Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
199
hcanaB Space Theory
dna sti snoitacilppA
Proceedings of the First Romanian-GDR Seminar
Held at Bucharest, Romania, August 13 - September ,6 1891
Edited by .A Pietsch, .N Popa, and .I Singer
galreV-regnirpS
Berlin Heidelberg New York Tokyo 1983
Editors
Albrecht Pietsch
Sektion Mathematik,.Friedrich-Schiller-Universit~t Jena
Universit~tshochhaus .71 OG, 6900 Jena, GDR
Nicolae Popa
Ivan Singer
Department of Mathematics
The National Institute for Scientific and Technical Creation
Bdul P&cii 220, ?9622 Bucharest, Romania
AMS Subject Classifications (1980): 46 B 05, 46 B ,51 46 B 20, 46 B 22,
46 B30, 46 B40, 46 M05, 46 M 35, 46A06, 46A10, 46A12, 47A70,
47 D 25, 47 D 30, 47 H 09, 47 H ,01 14 A 65, 54 D 30
ISBN 3-540-12298-2 Springer-Verlag Berlin Heidelberg New York Tokyo
ISBN 0-38?-12298-2 Springer-Verlag New York Heidelberg Berlin Tokyo
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Foreword
From the 31st of August to the 5th of September 1981, the
Department of Mathematics of INCREST (the National Institute for
Scientific and Technical Creation), Bucharest, in collaboration with
the Department of Mathematics of the University of Jena, organized
the First Romanian-GDR Seminar on Banach Space Theory and its Appli-
cations held in Bucharest.
The research contracts between the Mathematics Departments of
INCREST and the CNST (the National Council for Science and Technology
of Romania) constituted the generous framework which made possible the
organization of this Seminar. The Seminar also benefited from the
cooperation of the Romanian Academy and of the Mathematics Department
of the University of Bucharest.
We hope that this Seminar has contributed to international
cooperation and exchange of ideas, for the benefit of further research
in Banach space theory and its applications.
We are indebted to the Springer Verlag for including these
Proceedings of the Seminar in the series Lecture Notes in Mathematics.
Organizing Committee
Head of Mathematics Department Organizers,
of INCREST
Zoia Ceau~escu Albrecht Pietsch
Nicolae Popa
Ivan Singer
C O N T E N T S
J.Arazy Isomorphisms of unitary matrix spaces ......... I
M.Att~ia, Inf-compact potentials and Banachic
J.Audounet kernels ....................................... 7
R.Brigola On summability in conjugate Banach
spaces ........................................ 28
D. van Dulst, Fixed points of nonexpansive mappings and
B.Sims Chebyshev centers in Banach spaces with
norms of type (KK) ............................ 35
G.Godini Best approximation and intersections of
balls ......................................... 44
A.Gulisashvili Estimates for the Pettis integral in
interpolation spaces with some applications ... 55
S.Heinrich, Elementary equivalence of Ll-preduals ......... 79
C.W.Henson,
L.C.Moore
S.Heinrich, Some open problems in the nonlinear
P.Mankiewicz classification of Banach spaces ............... 91
J.Hennefeld M-ideals, related spaces, and some
approximation properties ...................... 96
H.U.Hess On Etcheberry's extended Milutin lemma ........ 103
R.C.James Extreme points and an unusual Banach space .... 111
K.John Tensor products and nuclearity ................ 124
H.Junek Linear operators in (F)-spaces ................ 130
K.D.K~rsten Local duality of ultraproducts of Banach
lattices ...................................... 137
R.H.Lohman On the existence of spheres and dual
spheres without gap points .................... 143
E.R.Lorch Problems in the classification of certain
compact spaces ................................ 149
R.Mennicken Eigenvalue problems depending nonlinearly
on the parameter .............................. 156
K.Musial The weak Radon-Nikodym property in
conjugate Banach spaces ....................... 182
C.Niculescu Order o-continuous operators on Banach
lattices ...................................... 488
N.Popa The primariness of rearrangement invariant
function p-spaces, 0<p~l ..................... ~02
G.Racher A proposition of A.Grothendieck revisited .... 215
IV
H.U.Schwarz Bounded operators in Banach lattices ......... 228
I.Stephani Generating topologies and quotients of
injective operator ideals .................... 239
P.Terenzi Sequences in Banach spaces ................... 259
D.Vuza The perfect M-tensor product of perfect
Banach lattices .............................. 272
D.Yost Intersecting balls in spaces of vector-
-valued functions ............................ 296
L I S T
OF PARTICIPANTS BST SEMINAR
A. ANTIPA Bucharest, Romania H. JUNECK - Potsdam, GDR
J.ARAZY Haifa, Israel G. KASSAY - Cluj, Romania
G R . ~ Bucharest, Romania H. KONIG - Kiel, West Germany
M.ATTEIA Toulouse, France J. L. KRIVINE - Paris, France
C. BANICA Bucharest, Romania TH.KOHN - Jena, GDR
M. BANULESCU - Bucharest, Romania K. D. KORSTE~ - Leipzig, GDR
A. BARAN Bucharest, Romania T. J. LAPRESTE - Aubi~re, France
B. BEAUZAMY Lyon, France W. LINDE - Jena, GDR
C . ~ Warszaw, Poland R. LO}94AN - Kent, USA
R. BRIGOLA P~egensburg, West Germany E. LORCH - New Yor, USA
M. CANEIA Barcelone, _Spain p. MANKIEWICZ - Warsaw, Poland
B. CARL Jena, GDR R.MENNICK~ - Regensburg, West Gerrmg_ny
P. CASAZ ZA Huntsville, USA K.F~JSIAL - Wroclaw, Poland
Z. CEAUSESCU Bucharest, Romania C. NICULESCU - Craiova, }~ania
T. CEAUSU Timi~oara, Rc~ania G. PALTINEANU - Bucharest, Romania
B. CHEVRFAU - Bucharest, Bc~nia A. PELCZYNSKI - Warszaw, Poland
I. OOLOJOARA - Bucharest, Romania A.PIETSCH - Jena GDR
R. CRI STESCU - Bucharest, Romania M. PIMSNER - Bucharest, ~ a
R.M. DANET Bucharest, Romania N.POPA - Bucharest, P~mnania
N. DANET Bucharest, Rcmania S.POPA - Bucharest, Rc~ania
J. DIESTEL Kent, USA ° G RACHER - Salzburq, Austria
G. DINESCU - Bucharest, Romania p. RICHTER - Leipzig, GDR
D.van DURST - Amsterdam, Netherlands H. U. SC~ARZ - Jena GDR
T. FIGIEL Sopot, Poland LU SHIJIE - Nard-in, China
D .GASPAR Timi ~)ara, Romania B. SIMS - Armidale, Australia
G.GODINI Bucharest, Rcmania I.SINGER - Bucharest, Rc~ania
C. GROSSU Bucharest, Romania I. STEPHANI - Jena, GDR
A. GULISHASVILI - Tbilisi, USSR S. T~.WMAN - Bucharest, Romania
U. HAAGERUP 0dense, Denmark P. TERENZI - Milano, Italy
S. HEINRICH Berlin, GDR L. TZAFRIRI - Jerusalem, Israel
° H H~NN~ ~uD - New York, USA D. VOICULESCU - Bucharest, Romania
H.U.HESS Regensburg, West Germany D.VUZA - Bucharest, Romania
R.C. JAMES ClaresDnt, USA D.YOST - Bundoora, Australia
K. JOHN Prague, Czechoslovakia C. ZALINESCU - Ia~i, Romania
C.ZANCO - Milano, Italy
SCIENTIFIC PROGRAMME OF THE SEMINAR
Monday, August 31
Section 1
Chairman: I.Singer
10.00-10.30 Opening session
10.30-11.15 L.Tzafriri: Uniqueness of unconditional bases
11.30-12.15 A.Pietsch: Approximation spaces
Chairman: A.Pietsch
15.30-16.00 B.Sims: Near uniform convexity
16.10-16.40 D.Yost: Intersecting balls in spaces of vector-valued
functions
16.50-17.20 H.V.Hess: On Etcheberry's extended Milutin lemma
17.30-18.00 R.Brigola: Some characterizations of conjugate Banach
spaces
Section 2
Chairman: J.Diestel
15.30-16.00 S.Teleman: Topological properties of the boundary
measures
16.10-16.40 N.Linde: Moments of measures on Banach spaces
16.50-17.20 T.Kuhn: Gaussian measures and entropy
17.30-18.00 B.Carl: Entropy and eigenvalues
Tuesday, September 1
Section 1
Chairman: R.C.James
9.30-10.15 A.Pelczynski: On Banach spaces of analytic functions
10.30-11.15 J.Diestel: Can the Pettis integral be used?
11.30-12.15 S.Heinrich: On uniform and Lipschitz classification of
Banach spaces
Chairman: E.R.Lorch
15.30-16.00 D.van Dulst: Some remarks on the fixed-point problem for
nonexpansive mappings
16.10-16.40 P.Casazza: Some remarks on Tsirelson's space
16.50-17.20 R.E.Lohman: On the existence of spheres without gappoints
JX
17.30-18.00 G.Racher: On a proposition of Grothendieck
Section 2
Chairman: H.K6nig
15.30-16.00 I.Stephani: Quotients of surjective operator ideals
16.10-16.40 K.D.K~rsten: On local duality of ultraproducts of Banach
lattices
16.50-17.20 P.Richter: On joint generalized eigenfunction expansions
17.30-18.10 H.Juneck: Linear operators in )F( and (DF) spaces
18.20-18.50 G.Kassay: The proximal point algorithm for reflexive
spaces
Wednesday, September 2
Section 1
Chairman: L.Tzafriri
9.30-10.15 R.C.James: Bushes and extreme points in Banach spaces
10.30-11.15 B.Beauzamy: Spreading models of Banach spaces
11.30-12.15 H.Konig: Some estimates for projection constants
Thursday, September 3
Section 1
Chairman: B.Sims
9.30-10.15 R.Mennicken: Non-linear eigenvalue problems
10.30-11.15 H.Schwarz: Bounded operators in Banach lattices
11.30-12.15 N.Popa: Some results concerning the rearrangement
invariant p-spaces, 0<p<l
Chairman: C.Bessaga
15.30-16.00 E.Lorch: ~ 1 and the structure of the homeomorphism
groups of certain compact spaces
16.10-16.40 P.Mankiewicz: Lipschitz and uniform classification of
Banach spaces
16.50-17.20 J.Hennefeld: M-ideals and related properties
Section 2
Chairman: R.Mennicken
15.30-16.00 M.Att4ia: Banachic kernels
16.10-16.40 J.T.Laprest4: Some properties of stable Banach spaces
16.50-17.20 C.Z~linescu: Uniformly convex functions
Friday, September 4
Section 1
Chairman: B.Beauzamy
X
9.30-10.15 C.Bessaga: Old and new results on the topological classi-
fication of metric linear spaces
10.30-11.15 T.Figiel: Discontinuous invariant functionals
11.30-12.15 G.Godini: Best approximation in certain classes of
normed linear spaces
Chairman: .D van Dulst
15.30-16.00 K.John: Tensor products and nuclearity
16.10-16.40 J.Arazy: Isomorphisms of unitary ideals
16.50-17.20 R.Cristescu: L'espace des op4rateurs r4guliers
17.30-18.00 D.Vuza: Perfect m-tensor products of perfect Banach
lattices
Section 2
Chairman: S.Teleman
15.30-16.00 R.Musial: The weak Radon-Nikodym property in Banach spaces
16.10-16.40 A.Gulisashvili: The estimates of Pettis integral in in-
terpolation spaces and applications
Saturday, September 5
Section 1
Chairman: A.Pelczynski
9.30-10.15 J.L.Krivine: On a theorem of Pisier
10.30-11.15 P.Terenzi: Sequences in Banach spaces
11.30-12.15 U.Haagerup: The Grothendieck inequality for bilinear
forms on C'-algebras
Chairman: I.Singer
12.30-12.45 Closinq session
ISOMORPHISMS OF UNITARY MATRIX SPACES
Jonathan Arazy
University of Haifa
Mount Carmel
Haifa, Israel
Let E be a symmetric sequence space and let E C be its associated unitary
matrix space, i.e., E C is the Banach space of all compact operators x on 2£ for
( 1
which the sequence s(x) = n s )x( n=l of s-numbers belongs to E, normed by
EcI~li
= IIs(x)ll E.
DEFINITION. E C is said to have a unique symmetric structure if whenever E C is iso-
morphic to another unitary matrix space F C then, necessarily, F E = C C (i.e., E = F,
with equivalent norms).
In Kent Conference (International Conference on Banach Space Theory and its Ap-
plications, Kent, Ohio, August 1979), A. Pelczynski raised the following problem:
PROBLEM A: Does every unitary matrix space have a unique symmetric structure?
At that time it was known that the Schatten-von-Neumann p-classes C (defined
p
by pC = C£p for 1 ~ p < ~, and C = Cc0) have unique symmetric structures. Indeed,
every symmetric basic sequence in Cp is equivalent to the unit vector basis of either
2£ or ~p c( , o if p = ~) , see ]i[ . Another information available at that time was
the following consequence of our earlier work on basic sequences in the spaces , E C
namely: if E C is isomorphic to F C then E is isomorphic to F (the converse is
false; see [2]).
In this paper we survey the progress made since Kent Conference in studying the
embeddings of one unitary matrix space into another, and the problem of the uniqueness
of the symmetric structure of unitary matrix spaces (Problem A).
Let us introduce some necessary notation and terminology. We denote by
o,
{~*,j }i,j=l the standard unit matrices (defined by ,le (k,£) j = 6i,k" 6j,£); they
form a Schauder basis of E C in the "shell orderinq"
e1~1' e1~2' e2j2' ~2e ' 1 e1~3' e2~3' e3j3' e3j2' e3jlt'''el,n , e2~nP'''en~l'''"
The sequence {Sn}~= 1 of subspaces, defined by n = S span{el, ; j max{i,j} = n} , is
called the shell decomposition; it is a reproducible, monotone finite dimensional
decomposition of E C (see [i]). [4], We use freely tensor products of operators on
£2' and identify naturally £2 with 2£ )~ £2' 2£ ~ £2 8 £2' etc.
To begin with, let E be any symmetric sequence space, and let b 6 CE, Hb~c~ =i.
