Seminarberichte Nr. 32

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Mathematik und

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Seminarberichte aus dem Fachbereich Mathematik der FernUniversität

32 – 1988

der Mathematik (Hrsg.)

Seminarbericht Nr. 32

Valuations of aomposition algebras

Holger P. Petersson

Congruenae relations on totalZy aonvex spaaes

Dieter Pumplün and Helmut Röhrl

La mesurabiZite de la famiZZe des droites

inaidentes saus un angle aonstant de Z'espsaae Eua Zidien E

3

Giulio Santoro

T0 -objeats and separated objeats in topoZogiaaZ aategories

Sibylle Weck - Schwarz

PositiveZy aonvex spaaes II

Andreas Wickenhäuser

1 -

7 -

- 39 -

- 45 -

- 53 -

Holger P. Petersson

Fachbereich :viathematik und Informatik Fern Universität

Lützowstraße 125

D-5800 Hagen, \Vest Germany

Non-archimedian valuations on composition algebras over the field Q of rational numbers have recently been studied in Balachandran, Rema and Sa- tyanarayanamurthy [1]. lt is the purpose of the present not~ to propose a different approach to the subject, leading at the same time to more general results and to shorter proofs as well.

1. Let k be a field. All k-algebras are supposed to be nonassociative with unit. By a (multiplicative) valuation on a k-algebra A we mean a map v, x H-

Jxl ,

from A to the nonnegative reals satisfying the usual conditions (Vl)

lxl =

0 {:::::::}

x =

^0,

(V2)

lx +

YI ~

lxl +

IYI ,

(V3)

Jxyj = lxl

IYI

for all x, y E A . For A to admit a valuation it is clearly necessary that it contain n_o divisors of zero (hence, in the finite-dimensional case, that it be a division algebra). Every valuation of A canonically induces a valuation on k.

2. Conversely, let k be a valued field with completion

k

and suppose A is a central simple k-algebra of finite dimension. We ,vish to set up a bijection between the valuations of A inducing the given valuation on k and the valuations of A 0k

k

inducing the given valuation on

k .

On the one hand, if we start with a valuation w of A 0

k

inducing the given valuation on k , restriction yields a valuation w1^A of A inducing the given valuation on k. On the other hand, if we start with a valuation v of A inducing the given valuation on k and denote by jv : A. --+

A

the canonical imbedding into

the corresponding completion. the composite map k - A - A extends to a homomorphism

k - .4 .

^giving

A

the structure of a k-algebra. Hence

Ju

extends to a homomorphism

)u :

A 2

k - A,

which, by central simplicity, is injective and so may be used to pull back the rnluation of

A

to yield a

\·aluation v

z l·

^{of A 0}

k

inducing the given valuation on

k.

3. Proposition Let k be a iialued field with completion

k

and A a crntral simple k-algebra of finite dimension. Then the assignments v ^1--+

v@k,

w 1--+ WjA defined in 2. yield inverse bijections between the set of valuation!

of A. inducing the given i•aluation on k and the set of valuations of A '8k k inducing the given valuation on

k .

PROOF. The relation ( vG

k

)1A

=

^v being straight forward to check, it remains to prove wlA 0

k =

^{w. Since A} has finite dimension, w is complete, so the natural map A. - A0

k

extends to a homomorphism :.p preserving valuations, as shown in the diagram

~

~ )wlA

y ~

~

- - - ( A 0 k , w ) ,

(.4., w1A) being the completion of (A., w1A) Hence )wi,1 is an isomorphism ,vith inverse r..p, and the assertion follows.

Proposition 3 has the following immediate application, generalizing

[l,

Lemma

:J.l].

4·. ·Cörollary Let k be a valued field with completion

k

and C a central composition algebra over k with norm n . Then the valuation of k extends to a valuation of C if and only if C 0kk is a division algebra ^1. In this case, such an extension is necessarily unique, given by the formula x ^1-+ \n(x)\½, and preserves automorphisms as weil as anti-automorphisms of C ; in particular, it preserves conjugation. lt is respectively archimedean, non-archimedean, discrete if and only if the valuation of k has the corresponding property .

1The author is indebted to W. Scharlau, who suggested this simple but important result during a bycicle ride to the mathematics department of the University of Münster in the fall of 1971.

PROOF. Proposition ;3 allows us to assume, whenever necessary, that k is complete and so is non-archimedean or agrees with R or C. In any event, we may either pass to the quadratic Jordan algebra associated with C and invoke the valuation theory developed in [4], particularly Satz ,5.1 and the subsequent Bemerkung 2, or use a weil known theorem of Springer [6]. to conclude the proof.

5. Remark (i) Granting the obvious adjustmends of Proposition 3 to the Jordan setting, the argument given above carries over verbatim to finite- dimensional absolutely simple quadratic Jordan algebras, producing a result which is completely analogous to Corollary 4 and contains the corresponding statement for finite-dimensional central simple associative algebras as a spe- cial case. \Ve omit the details.

(ii) In [l] only valuations preserving conjugation were considered. According to Corollary 4, this restriction is a fictitious one.

(iii) lt is not diffi.cult to obtain a version of Proposition 3 for non-associative algebras which may not contain a unit.

(iv) The fact that there exists at most one extension of the valuation of k to a valuation of C is originally due to Eichhorn [2, Satz 10].

( v) The follmving statement generalizes a result announced in [l].

6. Corollary Let C be an octonion algebra over a number field k. Then a non-real valuation v on k does not extend to a valuation of C.

PROOF. Indeed,

k

being either the field of complex numbers or a local field ,vith finite residue field, it is a standard fact that there are no octonion divi- sion algebras over

k.

Hence Corollary 4 applies.

7. We now turn to a question that has been discussed in [1] at length:

Given a prime number p and a quaternion algebra D over Q, what does it mean that the p-adic valuation of Q extends to a valuation of D ? Below we will g1ve a quick answer to this question by using Corollary 4 and the theory of local symbols as developed in Serre [ 5, Chap. XIV ] ^2• Adopting the usual notation, we let ( r, s), for non-zero rational mimbers r, s, be the rational quaternion algebra with norm

(1,

-r, -s, rs}

=

^{x2 - ry2 - sz2}

+

^rsw2•

(This seems to agree with the algebra D(-r, -s) in [l].) On the other hand, we have the p-adic symbol (r,s)p E {±1} [5, XIV§ 2, p. 21.5, with n

=

2The author, who originally had proceded in a slightly different manner, is indebted to W. Scharlau and M. Schulte for having drawn his attention to this.

2 ], \vhich is -1 if Qp does not split the quaternion algebra (r,s) (i.e., by Corollary --1, if the p-adic valuation of Q extends to a valuation of ( r, s) ) and 1 otherwise ( [5, XIV Proposition 7] and [:3, .57:9]). Similar to [1], we may assume, whenever necessary, that

D

=

^{(m,n) or} D

=

(m,pn) or D

=

(pm,pn), where m, n are integers not divisible by p [3, 57:10].

8. Suppose now that p is odd. \iVriting non-zero integers m, n as

c, I ß I

m=pm, n=pn,

\vith n, .8 E Z non-negative and m', n' E Z not divisible by p, we can express (m, n)p via

I I

( m, n )p

= (

-1

tß

~

(

~

t (

m )ß

p p

as a product of Legendre symbols [5, Chap. XIV § 4, p. 218]. Combining this with 7., we conclude

9. Corollary [1, Theorems 3.7, 3.11] Let p be an odd prime and m, n integers not dii,isible by p. Then the p-adic valuation of Q does not e:cte nd to a valuation of the quaternion algebra (m, n). lt extends to a valuation of the quaternion algebra (pm, pn) if and only if

10. We are left with the case p

=

2. Fixing odd integers m, n, we have

m-1 .!!.:l

(m,n)z

=

^{(-1) 2 2}

by [löc. cit., p. 219] and conclude

11. Corollary [1, Theorem 3.5] Let m, n be odd integers. Then the 2-adic valuation of Q extends to a valuation of the quaternion algebra ( m, n)

if and only if m

=

ⁿ

=

:3 rnod 4.

Since the 2-adic syrnbol rnay be viev,·ed as a symmetric bilinear form on the Frvectorspace

Q; /Q;

^2, we finally obta.in, m, n E Z still being odd,

m2-l m-1 n-1

(m, 2n)z

=

(m, 2)2(m, n)z

=

(-l)_s_ (-l)-2 -2

from [loc. cit.]. whence the 2-adic rnluation of Q extends to valuation (m, 2n) if and only if either m

=

^{3, .S} mod 8 or m

=

ⁿ

=

^:3 mod 4 (but not both). vVe thus end up with the following result.

12. Corollary Let m, n be odd integers. Then the 2-adic valuation of Q extends to a valuation of the quaternion algebra (m, 2n) if and only if one of the following conditions is fulfilled.

(i) m

=

^{:3 mod 8 and n}

=

^{1 mod 4.}

{ii) m

=

^{5 mod 8.}

(iii) m

=

7 mod 8 and n

=

^{3 mod 4.}

Since (2m, 2n) ~ (2m, -4mn) ~ (-mn, 2m) by [3, 57:10], Corollary 12 ag- rees with the theorem stated without proof in

[1,

p.

118].

References

[1] V. K. Balachandran, P. S. Rema and P. V. Satyanarayanamurthy.

Nonarchimedean valuations on rational composition algebras. J. Math.

Phys. Sei. 22 (1988), 101-1:30.

[2] W.

Eichhorn. Über die multiplikativen Abbildungen endlich-dimen- sionaler Algebren in kommutative Halbgruppen. J. Reine Angew. Math.

231 (1968), 10-46.

[3] 0. T.

O'Meara. "lntroduction to quadratic forms ". Springer-Verlag, Berlin-Heidelberg-New York, 1963.

[4]_

H. -P. Petersson. Jordan-Divisionsalgebren und Bewertungen. Math.

Ann. 202 (1973), 215-243.

[.5] J. P. Serre. ';Corps locaux". Hermann, Paris, 1968.

[6] T. A. Springer. Quadratic forms over fields with a discrete valuation.

Indag. Math. 17 (1955 ), 352-362.

Holger H. Petersson

Fachbereich Mathematik und Informatik FernUni vers ität

LUtzowstr. 125

0-5800 Hagen, West Germany

CONGRUENCE RELATIONS ON TOTALLY CONVEX SPACES.

Dieter Pumplün and Helmut Röhrl

Dedicated to Professor Dr. M. Koecher on the occasion of his 65 birthday.

ABSTRACT: Let X be a totally convex space and !et

~

be a congruence relation on X. In section l we give a simple and complete description of the congruenc:e c:lasses that are c:ontained in the interior. Sec:tion 2 deals with those c:ongruenc:e c:la.sses, which are not contained in the interior, and some general results are derived.

Sec:tion 3 gives a detailed desc:ription of the latter c:ongruenc:e c:lasses in case X is separated. In section 4 we enumerate all c:ongruenc:e relations on the free totally convex space on two generators. The last section gives a complete charac:terization of all twc>generated totally convex spaces and cloaes with a pictorial representation of eac:h of them.

1980 MATHEMATICS SUBJECT CLASSIFICATION: Primary 46 Ml5, 46B20

KEY WORDS AND PHRASES: Congruence Relation, Totally Convex Space, Banach Space.

§0 INTRODUCTION

Totally convex spaces were defined in [PRl] as

n

-algebras C satisfying a projection axiom and a big barycenter axiom. Here,

n

is the set of all sequences ( ai : i E N), ai E K for i E N, K

=

R or C, such that

E

^{lail ~ 1 holds.} If the effect of (ai : i E N) on the

i

sequence ( Ci : i E N) in C is denoted by

E

^aiCi then the projection axiom reads

i

where ~ is the Kronecker symbol, and the big barycenter axiom is

In this paper we investigate congruence relations ~ on totally convex spaces. The partitions induced on a totally convex space C by the various congruence relations are precisely the partitions of C into the fibers in C of the various surjective morphisms with domain C.

In section 1 we prove, quite in general, that the Banach space S(C/~) associated with the totally convex space C/~ is canonically isomorphic with S(C)/V where V is the subvector space generated by the set {0c(x) - Oc(Y) : x~y }. This implies that, for

0 0 0

any ~-congruence dass A and any a E An C, the isomorphism °'=:

c-

O(S(C)) maps

0 0

An C onto (°'=(a)

+

V)n O(S(C)).

In section 2 we define facets (sides) of a totally convex space C as convex (maximal convex) subsets of the boundary bdyC of C. We prove: Let a1 , • • • , an E bdyC such that the ccngruence dasses cf a1 , · · • , an are facets; then either the congruence classes of all points of the dosed simplex

<

a1 , • • • , an

>

are facets or the congruence classes of all points of the open simplex

<

^a1 , •••,an

>

⁰ fail to be facets. Similar statements, and more, hold for sides rather than facets.

Section 3 deals with congruence relations cn separated totally convex spaces C.

Since such spaces can be imbedded into the Banach space S(C), one can associate with each point a E C a real vectors space V ^a• lt is the smallest real vector space such that the congruence dass Aa of a is contained in a

+

V a• We show: If F is a facet of C such that for some internal point z of

F, Az

~

F

holds, then for all internal points a of

F,

( a

+ V

a)n inF ~

Aa,

where inF is the set of internal points of

F.

In section 4 we enumerate all congruence relations on the free totally convex space

F;

on two generators. Section 5 is a listing of all two-generated totally convex spaces.

They are obtained by passing to the quotients of

F;

modulo the congruence relations described in section 4. This means that one knows explicitly all mutual relations of two elements in an arbitrary totally convex space. A visual presentation of these spaces and cf the ccngruence relations generating them is given in the appendix.

§1 CONGRUENCE RELATIONS ON TOTALLY CONVEX SPACES:

THE INTERIOR.

In [PR3], 1.5 and 1.6, the following was proved (for notation see [PR!], (7.7), and [PR2], (10.1)):

(1.1) Lemma:

0

(i) Let C be a totally convex space and denote by C its interior. Then the universal

A O ~

morphism

°'= :

^{C-+ 0 o S(C)} induces a natural isomorphism C :!! 0 (S(C)).

(ii)

5 :

Ban₁ -+ TC, assigning to any Banach space its open unit ball is a left adjoint of S: TC-+ Ban1.

Both results

will

be important for the following investigations. (1.1), (ii), together with [PRl], (7.7), implies that S preserves limits and colimits, in particular congruence relations and coequalizers ( canonical projections).

(1.2) Proposition: Let ~ be a congruence relation on the totally convex space C and denote by

K(

~) the subvector space of S(C) generated by the set {

°'=

^{(x) - 4}

^{(y) :}

^x,

^y

^E

C and x~y}. Then K(-) is a closed subspace ofS(C) and the quotient map 11': C-+ C/~

induces an isomorphism

S(C)/K(~) :!! S(C/~).

0

Moreover, the K-vector space K(~) is generated by 4(Cnker(~)); where ker(~) :=

{x:x E C and x ~0} ( cp. [PRl], ( 4.2)).

Proof: Let E:= {(x,y):x,y E C and x~y} ~ C

x

C, which is a subspace of C

x

C, and denote by Pi:E - C the restriction of the canonical projections C x C - C to E, i

=

1, 2. Then 1r is the coequalizer of its kernel pair (p_{1 ,} p2). Hence, because of the remark following (1.1), S(1r) is the coequalizer of (S(p_{1 ),} S(p2)) in Ban1 and (S(p1), S(p2)) is the kernel pair of S(1r) in Ban1 • As ker S(1r)

=

{S(pi)(e) - S(p2)(e) : e E E} the first assertion follows due to

0

Denote by V the subvector space of S(C), that is generated by 0c(Cnker(~)). Since x E ker( ~) means x~0, we have trivially V ~ K ( ~ ). Conversely, let x,y E C satisfy x~y.

Then

Hence

1 1

°

4

^{x -}

4

^{y E} C n ker(~) and

¼ (0c(x)-

Oö(Y)) = Oe

(ix-¼y)

E Oe (c

^nker(~))

^{~ V}

Thus K(~) ~ V.

(1.3) Theorem: Let ~ be a congruence relation on the totally convex space C . Then K(~) is the unique Banach subspace V of S(C), which satisfies

0

^(V)=

^Oe (c

^nker(~)).

M oreover, for any ~-equivalence c/ass

A

and any a

E A

q;(A) ~ (q;(a)

+

K(~)) n O(S(C)).

ff

^a

e

A

n

C 0 then

0

(e7c(a) +

^K(~))

n O

(S(C)) ~

e7c(A).

Proof: The uniqueness of V is obvious. Due to (1.2) we know that K(~) is a Banach

0 0

subspace of S(C) and that q; (Cnker(~)) ~ O(K(~)) holds. Conversely, let 0

#

v E

0

O(K( ~ )). Then llvll

< 1

and

0

forsomea1,··•,am E Kandx1,···,Xm E Cnker(~).Let,\ =la1l+···+laml•

If ,\

s

^{1, then}

0

V= o;;(o1X1

+ · · · +

OmXm) E O'c(C nker(-)).

If ,\

>

1, put x

= T

^{X1 + .. ·}

+

^{~ Xm,} Then

0

x ^E C nker(~) and As a trivial consequence of (1.1), (i), we have

1 1

xllvll = llxvll

=

IIO"c(x)II

=

llxll-

choose e

>

0 such that llvll + ~

=

,\(llxll

+

e)

<

1 holds. Then llxll + e S .\(llxll + e)

<

1, and hence x

=

(llxll + e)y for some y E C. Since x-0, [PRl], ( 4.1), implies that (llvll + e,\) y~O. vVe have

v = ,\ O"c(x) = ,\ % ((llxll + e)y) = ,\

't: (~ ·

A(llxll + e)y) = % ((llvll + cA)y)

0

and (llvll + ~)y E C n ker( ~ ), which verifies the first equation.

Finally, let A be a ~-equivalence dass and let a E A. Choose an algebraic comple- ment W of K(~) in S(C). Then o;;(a)

=

v+w for some v E K(~) and w E "vV. If a' E A

0

and o;;(a')

=

v' + w' then, due to [PRl}, (4.3), ½a- ½a' E C n ker(-). Hence 1 ( ') 1 ( ') · 1 ( 1 ') ( 1 1 ') ( )

3

^{v - v +}

3

^{w - w}

⁼ 3

^{ct; a) -}

3

^0c(a

⁼

^Ob

3

^{a -}

3

^a

^{E K}

^{~ .}

Thus w

=

w' and we obtain

0c(a')

=

v' + w'

=

v' + w

=

(v' - v) + ct;(a) E O"c(a) + K(~),

which, in conjunction with [PRl], (i.3), shows the first inclusion relation. If a E A

0 0

n C and if ll0c(a) + vll

<

1 then, due to (1.1), (i), there is an element x E C with o-c(x)

=

0c(a)+v. Let ma.x(½, !lall, l10c(x)II)

< >.

~ 1. Then a = >.a' and x=

>.x.'

for some

0

a', x' E C, and we have

ct(

~x' -

~a')

=

2\

v

E

O

^(K(~)),

0

due to [PR!}, (7.2), as ¼x' - ½a' E C holds. By the first equality in (1.3) this means that ½a' - ½x' E ker(~)~ Hence

1 , 1 , 1 1 , 1 1 , 1 ') 1 , 4x

=

4x +

2° -

4x +

2(

2a -

rt ⁼

^4a'

and

[PRl], (

4.1), implies that

a.

=

>..a.' ~

AX =

X

as had to be shown.

(1.3) says that a.ny congruence rela.tion on any totally convex space C partitions

0 0

C into the intersection of O(S(C)) with the various transla.tes w+ V of a. suitable Ba.nach subspace V of S(C), where w ranges over an algebraic complement W of V in S(C).

Therefore the complexities of congruence relations on a totally convex space C are only encountered on the frame F(C) of C (see

[PR2,

§141).

One issue that remains ^is whether or not every closed subspace of a given Ba.nach space is the subspace associated via (1.3) with a. suitable congruence relation.

(1.4) Scholium: Let B be a Banach space and /et V be a closed subvector space of B.

0

Let

~

be the equivalence relation on O(B) given by:

x ~ x' if and only if x'

=

x + v for some v E V.

0

Then ~ is a congruence relation on O(B) and the subspace associated with it via (1.3) equals V.

0

Proof: For Xi,Yi E O(B), i EN, Xi-Yi is equivalent to Xi -Yi E V. This implies

E

^{ai(Xi - Yi) E V resp.}

E

^C'liXi~

L

"'iYi, for any "'• E

n.

Hence, - is a congruence

i i i

0

rela.tion on the totally convex space O(B) and, obviously, V= K (~) because ker(~)

=

0

O(B)

n

V. The same assertion holds for the totally convex spa.ce O(B), too.

We close this section with

(1.5) Proposition: Let - be a congruence relation on the totally convex space C and denote the quotient map

C - Cf -

by ^1r. Then, for any y E

C/ ~,

IIYII =

inf {llxll : y

=

^1r(x)}.

Proof:

[PRl],

(6.3), implies that

IIYII

~ inf {llxll : y

=

1r(x)}. Hence nothing need be shown

if IIYII =

1. So, assume

IIYII <

^0t

<

1. Then y

=

ay' for some y' E C/~. But y'

=

,r(x') for some

x'

E C. Put x

=

ax' to obtain

llxll

=

^llax'II

<"'

and y

=

ay'

=

^a,r(x')

=

^,r(ax')

=

^,r(x),

showing that inf {llxll : y

=

,r(x)} ~

IIYII•

§2 CONGRUENCE RELATIONS ON TOTALLY CONVEX SPACES:

THE BOUNDARY.

Let C be a totally convex space. Then bdyC, the boundary of C, is the set {x E C : llxll

= l}.

A subset S ~ C is said tobe convez (resp.superconvez (see

[R]),

if for all

X1, · · ·, Xn ES (resp.xi ,x2, ···ES) and all a1 ~ 0, •··,On ~ 0 (resp.a1 ~ 0, 02 ~ 0, • • •)

n 00 n oo

with

L

^Oi

=

1 (resp.I: Cl'i

=

^1),

L

^{Cl'iXi (resp.I: ai;q)} is in S. Obviously, each equivalence

1 1 1 1

class of a congruence relation on a finitely totally convex (resp. totally convex) space is convex (resp. superconvex).

A

convex (resp. superconvex) subset

S

~ C is called a facet (resp. superfacet) of C if it is contained in bdyC. A facet (resp. superfacet) of C that is maximal among all facets (resp. superfacets) of C with respect to inclusion is said tobe a side (resp.superside) of C.

(2.1) Lemma: Let C be a finitely total/y convex space and /et F be a facet of C. Then F is contained in some side of C.

Proof: The set of all facets of C that contain F is not empty. Hence Zorn's Lemma will do the job.

By comparison, it seems to be much more difficult to establish the existence of supersides in a totally convex space. However, we have

(2.2) Corollary: Let C be a totally convex space and /et S be a superside of C. Then S is a side of C.

Proof: Since S is a. fa.cet of C, (2.1) shows that S is contained in some side S0 of C. Suppose that

S

~ S0 • Choose

xo

E

So \S.

Then there are Xi E

S,

i

= 1, 2 · · ·,

and

00

Cl'i, i

= 0, 1, · · ·

with

ao

::j:.

0,

Cl'i ~

0,

and

E

^Cl'i

= ¹

such that

0

However,

L

00 ^Cl'jXj

^<

^1.

i=O

00 00

~ . Q' i

L...J Cl'iXi

= aoXo + (1 - ao) L ₁ ^_

^Xi

^{E So,}

i=O i=l OQ

contradicting the preceding norm inequality.

(2.1) and (2.2) shows that the boundary of a totally convex space C is the union of the sides of C and tha.t the supersides of C are to be found among the sides of C.

lt is not known, if the assertion analogous to (2.1) holds for superfacets and super- sides. For an important class of totally convex spaces it can be proved, however, as is shown by the following considera.tions. vVe first prove the

(2.3) Lemma: IJF is a facet of a normed (cp. (PR2], §13), totally convex space C, then

00 00

the superconvex hul/ ofF, <F>,c

= {E

^{OiXi: Xi E F, Cl'i} > 0, for i E N, and

E

^Cl'i

=

^1}

i=l i=l

is a superfacet of

C.

Proof: Obviou.sly <F>₁c is superconvex. Hence, it remains tobe shown tha.t <F>,c~

00

bdy C. Fore E<F>,c:,

e = L

Cl'jXj,~{j E F,i

e

N; we may assume that Cl'j >

0

for every

i : l

i E N because of [PRl], (2.4), (iii). For n E N, put

n

with an :

= L

^Cl11

^> o.

^{Then en E} F holds and

llenll ⁼

1. vVe have

11:l

1 n 1 1 ⁰⁰

= 2 L

av(l - ;- )xv

+ 2 L

^a11x11

^~

^{1 - an,}

11:l n 11:n+l

On the other hand, in any totally convex space C, one has for x, y E C

Hence, as C is normed, one gets

i ^{(llxll -}

^IIY

^{ID ~ ~ II ~ x - ~}

^Y

^{II ,}

and after interchanging the roles of x and y

This implies 1 11e11 - llenll l ~ 2(1 - an) and, because lim an= 1, we get 11e11 = 1. Thus

n-oo

<F>1c is a superfacet.

(2.4) Corollary: In a normed, totally conver space C any facet is contained in a su- perfacet and any superfacet is contained in a superside.

Proof: The fi.rst assertion follows directly from (2.3). To prove the second, we proceed as in the proof of (2.1). Take a superfacet F ~ C and consider the set of all superfacets of C containing F. Take a linearly ordered subset

Gi,

i E I, of superfacets containing F.

Then

u Gi

is a facet of C containing F, hence, by (2.3),

< u Gi >,c

is a superfacet containing F and the assertion follows from Zorn's Lemma. '

In the category

TC

of totally convex spaces, the free object on the set X is F(X) = Ö(l1(X)). The elements of F(X) are the mappings f: X - K with llfll

=

L

^{{lf(x)I : x EX} ~ 1.} F(X) is a normed, totally convex space, hence (2.4) yields the (2.5) Corollary: Any facet and any superfacet of F(X), X a set, is contained in a superside.

the facets by mappings to the Linton space as is shown in the following

(2.6) Proposition: Let F be a facet of F(X) Then there is a map 'f)p from X to the Linton space L(K)

=

{O} U

{z

E K :

lzl =

1}, such that for all f E F and all x E X

{

=

0,

i/

r.,,p(x)

=

0 f(x)

E [O, l]i,,p(x), if 1,'p(x)

#

0.

Proof: Suppose there is a Xo EX and f1, f2 E F such that f1 (xo)

#

^{0, f2(x}0 )

#

^{0, and}

argf1(xo)

#

argf2(xo) holds. Since (½f1

+

½f2) (xo)

=

_½f1(xo)

+

½f2(xo), we have

and hence

ll1r1 ⁺ 1r2II ⁼ I: {1c1r1 ⁺ ~f2)(x)I: ^x ex}<

L {

~lf1(x)I: x

EX}+ L { i1r2(x)I:

^x

EX}< 1,

which contradicts the assumption that F is a facet. Hence the function <pp defi.ned by

{ 0, 1,'p (x)

=

^t'(x)

lflx)i'

if f(x)

=

0, for all f E F

if f E F is any function with f(x)

#

0, satisfies the requirements of (2.6).

Let rp : X - L(K) be any map, and let F"" be the set of all f E F(X) such that f(x)

= ⁰

whenever rp(x)

= ^0,

and f(x) E

[0,1]

cp(x), whenever cp(x);=

0.

One checks easily · that F"" is a superfacet. Moreover, for any facet F, F~ F "Pp holds. Additionally, F"" is a superside (as weil as a side), if and only if

Ort

cp(X) holds. .

If

F is a facet of F(X) and r.p : X - L (K) is given by r.p(x) := r.pF(x), whenever cpF(x)

:/=

0, and r.p(x) := 1 eise, then F ~ F"", and F"" is a superside.

Let

~

be a congruence relation on the totally convex space C. Then we denote the

~-equivalence class containing a E C by Aa,

(2.7) Theorem: Let - be a congruence relation on the totally cont1ex space C and let a1, · · · , an E bdyC be such that Aa1 , • • • , Aa. are facets of C. Then either:

n

A,:,1a₁+ .. -+p.a. is a facet of C for all Pl ~ 0, · · ·, Pn ~ 0 with

L

^Pi

=

1,

1

or:

n

Ap1a1+ .. ·+p.a. fails to be a facet of C for all Pl

>

0, · · ·, Pn

>

0 with

L

^Pi

=

^1.

l

Proof: Let 1r be the quotient map C - C/~. Then (1.5) shows that Aa is a facet if and only if JJ1r(a)II

=

1. Hence, it suffices to prove (2.7) for

~

the identity relation on

n

C. Suppose that for some o-1

>

0, · · ·, ^O'n

>

0 with

L

^O'i

⁼

^{1, we have}

1

n

Then, for all ,o

>

0, Ti ~ 0, 1 ~ i ~ n, with

L

^Ti

=

1

i:O

n

However, ,o(u1a1

+ · · · +

O'nan)

+

T1a1

+ · · · +

Tnan

=

E(,oO'i

+

Ti)ai holds. lt is easy to see that suitable choices of 7j, 0 ~ j ~ n, as specified above, i=l will generate all the combinations of P1, · · · , Pn, that are referred to in the second alternative of (2. 7) by

n

Pi

=

^ToO'i

⁺

^Ti• On the other hand, if all "interior" points

L

^{Piai, p}1

>

^{0, · • •} Pn

>

^{0, with}

n i=l

I:

^Pi= 1, of the convex subset of C spanned by a1, a2, ···,an are in bdy C then so are

i=l

all the facets of this subset.

Remark: (2.7) has an "infinite" generalization that reads as follows: Let - be a congru- ence relation on the totally convex space C and let ai E bdy C, i E N, besuch that AB.i is

00

a facet ofC, i E N. Suppose that for some ^O'i ~ 0, i E N, with

L

^O'i

=

1, A.,,. fai/s to

i=l

L o-,a,

l 00

be a facet of C. Then for all ^Pi> 0, i E N,with

L

^Pi= 1 and inf {PiO'i-l: i E N}

>

0,

i=l

A ... f ai/s to be a Jacet.

L

^Pi-'i

l

The proof of this statement follows the proof of (2. 7).

N ext, let F be a face_t of the totally convex space C and let a E F. Call the, obviously convex, subset of F

Conea (F):

=

^{aa

+

(1- o:)y: 0

<

a

:5

l,y E F}

the open cone of F with vertez a. Conea (F) is a principal ideal in the sense of Flood ( c p. [F]). The subset

inF := n{ Conea(F) : a E F}

is called the interior of

F

and its element are called interna/ points of

F (

cp.[F]). Then we have

(2.8) Proposition: Let - be a congruence re/ation on the tota/ly convex space C. Let S be a side of C and /et a E S. ff Aa fails to be a facet of C then so does each Ay, where y is in the open cone of S with vertex a.

0

Proof: Let b E Aan C. Then, for every 0

<

o: ~ 1 and y in the open cone,

0

aa

+

(1 - o:)y ~ab+ (1 - o:)y EC,

due to [PRl], (6.2).

(2.9) Corollary: Let ~ be a congruence relation an the totally convex space C. Let _S1 and S:i be two distinct sides of C with _{a 1} E S1 and a₂ E S2 \ S1 such that Aa₁

=

Aa,.

Then there is an open cone of

S

₁ such that for every y in that cone, Ay fai/s to be a facet.

Proof: Let T:

=

{aa₂ + (1-a)y: 0 ~ a ~ 1,y E Si}. lt follows from the finite variant of [PRl], (2.12), that T is a convex set. Since S1 is ^a side of C and S1 C T, T cannot be a facet. Hence there is a O

< ß

~ 1 and a z E S, such that llßa:i

+ (

1 - ß)zll

<

1. Since

S1 3 ßa1

+

(1 - ß)z

~

ßa:i

+

(1 - ß)z

it follows that Aßai+(l-ß)z is not a facet, and the claim follows from (2.8).

(2.10) Corollary: Let - be a congruence relation on the totally convex space C. Let S be a side of

C

such that, for some internal point z of

S, A

2 is contained in

S.

Then, for any a E S, Aa is contained in S.

Proof: Let a E S and assume that Aa (/_ S. Then there is an element a' EAa with a' ~

S. By (2.1), a' is contained in some side S' of C. Thus (2.9) implies that for all internal points y of

S ,

Ay fails to be a facet, which contradicts the assumption.

The following sheds some light on the nature of internal points.

(2.11) Proposition: Let F be a facet of the totally convex space C. Then:

(i) if a and z are in F and there are O ~ a

5

1 and y E F with z

=

a a

+

(1 - a)y, then for any O

5

a'

<

a, there is a y' E F with z

=

a'a

+

(1 - a')y'.

(ii) if a E F and z E inF , and if O

<

a

5

1 and y E F satisfy z

=

a a

+

( 1- a) y, then every point a' a + (1 - a,')y, 0

<

cl

<

1, belongs to inF ; in particular, inF is the cone of F with vertex z.

(iii) inF is a facet.

Proof: (i) This is clear for a,

=

^1. For a

<

1, and

c/ <

a, put

Y,

=

^a-a' 1-a' a + .1.=.a.y 1-a' to get z

=

a'a + (1- a')y'.

(ii) Let

u =

ßa. + (1 - ß)z, 0

<

ß

<

1, and let b be any point of F. Then for some

0 5 ,\ < 1

and some w

E

F, z

= ,\ b

+

(1 - ,\)

w holds. Moreover

" (l->. )(l-ß) . . F d Yo

=

l->.+>..ß a

+

l->.+>.ß w 1s

m

an

we have u

=

,\(1-ß)b+(l--\+-\ß)w. This proves the assertion for u

=

a'a + (1- a')y and a,

<

a'

<

1. For the remaining choices of a,' reverse the role of a and y.

(iii) follows directly from the definition of inF.

Although the results of this section give some insight into the structure of a con- gruence relation on the boundary of a (finitely) totally convex space, they suffer from a

lack of information concerning the nature of the sides. The most detailed information known for special types of locally convex spaces is (2.6).

§3 CONGRUENCE RELATIONS ON SEPARATED TOTALLY CONVEX SPACES.

Separated totally convex spaces are characterized by the property that 4 : C - O(S(C)) is injective (cp. [PR2], (11.1)). Hence, C and X:= 4(C) are isomorphic via

0

De.

Since O(S(C)) ~ X

s;;

O(S(C)), a separated totally convex space is given, up to isomorphism, by a Banach space B and a. totally convex subspace X of O(B) containing

0

O(B); moreover, the norm on Xis induced by the norm of B.

In a totally convex space C , the closed interval determined by two points x,y E C is defined as [x, y] := {z :z = ax + (1 - a)y, 0

:5

a

:5

l}, the open interval as (x, y) :=

{z :z = ax + (1...;, a)y, 0

<

a

<

l} and the half-open intervals as (x, y] := {z :z

=

ax+

(1 - a)y, 0

:5

a < 1} and [x, y) :=

{z :z

= ax + (1- a)y, 0

<

a

:5

1}.

Fora Ba.nach space B, let E(B): = {b : b E Band llbll = l} denote the unit sphere

0

of B. Moreover, for any set X with O(B) ~ X

s;;

O(B) define bdy X: = X

n

E(B). If X is a totally convex space, this definition of bdyX coincides with that given in §2. Two points x 1,x2 ^E E(B) are said tobe E(B)-joined, if [x1,x2] ~ E(B) holds.

0

(3.1) Proposition: Let B be a Banach space and X be a set with O(B) ~ X ~ 0 (B).

Then X is a finitely totally convex subspace of O(B) if and only if (i)

I::(K)·

bdy X = bdy X,

(ii) bdy X is closed under closed E(B)-intervals, i.e., if x1,X2 E bdy X are I:(B)-joined, then [x1, x2] ~ bdy X.

X is a totally convex subspace of O(B), if, in addition

to

(i) and (ii), the

f

ollowing condition is satisfied:

(iii) bdy Xis a closed subset of B.

Proof: If X is finitely totally convex then, clearly, (i) and (ii) are satisfied. Con- versely we have to show that X1, · · · , Xn E X implies a1X1 + · · · + OnXn E X for all 01, •••,On with lo1¹ + • • · + lonl

:5

1. This is done by induction on n. Generally we

0

have a 1x1 + · • • + OnXn E O(B) ~ X if either loil + · · · + lonl

<

^{1 or Oi}

#

^{0 and}

0

Xi eO(B) for some i. Hence we may assume that lad+···+ lonl = 1 and Xi Ebdy X, i=l, ... , n. Additionally, [PR.1], (2.10) and (2.4), (iii), permit us to assume Oi

#

0, i = 1, • • •, n, whereas condition (i) shows that Oi

>

0, i

=

1, · · ·, n, and 01 +···+On = 1 may be stipulated. Suppose n=2. If X1 and x2 are not E(B)-joined, then (2.7) applied

~ 0

to the identity relation O(B) implies that a1x1 + a2x2 E O(B)

s;;

X for all O

<

01 < 1.

If x1 and x2 are E(B)-joined then condition (ii) implies that 01x1 + a2x2 E X for all 0

<

a 1 < 1. Finally assume that a1x1 + • • • + OkXk EX for all k<n under the above stip-

ulations. Furthermore, let xi,···, Xn EbdyX, 0

<

Oi, i

=

1, · • •, n, and a₁ +•••+an

=

1.

Put

ß

= l - a1 = a2 + · · · + O'n- Then

where y

= T

x2 + • • • + yxn. By induction hypothesis we have y E X. Hence the case n=2 shows that a 1x 1 + (1 - ai)y EX, and we are done with the first half of (3.1).

00

N ow suppose that bdy X is a closed subset of B. vVe have to show that

L

^O'iXi

l

00

is in X whenever Xi E X, i = 1, 2, · · · , and

L

^{lad ~ 1.} As before we may assume that

l

00

O'i

>

0, i

=

1, 2, · · ·, and

L

^{O'i = 1. Set}

l

ßn = a1 + · · ·

+

O'n and a1 O'n

Yn

=

- X i

+ · · · +

^{- X n .}

ßn ßn

Then

00 00

L

^{O'jXj = ßnYn + (1- ßn)}

I:

^{(l -}O'j ^{ß )Xj.}

1 n+l n

If IIYnll

< 1

for some n , then

00 0

L

00 ^O'jXj

^~

ßnllYnll + (1 - ßn)

<

¹

1

and hence, }: O'iXi EO(B)~ X. Otherwise, each Yn is in bdyX as X is a fi.nitely totally

1

convex subspace of O(B). Moreover, in B,

00

Lll'iXi-Yn

1

n oo oo

- L(l -

ß;

¹)aiXi +

L

^O'iXi

^~

^{(1 - ßn) +}

L

^O'j

1 n+l n+l

00

= 2Lai,

n+l

00

Thus the sequence Yn, n = 1, 2, • • •, converges to

L

O'iXi, which now is in bdy X as bdy X is closed. 1

Condition (iii) of (3.1) is not necessary for X to be totally convex as can be seen from the following example: let K

=

R, B = R 2 with the norm ll(x1,x2)II = lx1I

+

lx2l, and put X= {(x1,x2): 1l(x1,x2)1l ~ 1 and X1

:p

±1}.

0

Remark: Fora set X with O(B) ~ X ~ O(B) and 2)K)-bdy X= bdy X, the following statements are equivalent:

(i) bdy X is c/osed under c/osed E(B)-intervals, (ii) For any facet F of O(B), FnX is a facet, (iii) X is a finitely totally convex space.

Proof: The equivalence of (i) and (iii) follows from (3.1). (ii) obviously implies (i) and (ii) follows easily from (iii).

0

For a Banach space B, let O(B) ~ X ~ O(B) be a (separated) totally convex subspace with a congruence relation - on X. lt follows from (1.3) that for any congruence dass A the following is true:

0

(i) if A n O(B)

#

</,, then for any aEA

0

(a + K(-))n O(B) ~ A ~ (a + K(~)) n X,

0

(ii) if An O(B) = </,, then for any a E A A ~(a+K(~))n bdy X.

As it turns out, (i) is sharp while (ii) can be improved. Denote by Va, a E X, the R-subvector space of B generated by the set {a' - a: a' E Aa}, Then we have

(3.2) Lemma: (i) Va is the smallest R-subvector space of B that satisfies Aa ~ a+ Va,

(ii) a1 ~a2 implies Va₁ = Va2 ,

(iii) if

O 5 IPI 5 1

then V ^a ~ V pa; in particular, if

IPI

=

1

then Vpa

=

Va,

(iv) ~{VIil: i E supp

a.}

~ VEa,at' a. E

n,

1 i

(v) Va ~ V for all aEX, where V is the Banach space in (1.3),

0

(vi) i/aEO(B) then K®RVa = V.

Proof: (i)-(v) are obvious. As for (vi), let !lall

< ,\ <

1. Then a = ,\ b, for some bE X.

Hence x~y implies that

a

=

^,\b

~

,\b

+

(1 - ,\) ( ~x -

~y) =

a

+

~(1 - ,\)(x -

y).

Hence

½(1 -

,\)(x - y)

E

Va, In other words, Va and V have the same set of generators.

(3.3) Corollary: If z is an internal point of a facet F of X, then V a ~ V z for all aE F;

in particular, for all internal points w of

F, Vw = Yz,

Proof: Immediate from (3.2), (iv), and the definition of internal points.

(3.4) Lemma: (a + Va) n (b + Vb)

#

^<P implies ,\a

~

,\b, for all

I.\I <

^1.

Proof: Suppose

a

+ L

µi ( ai - a) = b

+ L

^{Vj (b j - b)}

1 j

where ai~a, for all i, and bj~b, for all j. Put ~

= ¼ ·

l+ t !µ,~+ t lvil Then

1-.(a+2rµi(ai-a))

=

(¼·1+tlµ;~+~l11il) a+½2r1+tlµ~l;+I:l11il (½ai-½a)

" J ~ J

~,-;a.

Therefore "'a -~b, and [PR.l], (4.1), implies the assertion.

(3.5) Proposition: Suppose that the closed unit interval [0, 1] and the open unit inte,val (0,1) are equipped with the canonical convex (resp. superconvex) structure. Then the congruence relations, given in terms of equivalence classes, on the interval are precisely the fo/lowing:

On

[0, 1]:

On (0, 1):

(i)

{o:},

(ii)

[0, 1],

(iii) {0}, {l}, (0, 1), (iv)

[0, 1), {l}.

(v) {0}, (0, l].

(i)

{o:},

(ii) (0, 1).

o:

E [O, 1],

o: E (0, 1),

Proof: Obviously the partitions above are congruence relations. Conversely, let ~ be a non-trivial congruence relation on

[O,

l]. Then there are a

<

b with a ~ b. If Aa

=

{x: x E [0, l] and x ~ a } is the congruence class of a, then for x, y E Aa and 0

:5 ,\ :5

1 we have

-\x+ (1- .X)y ~ .Xa+ (1- .X)a

=

a,

0

i.e. Aa ~ [0, l] is an interval with non-empty interior Aa. as b E Aa. and a

<

b. Put a:

=

^{inf Aa.,}

^{/3 :} =

^{sup Aa., then o:}

^{< /3.}

^{lf 0}

^<

^{o:, take 0}

^<

^xi

^<

^{o: and o:}

^<

^Xo

^{< /3,}

such that

½xo

+

½x1 >

o:. Then, chose x

2

^with

X2 >

o:, such that

½x1

+

½x2 <

o:. This leads to

1 1 1 1

2

^x1

⁺ 2

^x2

^~ 2

^x1

⁺ 2

^{Xo ~ a,}

which is a contradiction. Hence

o: =

0 and, analogously,

/3 =

^1. This proves the assertion for

[O,

l]. The proof carries over verbatim to the open interval (0,

1).

0

(3.6) Theorem:

Let

O(B)~ X

~

O(B)

be a separated totally convex space and let ~ be a congruence relation on X. Suppose that F is a facet of X and that z is an internal point of F with Az ~F. Then

Proof: Let z'~z with z' =:/:, z. Then z + ½(z' - z)

=

½z + ½z'~z. Assume that both z and z

+

a(z' - z), with a

>

0, belong to inF. Then there are w and w' in F such that both z and z

+

a(z' - z) are in the open segment {ßw

+

(1 - ß)w' : 0

<

ß

<

l}.

Hence (3.5) implies that z ~ z+ a'(z' - z) for all O

:5

et'

:5

a. Moreover we see that, for fi.x.ed z, there is a €

>

0 (depending on z') such that z~z + €'(z' - z), for all 1€'1

:5

€.

New kt ·

n

z +

L

,\(zi - z) E (z + Vz)

n

inF.

l

Then there is a €

>

0 such that z~z + €..\i(Zi - z), i

=

1, • • •, n. Hence

n 1 n

z ~

L;

^(z

⁺

€..\i(Zi - z))

=

z

+ .:. L

..\i(Zi - z),

l n ¹

and another application of (3.5) leads to our assertion.

(3.7) Corollary: Assumptions as in (3.6), except that F is a side S of X. ff for some interna/ point zo of

S,

A20 is contained in S, then for all internal points z of S,

(z

+

V2 ) n inS ~ A2 •

Proof: (2.10) and (3.6).

A (finitely) totally convex space X is called rotund, if every point x in bdy X is an extremal point, i.e. if x

=

ax₁ + (1 - a)x2, with x1,x2 E X and O

<

a

<

1, implies x

=

^x1

=

x2. We close this section by determining the equivalence classes of a congruence

0

relation ~ on a separate, rotund, (finitely) totally convex space O(B)~ X ~ O(B).

This is done by parametrizing the congruence classes by the elements of the open unit ball of a complement W of V

=

K( ~ ). We have S(X)

=

B and (1.3) shows that S(X/~) is isomorphic to B/V. Let W be an algebraic complement of V in Band equip it with the quotient norm with respect to V, i.e. one defines

lwl:=

inf {llv +

wll :

v E V}

for w E W. Let 7rW : B - W be the canonical projection given by 11"w(w+v):

=

w, w E W, v E V. Take an element (a, 11"(x)) E S(X/~) (cp. [PR!], §7) and define

<p

((a,

11"(x))) := 1rW(ax), then <p : S(X/~)

-w

is the isomorphism in Ban1 with

<p o S(11")

=

7rW. The open unit ball of W with the quotient norm

1- 1

will be de-

o

noted by Oq(W); in general the quotient norm on W is different from the norm induced in W by the norm of B.

0

- 0

Due to (1.2) (X/~) is isomorphic to Oq(W) via O(ip)ux/~ and ux is simply the inclusion X c....+ O(B). As 11"(x), x EX, are the congruence classes of ~ , we use O(ip)ux/~

0

to define, for w E Üq (W), Aw

:=

Ax_{0 ,} where Ax₀ is uniquely determined by w

=

<pux1~ (11'(Xo))

=

^ipux1-(Ax0)

=

11'w(xo), Thus the equivalence classes are bijectively

0 0

parametrized by w E Üq(W). For w E Oq(W) one defines Ew := Awn bdy X; Ew may be empty. With these notations one gets

0

(3.8) Theorem: Let O(B) ~X~ O(B) be a separated, rotund, (finitely) tota//y convex space, /et~ be a congruence relation on

X

with

V

and

W

as above. Then the sets

E...,,

w E

0

Oq(W), satisfy:

(i) The

E...,

are m'Utually disjoint,

0

(ii) E..., ~ w

+

V for w E Üq(W), (iii) Eaw

=

aE..., for a E E (K) , M oreover, the congruence c/asses are:

0

(w) Aw

=

((w

+

V)n O(B)) UE..., ,

0

w E Oq(W).

0

w E Oq(W),

(z)Az={z}, z E bdy X \ U { E..., : w E Üq(W)} .

0

Conversely, given any assignment w ^1-+ E..., ~ bdy

X,

for w

E

Oq(W), where q denotes the quotient norm on a subspace ^W of ^B with respect to an algebraic complement of V in B, which is a- closed subspace satisfying (i) , (ii), and (iii), the partition of X into the sets Aw and

A

₂ is a congruence relation on

X.

Proof: (i): If z E Ew

n

E...,,, then A...,

=

A...,,, i.e. w

=

w'. To prove (ii), take x E E...,, then x ~ Xo if w

=

1rw(xo), which implies x - Xo E V resp. 1rw(x)

=

1rw(Xo)

=

w, x E w

+

V. For (iii), we have

aw ⁼

1rw(0Xo), if ^w

=

1rw(Xo), and

lal ⁼

^1, hence x E Eaw if and only if llxll

=

1 and x~oXQ. This is equivalent to a-¹x~XQ and llo-¹xll

=

1, i.e. to o-¹x E Ew, or x E oEw. Next, (1.3) and the defi.nition of Ew imply that

0 0

A...,

=

((w

+

V)n O(B)) UEw for w E Üq(W). Now take x EX, llxll

=

1 and x ~ Ew

0

for all w E Üq(W). If there were y E Ax with

IIYII

< 1, we would have y E A..., with

0

w

:=

1r(y), w E Oq(W). This would lead to the contradiction x E Awn bdy X

=

Ew.

Hence, in this case, we get Ax ~ bdy X, which implies Ax

=

{x} due to the rotundity of X.

If

x E X, llxll

=

1 and x E Ew, we are finished. Hence, the only remaining case is

0

x EX, llxll< 1. But, with x

=

w

+

v, we have x E

(w +

V) n O(B) ~ A...,.

0

Now, conversely, given such a partition, we consider ^Xi, Yi E Aw₁₁ Wi E Oq(W), i E I, and ^Zj,j E

J,

with

Az

₁

=

{zj}, and I and

J

at most countable. One has to show that, with

X:= LOjXj

+

^LOjZj,

i j

y : =

L

^OiYi

⁺ L

^OjZj'

i j

o. E

n,

x and y belang to the same

Au,

u EX. The discussion is simplified by the fact that these sums are usual sums in a Banach space and we may therefore assume that a~ ::/: 0 for k E I U J and that the Xi, Yi and ^Zj are pairwise different.

Next, consider the case, where at least two terrns in the sum for x are different from zero. As X is rotund, this implies llxll

<

1. If the sum for y also contains at least two non-zero terrns, we have IIYII

<

1, too, and are fi.nished because of the above remark. If the sum for y contains at most one non-zero term, then there is an index k E I U J with ak

#

0, Yk

=

0 and hence

IIYII $

L

lad IIYdl +

L

^{lajl $ 1 - lakl}

^<

^1,

1 j

and we are finished.

Now, let the sum for x contain only one non-zero term, x

=

_{a 1x1.} If lail

<

1, then !lxll

<

1. If y also contains only one non-zero term or more than one, we get in each case IIYII

<

1 and are finished. Thus, the only case left is

x =

a1x1 with lo- 1¹

=

1.

But then, necessarily, we get y

=

a1Y1 and x, y E Aa₁u, if x1, Y1 E Au, This proves the assertion.

Remark: lt should be noted that, for a Hilbert space H, any totally convex subspace

0

X of O(H) with O(H) ~ X ~ O(H) is rotund. This follows, with O

<

a

<

1 and

x,

y E X with x

#

^y and llxll

= IIYII =

^1, immediately from the following inequality:

!lax+ (1 - a)yll²

=

a²llxll² + (1 - a)211Yll² + 2o(l - a)

<

x, y

>

<

a²

+

(1 -

a)2 +

2o(l - o)

=

1.

§4 CONGRUENCE RELATIONS ON F:i.

For convenience we denote the free (finitely) totally convex space on two generators by F; (see [PRl), (3.1)) and, for simplicity, we take K tobe R. Then

F,

is represented in R² as the square with vertices :1:: i, :1:: j, where i

=

(1,0), j

=

(0,1). If ~ is a congruence relation on the (finitely) totally convex space F, then the vector space V

=

K( ~) of (1.3) has dimension less than er equal to 2. We will now describe (up to obvious isomorphisms) all congruence relations on F, : in each case the description will be given by the associated partition cf

F,.

The dimension cf V is used for classifying the congruence relations.

If dima V=

0.

then (1.3) implies that - is the discrete or identity relation, and the congruence classes are given by

(Oll) {x}, XE

f,.

If dima V= 1, we may, without loss of generality, assume that V = (i + w j)R with

O

$ w $ l. Because,

if V is

not of this form it is mapped to a vector space of this type by the automorphism cf R ² interchanging i and j and this automorphism induces an isomorphism between the congruence relations inducing the respective vector spaces

(cp. (1.3)). .

In the following the points of the open interval (-j, j) (for notation see §3) are taken

0

as representatives of the ~-classes of the points in the open square F, . By (1.3) every