On the extension of isometries between the unit spheres of a C*-algebra and B(H)

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by  Francisco J. Fernández-Polo

ON THE EXTENSION OF ISOMETRIES BETWEEN THE UNIT
SPHERES OF A C -ALGEBRA AND B(H)
∗
7
1 FRANCISCOJ.FERNA´NDEZ-POLOANDANTONIOM.PERALTA
0
2
n Abstract. Given two complex Hilbert spaces H and K, let S(B(H)) and
a S(B(K)) denote the unit spheres of the C∗-algebras B(H) and B(K) of all
J boundedlinearoperatorsonH andK,respectively. Weprovethateverysur-
jective isometryf :S(B(K))→S(B(H)) admits anextension toasurjective
1
complexlinearorconjugatelinearisometryT :B(K)→B(H). Thisprovides
1
apositiveanswertoTingley’sprobleminthesettingofB(H)spaces.

]
A
F
h. 1. Introduction
t
a Let X and Y be normed spaces, whose unit spheres are denoted by S(X) and
m
S(Y), respectively. Suppose T : X Y is a surjective real linear isometry. The
→
[ restriction T : S(X) S(Y) defines a surjective isometry. The so-called
|S(X) →
Tingley’s problem, named after the contribution of D. Tingley [33], asks if every
1
v surjective isometry f : S(X) S(Y) arises in this way, or equivalently, if every
→
6 surjectiveisometryf :S(X) S(Y)admitsanextensiontoasurjectivereallinear
→
1 isometry T : X Y. Tingley’s achievements show that, for finite dimensional
9 →
normed spaces X and Y, every surjective isometry f : S(X) S(Y) satisfies
2 →
f( x)= f(x) for every x S(X) (see [33, THEOREM in page 377]).
0 − − ∈
. A solution to Tingley´s problem has been pursued by many researchers since
1
0 1987. Positiveanswersto Tingley’s problemhavebeen establishedfor ℓp(Γ)spaces
7 with 1 p (see [7, 8, 10] and [11]), Lp(Ω,Σ,µ) spaces, where (Ω,Σ,µ) is
≤ ≤ ∞
1 a σ-finite measure space and 1 p (compare [26, 27] and [28]), and C (L)
0
v: spaces (see [34]). Tingley’s prob≤lem≤als∞o admits a positive solution in the case of
i finite dimensional polyhedral Banach spaces (see [18]). The reader is referred to
X
the surveys [12] and [35] for additional details.
r
a Inthenon-commutativesetting,Tingley’sproblemhasbeensolvedforsurjective
isometriesbetweenthe unit spheresoftwofinite dimensionalC -algebras(see [31])
∗
and for surjective isometries between the unit spheres of two finite von Neumann
algebras [32]. A more recent contribution solves Tingley’s problem for surjective
isometries between the unit spheres of spaces, K(H), of compact linear operators
on a complex Hilbert space H, or more generally, for surjective isometries between
the unit spheres of two compact C -algebras [22, Theorem 3.14]. The novelties in
∗
[22]arebasedontheuse oftechniquesofJB -triples,andTingley’sproblemis also
∗
solved for surjective isometries between the unit spheres of two weakly compact
Date:December2016.
2010 Mathematics Subject Classification. Primary 47B49, Secondary 46A22, 46B20, 46B04,
46A16,46E40,.
Key words and phrases. Tingley’sproblem;extensionofisometries;C∗-algebras;B(H).
2 F.J.FERNA´NDEZ-POLOANDA.M.PERALTA
JB -triples of rank greater than or equal to 5. In [16] we establish a complete
∗
solution to Tingley’s problem for arbitrary weakly compact JB -triples.
∗
Tingley’sproblemforsurjectiveisometriesbetweentheunitspheresoftwoB(H)
spacesseemstobethelastfrontierinthestudiesonTingley’sproblem. Thispaper
is devoted to provided a complete solution in this case.
The results in [31, 22, 16] are based, among other techniques, on the results
describing the (maximal) norm closed proper faces of the closed unit ball of a C -
∗
algebra (see [2]) or of a JB -triple (see [13]). Throughout the paper, the closed
∗
unit ball of a normed space X will be denoted by . It is shown in [31, 22, 16]
X
B
that for a compact C -algebra A (respectively, a weakly compact JB -triple E)
∗ ∗
the norm closed faces of are determined by finite rank partial isometries in A
A
B
(respectively, by finite rank tripotents in E). However, for a general C -algebra A
∗
the maximal proper faces of are determined by minimal partial isometries in
A
B
A (see Section 2 for more details). This is a seriousobstacle which makes invalid
∗∗
the arguments in [22, 16] in the case of B(H).
Toavoidthedifficultiesmentionedinthepreviousparagraph,ourfirstgeometric
result shows that a surjective isometry f from the unit sphere of a C -algebra A
∗
onto the unit sphere of B(H) maps minimal partial isometries in A into minimal
partial isometries in B(H) (see Theorem 2.5). Apart from the just commented
geometric tools, our arguments are based on techniques of functional analysis and
linear algebra. In our main result we prove that given two complex Hilbert spaces
H and K, every surjective isometry f :S(B(K)) S(B(H)) admits an extension
→
to a surjective complex linear or conjugate linear isometry T :B(K) B(H) (see
→
Theorem 3.2). In the final result we show that the same conclusion remains true
when B(H) spaces are replaced by ℓ -sums of B(H) spaces (see Theorem 3.3).
∞
The next natural question beyond these conclusions is whether Tingley’s problem
admits or not a positive answer for Cartan factors and atomic JBW -triples.
∗
It should be remarkedhere that the solution to Tingley’s problem for surjective
isometries between the unit spheres of K(H)-spaces in [22, 16] and the solution
presented in this note for surjective isometries between the unit spheres of B(H)-
spaces are completely independent results.
2. Surjective isometries between the unit spheres of two
C -algebras
∗
In this section we carry out an study of the geometric properties of surjective
isometries between the unit spheres of two C -algebras with special interest on
∗
C -algebras of the form B(H). We begin by gathering some technical results and
∗
concepts needed for later purposes.
Proposition 2.1. ([3, Lemma 5.1] and [29, Lemma 3.5]) Let X, Y be Banach
spaces, and let T : S(X) S(Y) be a surjective isometry. Then C is a maximal
→
convex subset of S(X) if and only if T(C) is that of S(Y). Then C is a maximal
proper (norm closed) face of if and only if f(C) is a maximal proper (norm
X
closed) face of . B (cid:3)
Y
B
An interesting generalization of the Mazur-Ulam theorem was established by P.
Mankiewiczin [19], who provedthat, giventwo convexbodies V X andW Y,
⊂ ⊂
every surjective isometry g from V onto W can be uniquely extended to an affine
isometry from X onto Y. Consequently, every surjective isometry between the
ON THE EXTENSION OF ISOMETRIES BETWEEN UNIT SPHERES 3
closed unit balls of two Banach spaces X and Y extends uniquely to a real linear
isometric isomorphism from X into Y.
Let a and b be two elements in a C -algebra A. We recall that a and b are
∗
orthogonal (a b in short) if ab = b a = 0. Symmetric elements in A are
∗ ∗
⊥
orthogonalif and only if their product is zero.
For each element a in a C -algebra A, the symbol a will denote the element
∗
| |
(a∗a)12 A. Throughout this note, for each x A, σ(x) will denote the spectrum
∈ ∈
of the element x. We observe that σ(a) 0 = σ(a ) 0 , for every a A.
∗
| | ∪{ } | | ∪{ } ∈
Let a=v a be the polar decomposition of a in A , where v is a partial isometry
∗∗
| |
in A , which, in general,does notbelong to A (compare [24]). It is further known
∗∗
that v v is the range projection of a (r(a) in short), and for each h C(σ(a)),
∗
| | | | ∈ | |
with h(0)=0 the element vh(a) A (see [1, Lemma 2.1]).
| | ∈
Proposition 2.1 points out the importance of an appropriate description of the
maximal proper faces of the closed unit ball of a C -algebra A. A complete
A ∗
B
study was established by C.A. Akemann and G.K. Pedersen in [2]. When A is
a von Neumann algebra, weak -closed faces in were originally determined by
∗ A
B
C.M. Edwards and G.T. Ru¨ttimann in [14], who proved that general weak -closed
∗
faces in have the form
A
B
Fv =v+(1 vv∗) A(1 v∗v)= x A : xv∗ =vv∗ ,
− B − { ∈B }
for some partial isometry v in A. Actually, the mapping v F is an anti-order
v
7→
isomorphismfromthe complete lattice ofpartialisometriesinA ontothe complete
lattice of weak -closed faces of , where the partial order in the set of partial
∗ A
B
isometriesofAis givenby v u ifandonly if u=v+(1 vv )u(1 v v)(see [14,
∗ ∗
≤ − −
Theorem 4.6]).
However, partial isometries in a general C -algebra A are not enough to deter-
∗
mine all the norm-closed faces in , even more after recalling the existence of
A
B
C -algebras containing no partial isometries. In the general case, certain partial
∗
isometries in the second dual A are required to determine the facial structure of
∗∗
. WerecallthataprojectionpinA iscalledopen ifA (pA p)isweak -dense
A ∗∗ ∗∗ ∗
B ∩
in pA p (see [21, 3.11]and [25, III.6]). A projectionp A is said to be closed
∗∗ ∗∗
§ § ∈
if 1 p is open. A closed projection p A is compact if p x for some positive
∗∗
− ∈ ≤
norm-oneelementx A. Apartialisometryv A belongs locally to Aifv v isa
∗∗ ∗
∈ ∈
compactprojectionandthereexistsanorm-oneelementxinAsatisfyingv =xv v
∗
(compare [2, Remark 4.7]).
It is shownin [2, Lemma 4.8 and Remark 4.11]that “the partialisometries that
belong locally to A are obtained by taking an element x in A with norm 1 and
polar decomposition x = ux (in A ), and then letting v = ue for some compact
∗∗
| |
projection e contained in the spectral projection χ (x) of x corresponding to
{1} | | | |
the eigenvalue 1.”
It should be noted that a partial isometry v in A belongs locally to A if and
∗∗
onlyifitiscompactinthesenseintroducedbyC.M.EdwardsandG.T.Ru¨ttimann
in [15, Theorem 5.1].
The facial structure of the unit ball of a C -algebra is completely described by
∗
the following result due to C.A. Akemann and G.K. Pedersen.
4 F.J.FERNA´NDEZ-POLOANDA.M.PERALTA
Theorem 2.2. [2, Theorem4.10] Let A be a C -algebra. The norm closed faces of
∗
the unit ball of A have the form
F =(v+(1 vv ) (1 v v)) = x : xv =vv ,
v ∗ A∗∗ ∗ A A ∗ ∗
− B − ∩B { ∈B }
for some partial isometry v in A belonging locally to A. Actually, the mapping
∗∗
v F is an anti-order isomorphism from the complete lattice of partial isometries
v
7→
in A belonging locally to A onto the complete lattice of norm closed faces of .
∗∗ A
(cid:3) B
A non-zero partial isometry e in a C -algebra A is called minimal if ee (equiv-
∗ ∗
alently, e e) is a minimal projection in A, that is, ee Aee = Cee . By Kadison’s
∗ ∗ ∗ ∗
transitivity theorem minimal partial isometries in A belong locally to A, and
∗∗
hence every maximal proper face of the unit ball of a C -algebra A is of the form
∗
(v+(1 vv ) (1 v v))
∗ A∗∗ ∗ A
− B − ∩B
forauniqueminimalpartialisometryv inA (compare[2,Remark5.4andCorol-
∗∗
lary 5.5]).
Our main goal in this section is to show that a surjective isometry f : S(A)
→
S(B)betweenthe unit spheresoftwo C -algebrasmaps minimalpartialisometries
∗
intominimalpartialisometries. Inafirststepweshallshowthat,foreachminimal
partial isometry e in A, 1 is isolated in the spectrum of f(e).
| |
Theorem 2.3. Let A and B be C -algebras, and suppose that f :S(A) S(B) is
∗
→
a surjective isometry. Let e be a minimal partial isometry in A. Then 1 is isolated
in the spectrum of f(e).
| |
Proof. Since e alsoisa minimalpartialisometryin A andbelongs(locally)to A,
∗∗
the set F = e+(1 ee ) (1 e e) is a maximal proper face of . Applying
e ∗ A ∗ A
− B − B
Proposition 2.1 and Theorem 2.2 we deduce the existence of a minimal partial
isometry w in B such that
∗∗
(1) f(Fe)=Fw =(w+(1 ww∗) B∗∗(1 w∗w)) B= x A :xw∗ =ww∗ .
− B − ∩B { ∈B }
Since f(e) f(F )=F we have f(e)=w+(1 ww )f(e)(1 w w).
e w ∗ ∗
∈ − −
Arguing by contradiction, we assume that 1 is not isolated in σ(f(e)). Let
| |
f(e)=rf(e) denote the polar decomposition of f(e).
| |
By assumptions we can find t σ(f(e)) satisfying 3 < t < 1. Let us
0 ∈ | | √10 0
consider the functions h and h in the unit sphere of C (σ(f(e))) given by
1 2 0
| |
t , if 0 t t 0, if 0 t 1(t +1)
h (t):= at0ffine, if t ≤ t≤ 01(t +1) ; h (t):= affine, if 1(≤t +≤1)2 0t 1
1  0, if 10(t≤+≤1)2 0t 1 2  1, if t2=01. ≤ ≤
2 0 ≤ ≤
 
We set x = rh (f(e)) and y = rh (f(e)). Obviously h (f(e)) and h (f(e))
1 2 1 2
| | | | | | | |
are positive elements in S(B) satisfying h (f(e))h (f(e))=0. Since
1 2
b b | | | |
xy∗ =rh1(f(e))h2(f(e))r∗ =0,
| | | |
and bb
y∗x=h2(f(e))r∗rh1(f(e))=h2(f(e))h1(f(e))=0,
| | | | | | | |
it follows that xˆ yˆ.
b b⊥
ON THE EXTENSION OF ISOMETRIES BETWEEN UNIT SPHERES 5
Let x = f 1( xˆ) S(A) and y = f 1(yˆ) S(A). Since f is an isometry we
− −
− ∈ ∈
deduce that
1= xˆ+yˆ = yˆ ( xˆ) = f(y) f(x) = y x ,
k k k − − k k − k k − k
and
1+t = f(e)+xˆ = f(e) f(x) = e x .
0
k k k − k k − k
We recall that, from (1), f(e) = w + k where k = (1 ww )f(e)(1 w w)
∗ ∗
− −
satisfies k w = 0 = wk , which proves that k w. Let r denote the (unique)
∗ ∗ 0
⊥
partial isometry appearing in the polar decomposition of k. Since r is the partial
isometryinthepolardecompositionoff(e),w k,andf(e)=w+k,itfollowsthat
⊥
r=w+r with r w. We alsoknow that f(e) =w w+ k , and hence a simple
0 0 ∗
⊥ | | | |
application of the continuous functional calculus (having in mind that h (1) = 1)
2
shows that h (f(e))=w w+h (k ), with w w h (k ). We therefore have
2 ∗ 2 ∗ 2
| | | | ⊥ | |
(2) yˆw =rh (f(e))w =rw ww +rh (k )w =rw =(w+r )w =ww ,
∗ 2 ∗ ∗ ∗ 2 ∗ ∗ 0 ∗ ∗
| | | |
which implies that yˆ F , and consequently y =f 1(yˆ) F (see (1)).
w − e
∈ ∈
We claim that
(3) ee (e x)e e >1.
∗ ∗
k − k
The element e x has norm 1 +t > 1. Suppose that H is a family of
0 i I
− { }
complex Hilbert spaces and π : A ℓ∞B(H ) is an isometric -homomorphism
→ i i ∗
withweak∗-denserange(wecanconsidLer,forexample,theatomic representation of
A [21, 4.3.7], where the family I is precisely the set of all pure states of A and π is
the directsumofallthe irreduciblerepresentationsassociatedwith the pure states
[21, Theorem 3.13.2]). For each j I, let P denote the projection of ℓ∞B(H )
∈ j i i
ontoB(Hj)andletπj =Pj π. Clearly,πj isa∗-homomorphismwithwLeak∗-dense
◦
range. Since e is a minimal partial isometry, there exists a unique i I such that
0
∈
π (e) is a non-zero(minimal) partialisometry and π (e)=0, for everyj =i . We
i0 j 6 0
also know that x =1, and thus π (e x) =1+t .
k k k i0 − k 0
Letπ (e x)=uπ (e x) bethepolardecompositionofπ (e x)inB(H ).
i0 − | i0 − | i0 − i0
Take0<ε<t 3 . Since π (e x) =1+t wecanfindaminimalprojection
0−√10 k| i0 − |k 0
q =ξ ξ B(H ) with ξ =1 in H satisfying q u u and
⊗ ∈ i0 k k i0 ≤ ∗
(4) 1+t ε< π (e x)(ξ)/ξ ,
0− h| i0 − | i
and
(5) 1+t0−ε<h|πi0(e−x)|(ξ)/ξi≤h|πi0(e−x)|(ξ)/|πi0(e−x)|(ξ)i12 kξk
=h|πi0(e−x)|2(ξ)/ξi12 =hπi0(e−x)∗πi0(e−x)(ξ)/ξi12.
The element v =uq is a minimal partial isometry in B(H ).
i0
Weobservethatπ(e)=π (e)andvarenotorthogonal. Otherwise,π (e) π (e)
i0 i0 ∗ i0 ⊥
v v =q, and hence π (e)q =0=qπ (e) , which, by (5), implies that
∗ i0 i0 ∗
(1+t0−ε)2 <hπi0(e−x)∗πi0(e−x)(ξ)/ξi=hqπi0(e−x)∗πi0(e−x)q(ξ)/ξi
=hqπi0(x)∗πi0(x)q(ξ)/ξi≤kπi0(x)∗πi0(x)k=kπi0(x)k2 ≤kxk2 =1,
which is impossible.
Therefore, π (e) and v are two minimal partial isometries in B(H ) which are
i0 i0
not orthogonal. They must be of the form π (e) = η ξ and v = η ξ for
i0 1 ⊗ 1 1 ⊗ 1
e e
6 F.J.FERNA´NDEZ-POLOANDA.M.PERALTA
suitable ξ ,η ,ξ ,η S(H ) with ξ /ξ + η /η = 0. Let us consider two
1 1 1 1 ∈ i0 |h 1 1i| |h 1 1i| 6
orthonormalsystems η ,η and ξ ,ξ such that
e e { 1 2} { 1 2}e e
v =απ (e)+βv +δv +γv ,
i0 12 22 21
where π (e) := v = η ξ , v = η ξ , v = η ξ , v = η ξ , α =
i0 11 1 ⊗ 1 12 2 ⊗ 1 21 1 ⊗ 2 22 2 ⊗ 2
ξ /ξ η /η , β = ξ /ξ η /η , γ = ξ /ξ η /η , δ = ξ /ξ η /η C. It
1 1 1 1 1 1 1 2 2 1 1 1 2 1 1 2
h ih i h ih i h ih i h ih i ∈
is easy to check that α2+ β 2+ γ 2+ δ 2 = ξ /ξ 2 η 2+ ξ /ξ 2 η 2 =
e e | |e |e| | | | | e |he1 1i| k 1k e|h 2e 1i| k 1k
ξ 2 =1, and αδ =βγ.
k 1k e e e e
For each (i,j) 1,2 2, let ϕ S(B(H ) ) be the unique extreme point of
e ∈ { } ij ∈ i0 ∗
the unit ball defined by ϕ (z):= z(ξ )/η (z B(H )). We shall also
BB(Hi0)∗ ij h i ji ∈ i0
consider ϕ S(B(H ) ), defined by ϕ (z):= z(ξ )/η (z B(H )). Each ϕ
v ∈ i0 ∗ v h 1 1i ∈ i0 ij
is supported by v S(B(H )), while ϕ is supported by v.
ij ∈ i0 v e e
Clearly, the identity
ϕ (π (e))= π (e)(ξ )/η = ξ /ξ η /η =α¯
v i0 h i0 1 1i h 1 1ih 1 1i
holds, and similarly we have e e e e
ϕ π (x)=α¯z +β¯z +δ¯z +γ¯z ,
v i0 11 12 22 21
where z = ϕ π (x) for all (i,j) 1,2 2. We also know that ξ ξ = v v =
ij ij i0 ∈ { } 1 ⊗ 1 ∗
q =ξ ξ u u,andthusξ =µ ξ,forasuitableµ Cwith µ =1. Wededuce
⊗ ≤ ∗ 1 0 0 ∈ | 0|e e
from (4) that
e
1+t0−ε<h|πi0(e−x)|(ξ)/ξi=hu∗u|πi0(e−x)|(ξ)/ξi=hu|πi0(e−x)|(ξ)/uq(ξ)i
= π (e x)(ξ)/v(ξ) = π (e x)(ξ )/v(ξ ) = π (e x)(ξ )/η =ϕ (π (e x))
h i0 − i h i0 − 1 1 i h i0 − 1 1i v i0 −
=ϕ (π (e))+ϕ (π (x)) α + ϕ (π (x)) α +1,
v i0 v i0 e ≤e| | | v i0 |e≤|e|
which proves
t ε< α 1.
0
− | |≤
Now, the equality α2+ β 2+ γ 2+ δ 2 =1 implies that
| | | | | | | |
1
β 2, γ 2, δ 2 1 (t ε)2 < ,
0
| | | | | | ≤ − − 10
and since z 1, we have
ij
| |≤
1+t ε<ϕ π (e x)= ϕ π (e x) = α¯ (α¯z +β¯z +δ¯z +γ¯z )
0− v i0 − | v i0 − | | − 11 12 22 21 |
3
α 1 z + z β + z γ + z δ 1 z + β + γ + δ 1 z + .
11 12 21 22 11 11
≤| || − | | || | | || | | || |≤| − | | | | | | |≤| − | √10
Letusobservethatπ (e)π (e) =v v =η η andπ (e) π (e)=v v =
i0 i0 ∗ 11 1∗1 1⊗ 1 i0 ∗ i0 1∗1 11
ξ ξ . Therefore
1 1
⊗
3
1<1+t ε 1 z = ϕ π (e x)
0− − √10 ≤| − 11| | 11 i0 − |
=(cid:12)ϕ11(cid:16)(πi0(e)πi0(e)∗)πi0(e−x)(πi0(e)∗πi0(e))(cid:17)(cid:12)
(cid:12) (cid:12)
(cid:12)≤(cid:13)(πi0(e)πi0(e)∗)πi0(e−x)(πi0(e)∗πi0(e))(cid:13) (cid:12)
= π(cid:13)((ee )(e x)(e e)) (ee )(e x)(e (cid:13)e) ,
k i(cid:13)0 ∗ − ∗ k≤k ∗ − ∗(cid:13) k
which proves the claim in (3).
ON THE EXTENSION OF ISOMETRIES BETWEEN UNIT SPHERES 7
Finally, since y F =e+(1 ee ) (1 e e) we can write
e ∗ A ∗
∈ − B −
y =e+(1 ee∗)y(1 e∗e),
− −
and we deduce from (3) that
1= y x ee∗(y x)e∗e = ee∗(e+(1 ee∗)y(1 e∗e) x)e∗e
k − k≥k − k k − − − k
= ee (e+x)e e >1,
∗ ∗
k k
leading to the desired contradiction. (cid:3)
The problem of dealing with minimal faces of the unit ball of a C -algebra A
∗
is that we need to handle minimal partial isometries in A (compare Theorem
∗∗
2.2). We present now a technical result which will be used later to facilitate the
arguments depending on the facial structure of .
A
B
Lemma 2.4. Let A be a C -algebra. The following statements hold:
∗
(a) Every minimal projection p in A A is orthogonal to all minimal projections
∗∗
\
in A;
(b) Every minimal partial isometry u in A A is orthogonal toall minimal partial
∗∗
\
isometries in A.
Proof. (a) Suppose p is a minimal projection in A A. Let q denote a minimal
∗∗
\
projection in A. Arguing by contradiction we assume that pq =0.
6
As in the proof of Theorem 2.3 let π : A ℓ∞B(H ) be an isometric -
→ i i ∗
homomorphismwithweak∗-denserange,where HiLI isafamilyofcomplexHilbert
{ }
spaces (consider, for example, the atomic representation of A [21, 4.3.7]). By
the weak -density of A in ℓ∞B(H ) and the separate weak -continuity of the
∗ i i ∗
productofeveryvonNeumaLnnalgebra,π(q)isaminimalprojectionin ℓ∞B(H ).
i i
Clearly, the images of the mappings Lπ(q) : x π(q)x and Rπ(q) :Lx xπ(q)
7→ 7→
( x ℓ∞B(H )) arecontainedin suitable Hilbert spaces. It followsthat the left
∀ ∈ i i
and rigLht multiplication operators Lq and Rq by q on A factors through a Hilbert
space, and thus they are weakly compact (compare [6]). Consequently, the spaces
(1 q)Aq, qA(1 q) and qAq = Cq are all reflexive. Applying the Krein-Sˇmulian
the−orem we dedu−ce that (1 q)Aq, qA(1 q) and qAq = Cq are weak -closed in
∗
− −
A , showing that
∗∗
(1 q)A q =(1 q)Aq, qA (1 q)=qA(1 q) A, and qA q =qAq =Cq.
∗∗ ∗∗ ∗∗
− − − − ⊆
We recall now an useful matricial representation theorem. Let C denote the
C -subalgebra of A generated by p and q. Since p and q are minimal pro-
∗ ∗∗
jections in A , Theorem 1.3 in [23] (see also [20, 3]) assures the existence of
∗∗
§
1 0
t [0,1] and a -isomorphism Φ : C M (C) such that Φ(q) = and
∈ ∗ → 2 (cid:18) 0 0 (cid:19)
t t(1 t)
Φ(p) = − . Since pq = 0 we know that t = 0. Clearly,
(cid:18) t(1 t) p1 t (cid:19) 6 6
− −
0 1p 1 0
Φ 1 qC(1 q) (1 q)Cq A, and Φ 1 =q A. Then
− (cid:18) 1 0 (cid:19)∈ − ⊕ − ⊂ − (cid:18) 0 0 (cid:19) ∈
0 0 0 1 1 0 0 1
Φ−1(cid:18) 0 1 (cid:19)=Φ−1(cid:18)(cid:18) 1 0 (cid:19)(cid:18) 0 0 (cid:19)(cid:18) 1 0 (cid:19)(cid:19)∈Φ−1(Φ(A∩C))=A∩C.
t t(1 t)
By linearity p=Φ−1(cid:18) t(1 t) p1 −t (cid:19)∈A which is impossible.
− −
p
8 F.J.FERNA´NDEZ-POLOANDA.M.PERALTA
(b)SupposenowthatuisaminimalpartialisometryinA Aandvisaminimal
∗∗
\
partial isometry in A. We shall first show that uu ,u u A A. Indeed, since
∗ ∗ ∗∗
∈ \
every minimal partial isometry in A belongs locally to A (compare Kadison’s
∗∗
transitivity theorem and [2, Remark 5.4 and Corollary 5.5]), there exists a norm
oneelementx Asatisfyingx=u+(1 uu )x(1 u u). Ifuu (respectively,u u)
∗ ∗ ∗ ∗
∈ − −
lies in A then u=uu x A (respectively, u=xu u A) which is impossible.
∗ ∗
∈ ∈
We have therefore shown that uu ,u u A A are minimal projections, while
∗ ∗ ∗∗
∈ \
vv ,v v areminimalprojectionsinA. Itfollowsfrom(a)thatuu ,u u vv ,v v.
∗ ∗ ∗ ∗ ∗ ∗
⊥
Finally, the identities u v = u uu vv v = 0 and vu = vv vu uu = 0 prove that
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
u v. (cid:3)
⊥
We are now in position to show that a surjective isometry between the unit
spheres of two C -algebras maps minimal partial isometries to minimal partial
∗
isometries.
Theorem 2.5. Let A be a C -algebra, and let H be a complex Hilbert space. Sup-
∗
pose that f :S(A) S(B(H)) is a surjective isometry. Let e be a minimal partial
→
isometry in A. Then f(e) is a minimal partial isometry in B(H). Moreover, there
exits a surjective real linear isometry
T :(1 ee )A(1 e e) (1 f(e)f(e) )B(H)(1 f(e) f(e))
e ∗ ∗ ∗ ∗
− − → − −
such that
f(e+x)=f(e)+T (x), for all x in .
e (1 ee∗)A(1 e∗e)
B − −
In particular the restriction of f to the face F =e+(1 ee ) (1 e e) is a real
e ∗ A ∗
− B −
affine function.
Proof. Arguing as in the beginning of the proof of Theorem 2.3, the set
F =e+(1 ee ) (1 e e)
e ∗ A ∗
− B −
is a maximal proper face of , and thus, by Proposition 2.1 and Theorem 2.2,
A
B
there exists a minimal partial isometry w in B(H) such that
∗∗
(6) f(F )=F = w+(1 ww ) (1 w w) .
e w ∗ B(H)∗∗ ∗ B(H)
− B − ∩B
(cid:0) (cid:1)
We claim that w B(H). Suppose, on the contrary that w B(H) B.
∗∗
∈ ∈ \
Theorem2.3 implies that 1 is an isolatedpointin σ(f(e)), andhence the func-
| |
tion χ belongs to C (σ(f(e))). Let f(e)=rf(e) denote the polar decomposi-
{1} 0 | | | |
tion of f(e). An application of the continuous functional calculus proves that vˆ=
r χ (f(e))isapartialisometryinB(H).Furthermore,sincef(e) f(F )=F ,
e w
{1} | | ∈
we deduce that vˆ F and
w
∈
(7) vˆ=w+(1 ww )vˆ(1 w w)
∗ ∗
− −
(compare the arguments in the proof of (2) in page 5).
In B(H) we can always find a minimal partial isometry wˆ B(H) satisfying
∈
(8) vˆ=wˆ+(1 wˆwˆ )vˆ(1 wˆ wˆ).
∗ ∗
− −
Since, by assumptions w B(H) B(H), Lemma 2.4 implies that w wˆ,
∗∗
∈ \ ⊥
wˆ =(1 ww )wˆ(1 w w)
∗ ∗
− −
and hence, by (7) we get
vˆ wˆ =w+(1 ww∗)(vˆ wˆ)(1 w∗w).
− − − −
ON THE EXTENSION OF ISOMETRIES BETWEEN UNIT SPHERES 9
By hypothesis,
2= f(e)+wˆ = f(e) ( wˆ) = e f−1( wˆ)
k k k − − k k − − k
where e is a minimal partial isometry in A. Proposition 2.2 in [16] proves that
eˆ=f 1( wˆ)= e+(1 ee )eˆ(1 e e).
− ∗ ∗
− − − −
By construction f(e)=vˆ+(1 vˆvˆ )f(e)(1 vˆ vˆ), and by (8),
∗ ∗
− −
f(e)=wˆ+(1 wˆwˆ∗)vˆ(1 wˆ∗wˆ)+(1 vˆvˆ∗)f(e)(1 vˆ∗vˆ),
− − − −
and consequently f(e) wˆ 1. Having in mind that
k − k≤
f(e) wˆ =f(e) (1 ww )wˆ(1 w w) F +(1 ww ) (1 w w)
∗ ∗ w ∗ B(H) ∗
− − − − ∈ − B −
we get
f(e) wˆ =w+(1 ww )(f(e) wˆ)(1 w w) F .
∗ ∗ w
− − − − ∈
Wededucefrom(6)thatz =f 1(f(e) wˆ) F ,andthusz =e+(1 ee )z(1 e e),
− e ∗ ∗
− ∈ − −
which leads to
1= f(e) = f(e) wˆ+wˆ = (f(e) wˆ) ( wˆ) = f(z) f(eˆ)
k k k − k k − − − k k − k
= z eˆ = e+(1 ee )z(1 e e) ( e+(1 ee )eˆ(1 e e))
∗ ∗ ∗ ∗
k − k k − − − − − − k
= 2e+(1 ee )(z eˆ)(1 e e) =2,
∗ ∗
k − − − k
and hence to a contradiction. Therefore w B(H) and
∈
F =w+(1 ww ) (1 w w).
w ∗ B(H) ∗
− B −
We canarguenowasinthe proofof[22,Proposition3.1]to conclude. We insert
a short argument here for completeness reasons. We have established that
f e+ =f(F )=F =w+ .
(1 ee∗)A(1 e∗e) e w (1 ww∗)B(H)(1 w∗w)
B − − B − −
(cid:0) (cid:1)
Let denote the translation with respect to x , that is (x) = x+x . The
Tx0 0 Tx0 0
mapping f = 1 f is a surjective isometry from
e Tw− |f(Fe) ◦ |Fe ◦ Te|B(1−ee∗)A(1−e∗e)
onto . Mankiewicz’s theorem (see [19]) im-
(1 ee∗)A(1 e∗e) (1 ww∗)B(H)(1 w∗w)
B − − B − −
plies the existence of a surjective real linear isometry T : (1 ee )A(1 e e)
e ∗ ∗
− − →
(1 ww )B(H)(1 w w) such that f =T and hence
∗ ∗ e e S((1 ee∗)A(1 e∗e))
− − | − −
f(e+x)=w+T (x), for all x in .
e (1 ee∗)A(1 e∗e)
B − −
In particular f(e)=w.
For the final statement we simple write
f|Fe =Tw|B(1−ww∗)B(H)(1−w∗w) ◦fe◦Te−1|Fe =Tw|B(1−ww∗)B(H)(1−w∗w) ◦Te◦Te−1|Fe
as a composition of real affine functions. (cid:3)
The next technical lemma is obtained with basic techniques of linear algebra.
Lemma 2.6. Let H be a complex Hilbert space, and let v ,v ,v , e and e be
1 2 3 1 2
minimal partial isometries in B(H) satisfying e e , v v , v v ,
1 2 1 2 1 3
⊥ ⊥ ⊥
v +v =e e , and v +v =e +e .
1 3 1 2 1 2 1 2
− −
Then v =e and v =e =v .
1 2 2 1 3
10 F.J.FERNA´NDEZ-POLOANDA.M.PERALTA
Proof. Since v v ,v , by multiplying the identities v + v = e e and
1 2 3 1 3 1 2
⊥ − −
v +v =e +e on the left by v we get
1 2 1 2 1∗
v v =v e v e , and v v =v e +v e ,
− 1∗ 1 1∗ 1− 1∗ 2 1∗ 1 1∗ 1 1∗ 2
which shows that v e =0. Multiplying by v on the right we prove e v =0. We
1∗ 1 1∗ 1 1∗
have therefore shown that v e .
1 1
⊥
Applying that e e and v v we get e e +e e = v v +v v where
1 ⊥ 2 1 ⊥ 2 1 ∗1 2 ∗2 1 1∗ 2 2∗
e e and v v are orthogonalrank one projections. Thus, e e =e e v v , and by
1 ∗1 1 1∗ 1 ∗1 1 ∗1 2 2∗
minimality v v =e e . We can similarly prove v v =e e . Finally
2 2∗ 1 ∗1 2∗ 2 ∗1 1
v =v v v =v v (v +v )=v v (e +e )=e e (e +e )=e ,
2 2 2∗ 2 2 2∗ 1 2 2 2∗ 1 2 1 ∗1 1 2 1
and the rest is clear. (cid:3)
Next, we shall establish several consequences of the above theorem.
Theorem 2.7. Let f :S(B(K)) S(B(H)) be a surjective isometry where H and
→
K are complex Hilbert spaces with dimension greater than or equal to 3. Then the
following statements hold:
(a) For each minimal partial isometry v in B(K), the mapping
T :(1 vv )B(K)(1 vv ) (1 f(v)f(v) )B(H)(1 f(v) f(v))
v ∗ ∗ ∗ ∗
− − → − −
given by Theorem 2.5 is complex linear or conjugate linear;
(b) For each minimal partial isometry v in B(K) we have f( v) = f(v) and
− −
T = T . Furthermore, T is weak -continuous and f(e) = T (e) for every
v v v ∗ v
minima−l partial isometry e (1 vv )B(K)(1 v v);
∗ ∗
∈ − −
(c) For each minimal partial isometry v in B(K) the equality f(w)=T (w) holds
v
for every partial isometry w (1 vv )B(K)(1 v v) 0 ;
∗ ∗
∈ − − \{ }
(d) Letw ,...,w bemutuallyorthogonalnon-zeropartialisometriesinB(K),and
1 n
let λ ,...,λ be positive real numbers with λ =1. Then
1 n 1
n n
f λjwj= λjf(wj);
Xj=1 Xj=1
 
(e) For each minimal partial isometry v in B(K) we have f(x) = T (x) for every
v
x S( );
(1 vv∗)B(K)(1 v∗v)
(f) Fo∈r eacBh p−artial isom−etry w in B(K) the element f(w) is a partial isometry;
(g) Supposev ,v aremutuallyorthogonalminimalpartialisometriesinB(K)then
1 2
T (x)=T (x) for every x (1 v v )B(K)(1 v v ) (1 v v )B(K)(1
v1 v2 ∈ − 1 1∗ − 1 1∗ ∩ − 2 2∗ −
v v );
2 2∗
(h) Supposev ,v aremutuallyorthogonalminimalpartialisometriesinB(K)then
1 2
exactly one of the following statements holds:
(1) The mappings T and T are complex linear;
v1 v2
(2) The mappings T and T are conjugate linear.
v1 v2
Proof. (a) Let v be a minimal partial isometry in B(K). Suppose that
Tv :(1 vv∗)B(K)(1 v∗v) (1 f(v)f(v)∗)B(H)(1 f(v)∗f(v))
− − → − −
isthesurjectivereallinearisometrygivenbyTheorem2.5. Havinginmindthat(1
−
vv )B(K)(1 v v)=B((1 v v)(K),(1 vv )(K)) and(1 f(v)f(v) )B(H)(1
∗ − ∗ ∼ − ∗ − ∗ − ∗ −
f(v)∗f(v))∼=B((1−f(v)∗f(v))(H),(1−f(v)f(v)∗)(H))areCartanfactorsoftype