!

n

SI is a neighborhood of p' and

1 J.

h(pl) E h(S!

n

SI) • g.(f~(S!

n

SI» c h(SI).

J. 1 1 1

Hence, h is an open mapping. On the other hand, let S be an open set containing h(pl), pI E SI, then h(pl) E S

n

S. and pi E h-1(S n S.) =

1 1 1

-1( -1 (

»

-1() -1

=

f I g . S n S. ch S, so h is open.

1 1 1

2) We have Ui E I h(Si)

=

hen), so h(T) satisfies (5.2.1).

I f h(SJ) nh(S!)

:f.

¢,

then (fI 0 h-1)(h(Sn nh(S!)

=

f?(S! n

sO

=

J. J 1 1 J 1 1 J

=

M(a(f!(SI

n

Sn»,and (f! 0 h-1) o(f! oh-1)-1 ::: fl 0

n-

1

1 1 J J 1 j 1

is strongly schlicht.' This proves (5.2.2).

As for

(5.2.3),

let (S,f) be a coordinate system in hen) satisfying the conditions (i) and (ii) from (5.2.3) for h(T).

Consider (h-1(S), fo h). This is a coordinate system on n, we shall show that it is in T; then the proof will be completed, since then (S,f) E h(T). (f 0 h)(h-1(S» ::: f(S)

=

M(o(f(S»). ii) If h-1(S)

n

S!

:f.

¢,

then 1 f!(h-1(S) n 8!)

=

(fI 0 h-1)(S

n

h(S!» = 1 1 1 1 ::: M(o·«f! oh-1)(S n h(Sn»), ::L :I.

and (f 0 h) 0 f!-1 ::: f 0 (fl 0 h-1 )-1 is strongly schlicht since

1 1

(S,f) satisfies (5.2.3)(ii) for h(T).

Since T is strongly analytic structure, this yields that (h-1(S),f 0 h)E T. The proof is completed.

Remark

The embedding construction described above, when applied to

is minimal in the following sense. I f (6',8') is a Riemann surface such that (D,T) can be embedded in (M(6'),M(8')hthen (6,8) can be embedded in (6',8').

Theorem 5.16

1) I f (D,T) is ana~tic equivalent to (D',T'), then the atlasses of (D,T) and (D₁,T₁) are identical.

2) If (M(6),M(e» is analytic equivalent to (M(6'),M(8'»,then (6,8)

is ana~tic equivalent to (6',8').

Proof

1) If T == {(S.,f.); i E I} , then the atlas of (D,T) consists of the

. J. J.

sets f.(S.) and

strong~

schlicht functions f. 0

f~1.

If h is a

J.J. J ).

homeomorphism of D onto D' with h(T) == T', then the atlas of (D',T') has the sets (f. 0 h-1)(h(S.»

=

f.(S.) and strongly schlicht

). ). J. J.

. ( -1 ) ( -1)-1 -1 )

functJ.ona f. 0 h 0 f. 0 h == f. 0 f. • Furthermore, 2 is

J ). J J.

an immediate consequence of 1).

We conclude this section with an example showing that it is not true that every strong Banach manifold is ana~tic eqUivalent to some (M(6),M(8».

Example 5.17

Let A :=PxP; (see emmple 3.6) D:= Ax {1,2} ; A₁ :== Ax {1} ==

= {(a, 1); a E A} ; A,2 := _{A x {2} • The canonical mappings A - A1, A - A2} are denoted by g1 and g2' respective~,sog1(a)

=

(a,1); g2(a)== (a,2). 8₀ := {(Q,~); Qc P, Qopen, ~ schlicht on Q}.

To := {(O,F); ° = M(o), F ... M(q; ),(0 tI» € a_o }

T := {(g1(0),Fg1'1); (O,F)€ To}

U{C

g2(0),Fg;1); (O,F)€ To}.

We shall show that (D,T) is a strong Banaoh manifold. Exoept for the

fact that T satisfies (5.2.3), this is obvious. Let (S,f) be a

coor-dinate system on D satisfying conditions (i) and (ii) from (5.2.3)

for T. We shall prove that (S,f) €"T.

Let S

n

_A1

= :

51; S

n

A

2

= :

S2; f(S1)

= :

01; f(52)

= :

02. Since f

is & homeomorphism, 01 n ° 2

=

~"

Since (A

1,g1'1) and (A2,g;1) are in T,

condition (ii) yieldS: M(0(01»

=

01' M(0(02» ...

°

₂- Moreover,

condition (i) gives M{0{01 U 02»

=

01 U 02. Let Q

1 : ... 0(°1), Q2:= 0(02)' then Q

1

n

Q2

=

~ and M(Q1 U Q2) ... M(Q1)U M(Q2). This, however, is

impossible in

r

x runless 01 =

¢

or Q

2 = ~. So S c A1 or S c A2•

Again using that (A

1,g1'1) E T and

(A2,g~1)€

T, we find that either

f 0 g1 or f 0 g2 is strong~ schlicht; hence, (S,f) € T~

The construction of this section yields two disjoint copies of r for

a,

and we denote them by

r

1 and

r

2. The analytic structure on A also

contains parameters which are defined partly on

r

1 and partly on

r

2.

Furthermore, M(A) consists of the four components

r

1 x

r

l '

r

1 x

r

2'

r 2 x r

1, r2 x r2" Since D has only two components, (M(A), M(a»

:f.

(D,T).

Since (M(A),M(a» is the smallest principal extension in which (D,T)

can be embedded, (D,T) is not a principal extension.

6. The operational calculus in (M(A },M( 8»

In this section we first discuss some of the spectral and topo-logical properties of the principal extension of Riemann surfaces. The operational calculus which deals with principal extension of

ana~tic functions (see[6]ch. 5) is extended to funotions defined on a Riemann surface.

Let (~,e) be a Riemann surface where

e

== {(r.,~.); i E I} ,

~ 3.

let ~.(~.) ==:_{) . ) .} Q.• Now_J. (M(~),M(e» denotes the principal extension of

(6,e)

as introduced in sec.

4.

We first define the spectrum of a point of 111:(6).

Definition

6.1

Let p E M(6), let (M(Q.),a) E p, then

).

o(p) :==

{~~1

(a);o;C o(a)} •

The subset o(p) of 6 is called the speotrum of PI however, to justify this definition it is necesS&r,y to verify that the set

a(p)

is

in-dependent of i. If also (M(Q .),b) E P then b == F .. (a) and a(b) == q'f ••(o(a»

J ~J ~J

-1 .

by theorem 2.2. As ~i' == ~. 0 , . we have

J J J.

{,~1(O:)'O:Eo(a)}

:::

h~1(~); ~Eo(b)};

by this remark the independence of i and

a(p)

is established.

The spectrum of each point p E M(~) can be covered by one coordinate system of (h,e); also it is clear that

a(p)

is a non-void compact subset of 6.

Remark

For points on a strong Banach manifold (D,T) which is net necessarily a principal extension one can define the spectrum to be the spectrum of the corresponding point in the smallest principal extension in which

(D,T) can be embedded. If (6,8) is the Riemann surface constructed in sec.

5,

if p E D, then

a(p)

is a compact subset of h.

I f (S.,f.) E T, pES., then a(p) ={cp-:-1(a); a Ea(f.(p»} •

3. ~ ~ 3. 3.

Definition 6.2

Let n EIJ.; nEi::., then lte := [(M(Q.),(Jli(n)e)] •

~ ~

The points n8 are called the scalar points of M(6). Since the value of any analytic function in a scalar point a-e E A is scalar, the notion of scalar point of M(/::") is independent of i.

Obviously

aCne)

=

{n}.

Definition

6.3

If S is a subset of

M(IJ.);

then

a(8)

:= Up E S a(p). If i:: is a subset of /::,., then

M(i::) := {p; p E

M(IJ.),

a(p) c L:}.

The set M(i::) is called the principal extension of E in

M(IJ.).

The fact that the set Ee := {nelnEi::} is a subset of M(E) justifies the term extension.

The properties of principal extensions of subsets of 6 in M(/::,.) are analogous to those of principal extensions of complex sets in the Banach algebra (see [1J ).

Without proofs we mention:

M(i:: UE') ::> M(E) U M(EI ) ;

M(i::

n

E')

=

M(E)

n

M(EI);

i f E eEl, then M(L:) c M(L:I) ; a(M(L:»

=

E;

M(a(S»

::>

S.

These statements hold for all subsets E, E' of 6 and all subsets S of

M(l\.) •

MOreover, M(~) as defined in sec.

4

is the principal extension of ~; S.

=

M(Z.) for all i E I. 1. 1. I f Z c Z. then g.(M(ql.(Z»)

=

M(E). 1. 1. 1. Theorem 6.4

I f E c 6 is an open set, then M(E) is open.

Proof

Let P E M(E); (M(Q. ),a)E p, then En E. is open in ~ a.nd a(p)c E n2:

i

-1. 1.

Now M(ql.(L: n

L:.»

is an open subset of M(Q.) and a. E M(q'Ji(L: n

Z.».

1. 1. 1. 1.

So g. (M(ql.(E

n

E.») =: Mer.

n

2:.) is an open subset of M(Z) containing p.

1. 1. 1. 1.

Henoe M(E) is open.

For the definition of strong spectral continuity in the following

theorem we refer to [1J •

Theorem

6.5

If the algebra A has the property of strong spectral continuity and

if r. is closed in 6, then M(Z) is closed.

Proof

Let p E MeL:) (n

=

1,2, ••• ), p - p (n

_00).

We shall prove that

n n

p E M(l.::). If (M(Q.) ,a) E p, then from a certain n onward there exist

1.

elements an in M(Qi) such that

Pn = [(M(Qi) ,an) ]

and a - a (n - co). For such a we have a(a ) c m.(l.::

n

E.), and the

n n n""1. 1.

latter set is relatively closed in Q.• Sinoe aCa ) and a(a) are in Q.,

1. n 1.

and by the strong spectral continuity, a(a

n) _ a(e.) (n - 00) in the

Hausdorff - Frechet sense, we have

a(a.) c ip.(L:

n

l.::.) and hence p E M(L:).

Theorem 6.6

If S is open in M(6), then o(S) is open in 6.

Proof

Let n€o(S), then there exists a p € S with nE:o(P). I f (M(Q.),a) E: p

~

then there is an a E: o(a) such that

~~1(~)

== 1t. f.(S

n

Si) is open and

~ ~

contains a. o(f.(S

n

S.» is open in r(by sec.

2.6),

hence

~ ~

~~1(a(f.(S

n

S.») is open in 6. The latter set, however, is a subset

~ ~ ~

of o(S) and it contains 1t.

We now proceed to the definition of the principal extension of analytic fu.'1ctions on 6.

Definition

6.7

*)

Let L: be an open set in L'.I; let X be a locally-analytic r-valued function on E. If p E: M(E); (M(Q.),a) E: p then o{p) c (L: nE.).

~ 1

MOreover X. := X0 <:p:-1 is locally analytic on the open set q>.(E

n

L:.).

~ 1 1 J.

Let K. := M(X.) and set k(p) := K. (f. (p») :: K. (a).

~ 1 ~ 1 ~

then b

=

F ..(11.) where F ..

=

M(il> .. )

=

M(m. 0 q>71).

J.J ~J ~ J J ].

then in a neighborhood of o(e.) we have :

If also (M(Q.),b)E:p,

J

-1 I f X_j :::: X0 q>j ,

By theorem 2.3 this yields (K. 0 F .. )(a)::: K.(a), hence K.(b) == K.(a).

J 1J ~ J J.

So the value of k(p) is independent of the chosen coordinate system.

*)

_{In the present paper we only deal with A-valued functions on a}

strong Banach manifold. However, a theory of analytic mappings of one strong Banach manifold into another one can be given in a completely analogous way.

The above procedure defines an A-valued function k throughout M(E).

k is called the ;grincipal extension of X and denoted by M( X). It is

clear that k 0 f:-1 on f.(M(L:)

n

S.) is the ordinary principal

~ ~ ~

. -1 ( )

extens~on of X0 ('jli on ('jli E

n

L:

i •

The following theorem is the generalization of Dunford1s spectral

mapping theorem (sec. 2.2) for principal extensions of functions

defined on a Riemann surface.

Theorem 6.8

If X is locally_ analytic on I: cl\; k:

=

M( X) and p E M(L:),

then a(k(p» • x(a(p».

Proof

If (M(Qi),a)Ep, then k(p) = K

i(a) (in the notation introduced above).

By theorem 2.2 we have

Remarks

If the function Xis constant, x(n) _ A, then k(p)

=

Ae. Obviously

f.

=

M«'jl.) for all i E I.

~ 1.

Let L: cl\; let ZL: denote the ring of the locally analytic r-valued

functions on L:; let M(~) := {M(xh X E ~}, then we have

Theorem

6.9

M(~) is a ring of continuous functions, and principal extension

X - M(X) is an isomorphism of ~ onto M(Z'L;). Proof

It is easy to verify that principal extensions are continuous, that

onto its principal extension is a homeomorphism. Since M(X1)

=

M(X2)

implies X1(n)

=

a(M(X1)(ne))

=

~(n), it is an isomorphism.

We shall now discuss some topological properties of the rings Z~ and

( ) { } . -1 -1 ( )

M ZL: • I f _{Xn; n}

=

1,2,... c _{ZL: and J.f Xn} 0 <Pi - X0 <Pi n - oc

locally uniformly on q>.(~.n~) for all i E I for which the latter

J. 1.

set is non-empty then XEZ~. In this case M(X

n) and M(X) are in M(Z~)

whereas M(X )_n 0 g. - M(X)_J. 0 g.(n -_1. 00) locally uniformly on f.(S._{1. J.}

n

M(~)).

On the other hand we have the following

Theorem 6.10 If {k

n; n

=

1,2, ••• } E M(~) and if k is defined on M(~) in such a

way that k_n 0 g. - k_J. 0 g. (n -_1. 00) locally unifonnly on f. (S.

n

M(~))

1. 1.

for all i E I for which the latter set is non-empty, then k E M(Z~).

If, moreover, k_n

=

M(X ) (n_n == 1,2, ••• ); k = M(X) then

X_n 0 cp:-1_ X_J. 0 cp:-1 (n -_1. 00) locally uniformly on m._YJ.

(~.

_1.

n

~).

Proof

First we remark that f.(S.

n

M(~))

=

f.(M(~.

n

~))

=

M(m.(~.

n

~)) and

J. J. J. J. Y1. J.

that the functions X 0 <p:-1 are defined on

cp.(~.

n

~).

Their principal

n 1. J. J.

( ) -1 .

extensions are k 0 g., hence b.Y sec. 2.1 the sequence X 0 m. J.S

n J. n TJ.

locally uniformly convergent on cp. (~.

n

~). The limit is therefore a

J. J.

locally analytic function on cp. (L:.

n

L:). We denote this limit by XJ.'

J. J.

and we prove that (X. 0 <p.)(n) == (X. 0 cp.)(n) if nE~.

n

L:.

n

L:.

J. J. J J J. J

Now (X. 0 cp.)(n)

=

lim (X 0 m":"1 om.)(tt)

=

lim X (n).

J. J. n_oo n TJ. YJ. n-.oo n

This limit, of course, exists as lim_{n_oo} k (ne) exists and is scalar,_n

and k (n e) = X (n ) e • Likewi se (X. 0 m. )(n) == lim X (n). If we

n n J Y J n...:= n

define a function Xon ~ by setting X(n) = (X. 0 l'j).) (n) (n E: ~ n E.)

then X E:

7~

as X0 <p:-1 = X. is locally analytic. Let k* := M(X) then

1.: ]. 1

( -1) -1 -1 ( )

k* 0 g.

=

M X<) cp. • As 'If 0 <po .... X 0 <po n .... 00 locally uniformly

1 ]. '~]. ].

on <po(E

i

n

E) we have k 0 g ... k* <) g. (n .... (0) locally uniformly on

]. n ] . ] .

f.(S.

n

M(E».

80 k*

=

k and the proof is completed.

J. ].

Definition 6.11

An A_valued function k defined in p E: M(~) is called analytic in p, if there is a neighborhood 8 of p such that k is defined on S and that the restriction of k to 8 equals the restriction to 8 of a function from M(Za(S». A function k is called analytic on a set 8

if it is analytic in ever,y point of S.

It should be noticed that this definition is the same as for A .... A functions : a function is analytic means that it locally equals the principal extension of an ordinar,y analytic function. AmJther obvious way, however, for defining analyticity on manifolds is calling k

analytic if k <) g. is analytic whenever it is defined. As principal

J.

extensions themselves are defined by means of local coordinates, it is no surprise that both possible definitions are equivalent. This is expressed in the following

Theorem 6.12

I f S is an open set in M(~); k an A-valued function on S, then k is analytic on S iff for ever,y coordinate system Si with S

n

Si

f.

~ k <) g. is analytic on f.(S n S.).

]. ] . ] .

Proof

Let k be analytic on S, S

n s.

I~; we shall show that k 0 g. is

J. ].

analytic on f.(S

n

S.). Let a E f.(S.

n s);

p := g.(a). Now there

] . ] . ]. 1 J. .

exist E a.nd X such tha.t a(p) cE, XE: ~ a.nd k

=

k

analytic b,y the construction of g. on f.(M(il)

). 1.

on f. (S

n

S.).

1. 1.

conclude that k 0 g. is analYtic

1. .

On f.(M(E)

n

S.) is k

1 0 g. stronglY

1. 1. 1.

M(ZE). As k₁ 0 gi coincides with k 0

On the other hand, let k 0 g. be analytic wherever they are defined.

1.

Let pES, (M(Q.),a) E p. As k 0 g. is analytic in a there exists an

1. 1.

M(Q) n f.(S.

1. 1.

P : g.(M(Q»

1.

open set Q and a function X such that a(a) c Q c Q. and that X is

1.

analytic on Q whereas K := M(X) coincides with k 0 g. on

).

n

S). So k coincides with K 0 f. on the neighborhood of

J.

n

8. As K0 f.

~

M(X 0

~.)

on

~-i1(Q),

K 0 f. E M(Z

-1(n»

1. 1. 1. ~i ~~

and k is analytic.

We confine ourselves to one theorem on analytio functions and some applications.

Theorem 6.13 (Identity theorem).

If k and k* are analytic on a domain S c M(~) and if k

=

k* in a non-void open subset S1 of S, then k

=

k* ever,rwhere in 8.

Proof

Sinoe the functions analytic on 8 form a ring, it obviously suffices to consider the case in which k*

=

O. The set S2 of the points p with the property that k

=

0 in a neighborhood of p is an open subset of S and it is non-empty as S1 c S2. We shall prove that also S\S2 is open. Let q E S\ 8

2• then in every neighborhood of q there are points where k is unequal zero. Let (M(a.),a) E q; now K := k 0 g. is analytio in

1. 1.

f.(S.

n

S), whereas a E f.(S.

n

S) and in every neighborhood of a there

1. 1. 1. 1.

are points b with K(b) ~ O. B,y theorem 2.8 it follows that a has a neighborhood 0 c f.(S. n s) such that every point c of 0 has the

1. 1.

property that K does not vanish identically on any neighborhood of c.

Hence, the same j s true for q

=

g. (a) and K0 f. - k.

As an application of the previous theor,y, we conclude this section with the discussion of some topological properties of principal extensions. The following theorem holds even if A lacks strong spect:ral continuity. Theorem 6.14

Let (61,61) and (6,e) be Riemann surfaces. If ~ is open and closed in 6 then M(6₁) is open and closed in M(6).

Proof

Obviously, M(6₁) c M(6). The function X defined by x(n) := 0 (n € 6₁), x(n) := 1 (n € 6\6₁) is analytic. Let k:== M(X); now p€(M(6)\M(Li

1» iff 1E a(k(p» == x(a(p», but in the present situation this is equiva-lent to k(p) ~ 0; hence, M(6)\M(6₁) is open.

Corrola;ry 6.15

If S is a domain in M(6) then a(S) consists of finitely many components and for ever,y p € S, a(p) has non-empty intersection with each of these

components. Proof

Let ~ be a component of a(S); ~* := a(S) \

r.,

then S

n

M(~*) and S\M(~*) are disjoint open sets by the preceding theorem; since S\M(~*) is

non-empty, we conclude S

n

M(~*)

=

¢.

Hence, for ever,y p € S, a(p)n~

t

¢

and this holds for ever,y component of o(S). Since spectra are compact, there are only finitely ma~ components.

Remark

This corrolar,y also is an immediate consequence of the identity theorem. We consider the question whether M(6) is connected or not. By con-sidering manifolds of the form (M(Q),M(e», where Q is open in

r

and e consists of all schlicht functions in Q, it is already clear that if 6

is connected, M(6) need not be. The following lemma serves to yield the

manifold analogue of theorem 6.1. from [1] •

Lemma. 6.16

The principal extension of the Riemann sphere is connected.

Proof

Let (6,8) denote the Riemann sphere; t:; can be represented by

r u

{oe}

_•

Let P1 ,P2 E M(6). As a(P1) is contained in one coordinate system of e,

there is a point n₁E

r

wi th ~

1-

a(P1). In

e

there is the coordinate

system

(~1

,CP1) where

~1

:= t:;\

{~}

and CP1(I:;) == (I;; -

~

)-1 if I;; E r \

{"'1 }

and '1(00)

=

0_ _{So we have Q1} :== CP1(~1) ==

r,

a(P1) c ~1' P1 E M(~1) == S1'

Hence (M(Q1)' f₁(P1» E P1- Starting with P2 we find ~'~2'CP2 analogously,

Take nEr such that 1t

-I

~' 1t

-I

~' then ne E M(6) and 1te E S1 n S2'

Now M(Q1) == A and A is connected, so there is a continuous mapping w₁('t') of [0,1] into A with w₁(0) == £1 (P1)' w

1(1) == f1(ne) == (n - 1t1)-1 e,

Analogously we find w

2

(

't), Now the mapping of

[0,1]

into M(6) defined by

{

g1(w1 (21: ) ) (0 E; 1: E;

t)

g(1:) : ==

. g2(w₂(2 - 2't'»

(t

< 't E; 1)

describes a continuous curve in M(6) joining

g(O)

== P1 and

g(1)

== P

_2,

Theorem 6.11

I f (6,8) is a. Riemann surface a.nd 6 is simply connected, then M(6) is connected,

Preof

By Riemann's mapping theorem(theorem 1.10) 6 is analytic equivalent with

either the unit disk or the complex plane or the Riemann sphere.

By theorem 4.8 (M(/:;),M(e» is the principal extension of one of these Riemann surfaces. In eaoh case M(/:;) is connected.

Technological University Eindhoven, The ~retherlands.