GLOBAL CROSS SECTIONS OF UNITARY AND SIMILARITY ORBITS OF HILBERT
SPACE OPERATORS
MARTA PECUCH HERRERO
INTRODUCTION
In [2] D Deckard and L Ạ Fialkow characterized the operators on a sepa- rable Hilbert space which have local] unitary cross sections These are the operators of the form A ® B@ 1, where A and B are operators on finite dimensional spaces and 1 is the identity operator on a complex separable infinite dimensional Hilbert space #
The purpose of this article is to characterize the operators which have global unitary cross sections or global similarity cross sections To make these statements more precise, it is necessary to introduce some standard notation, which will be used throughout the paper: Y(#) denotes the algebra of all bounded linear ope- rators on #, &(#’) denotes the group of unitary operators in YH), UT) = = {U*TU: Ue U(#)} is the unitary orbit of an operator T in ⁄() and +:9() + U(T) is the (norm continuous) function defined by 1(U) = U*TỤ A local unitary cross section for t is a pair (y, #) such that @ is a relatively open subset of (7) which contains T and »: #@ + Y(#) is a norm continuous function such that tom =lg and o(T)=1; @ is a global unitary cross section when B=UT)
Trang 2The second part of the paper is devoted to the study of similarity cross sec- tions Let G(#) denote the group of invertible operators in @(#) and let ĂT) =: -= {W-1TW :W eG(#)} be the similarity orbit of an operator T in ¥Y(#) The (norm continuous) function s:¢(9) + F(T) is defined by s(W):-: M-1TM, L Ạ Fialkow and D Ạ Herrero ({5], [6]) proved that s has a local cross section if and only if Tis similar to a nice Jordan operator, that is, an operator of the form
n ay (a, ) @® Aly, + @® q„” >
ial ‘ gạ Ch
where 4q, 2s, ., 2„ are distinct complex numbers, g, is the k x & nilpotent Jordan cell, and for each ?, kj, < kip << < Kim, and a;; = co for at most one value of / In Theorem 17 it is shown that s has a global cross section if and only if 7 is similar to a very nice Jordan operator, that is, a nice Jordan operator of the form
„
® Gz + a)
ie
For operators on finite dimensional spaces and for the Calkin algebra, it is shown that there are no global cross sections except in the trivial case when the operator T, or respectively its image T in the Calkin algebra, is a multiple of the identitỵ As in the case of the unitary orbits, it follows from the Calkin algebra result that if 7 has a global similarity cross section ¢, then ¢ cannot be constructed so that ¢(7,) —
~— ¢(1,) is compact whenever 7,, T,¢ (T) and 7, — T, is compact, except, of course, in the trivial case when T is a multiple of the identity operator
[14] and [15] are used as standard references for Algebraic Topology throughout the paper
The author wishes to thank Professors N Salinas and D Voiculescu for pro- viding several helpful references, and to Professor D Ạ Herrero for suggesting the problem and for encouragement
PART 1 1 A CHARACTERIZATION OF THE UNITARY
ORBIT AS A HOMOGENEOUS SPACE
Let T be an operator in Y(#) and let »/'(T)= {Ae £(#): AT: TA} be the commutant of T
Trang 3Proof Let p:U(H#) ¬ ()/()n/'(T) be the canonical projection Since «71 ({T}) = UH) 0 f(T), there is a bijective continuous map ¢ : U(#)/ [U(#) 0 ĂT) > UT) which makes the following diagram commutative:
2() ——> #()/2() n.#'(T)
(1.1) 4 <a
UIT) °
If T satisfies the hypothesis, then t is an open mapping [2], [4], and it follows easily that g is indeed a homeomorphism
Remark Since %(T) is pathwise connected, W(H#)/WH) 1 x'(T) has the
same propertỵ
Consider the commutative diagram (1.1) In [2] it was shown that t has a local cross section; this implies that p also has a local cross section via the homeo- morphism ¿ Since 2()n Z (7) 1s a closed subgroup of %(#), it follows [1S, p 57) that
2(#)n #'(T) ——> %() —”—> #()J2() n (1)
is a fibre bundle (¡: 2(⁄)n.Z{(T) > UA) is the inclusion map.) Via g, the existence of a global (norm continuous) cross section for t is equivalent to the existence of a global cross section for p
From now on we will only consider the latter question, that is, we will work with the above mentioned fibre bundlẹ
First of all we need to determine the homotopy type of #(3/)n '(T) This is done in the next section
2 CHARACTERIZATION OF 2(⁄)n (7)
PROPOSITION 2 If A = N° ~ N@1,, for some irreducible operator N in L(C"), then the unitary operators which commute with A are the operators of the jorm 1,@ U with Ue @(C”)
Let A=4,@4,@ @A, and B= BOB @ @OB,, where 8; and A, are irreducible operators for each i,j, 1 <igm, 1 <j<n lt BFA, for every i,j, we say that A and B are disjoint In this case, it follows from [3, p 8] and a simple calculation that if U(A ® B) = (A ® B)U for some unitary operator
U, then U = U, ® U,, where U, commutes with A and U, commutes with B
Let T=A@BO@1 with A=A OAV @ @APM ELC), B=
= Bỉ @ B® đ Ba ô LCD, where 4,ẹZ(C) and B,ẹZ(C”) are
Trang 4We can assume that A and B are disjoint: if A; ~ B; for some pair i,j, then we can write (s,) T= Ả ® OAL OA @ OAM © BY O10 OB @1@ ©{4?°@ B/” @1) © B7 @ 1@ @ BH” @1~ jy) ~ A ® OAL? @ AH © @A @ BOỊ
(That is, all the common irreducible terms can be “absorbed” in B) Also notice
that BO1~B@1@ @B,@1, because Be @1~ B® 1, @1
~ B; @ 1 Hence, we can directly assume that ¢; = 1 forall j, 1 <j < m
From Proposition 2, the above remarks and the fact that m(#(C®) == Z for all k > 1, and 2(#) is contractible [11], we obtain:
PROPOSITION 3 For an operator T= A ® B@ 1 of the above form, UH) 0 f'(T)={1,,® U,@® @®1, @U,@1,@W;@ @1, @W„:
:U,e(C”), V,e#(,}
and
m(UH) 1 ĂT) =n (WUC) Ox (WC ))@
Om (UH) On (UH,))=Z
3 THE EXISTENCE OF GLOBAL CROSS SECTIONS
THEOREM 4 Let T= B@ 1, where B is an operator on a finite dimensional spacẹ Then the fibre bundle
(3.1) UH) 0) tT) —> UH) ——> U(#)|UH) 0 f(T) has a global (continuous) cross section
J=1
Proof By Proposition 3, &(#)n#(T) = lê 1, @ E;,: Vị;e#(#,) je24,2, , my It follows that %#(3/) n f’(T) is isomorphic with a (finite) pro-
duct of copies of #(#) and therefore it is contractible [11]
Trang 5(which obviously has global cross sections), it will follow that p has a global cross section
Consider the exact homotopỵ sequence
—>m(0#) n #'(1)) “> (UG) A> (UA YUH) 1 f'(T)) ~>
(3.2)
2 (UH) 0 #'(1)) <> > (UH) ND)
corresponding to the fibre bundle (3.1) Since x„(2(2)) = 0 for all z and x„(()n nZ(T))=0 for all ø, we obtain z„(()/2(⁄)n ZŒT)) =0 for all n > 1 Since UH UH) Nn '(T) ~ X(T) is pathwise connected, we also have 2,(U(#)/ [U(H) 0 L'(T)) = 0
This means that @()/W@(#) N x'(T) is weakly homotopy equivalent [14] to a contractible spacẹ To prove that it is actually contractible, it will be enough to show that it is homotopy equivalent to a CW-complex [14] This is done in the following auxiliary result
If Z is a subset of L(H#), then we define Z* = {R*: Re BZ}
Lemma 5 (4(#),G(#) 0 LT), HH) (GA) 0 S(T) Nn [#(⁄) n N'(T))}*) and (U(#), UH) '(T)) are homotopy equivalent relative to CW-complexes
Proof Since G(#) is paracompact and (4) 0 x'(T) is a closed subset of G(#), according to [13] it is enough to prove that the following property is satisfied: there is a neighborhood Q of the diagonal in (#7) x GH), a (continuous) function 2:2x[0, 1] ~ G&A) and an open covering {4,} of YH) such that the following properties are satisfied:
ĂV,W,0) = W for every (V,W) in Q, 2(V,W,1)—=W_ for every (V, W) in Q,
ÀĂV,V,!)=V_ for every Vin (7) (0 < t < 1),
AV Wi teGA) nf (T) if VW) is in HH)0'(T) X OP) ĂT) (<< i),
Py X Py <Q and
AG, < Sy X [0,1)) = S, for every ©, in the covering
Let Q= ((V,W)e GH) x WH) = \|\V— WI < min{\| VU, WU -H},
let O, = {We GH): ||V — W|| < 1/3||V-1|)} for each Win #(Z) and let 2: Qx <[0, 1] #2) be defñned by 2(V, M, £) = 0V +- (1 — OW
Trang 6To prove the second statement of the lemma, we notice that 2(V, W, t) belongs to [F(F)n ĂT) N{[FA)n we (T) if V and W are in [GH)N WV (TIS
1 [G(H#H) 1 Z/'{T)]*, so that the same proof holds in this casẹ
For the last statement, consider the map r: G() > W@(#) defined by
r{ƒ⁄)= V*V)~!* for V in Ø(#)
If 7: (2) ¬ #() denotes the inclusion map, then ro/ = lacy) Thus r is a retraction
TẾ Ve[#(⁄Z)n Z(Tjln[Z(⁄)n./'(T))*, then both V and V* commute with T and, a fortiori, the whole W*-algebra W*(V) generated by V is contained in «/(T) In particular, (V*V)—1? € ó(T) Since V is invertible, so is (V7V)~2?, and therefore r(V) == V(V*V)~¥?2 e€ UMA) 1[FA) 1 ĂT) NGM) 0 (TI) = == U(H) 1 f(T) Hence,
r|[[Z(#)n 4()1n [#(C#)n 01]: :12()n Z')1n1Ø()n
Nn f(T)" > (⁄)n 0) Therefore r induces a map of pairs
r: (#(#), [2(#)n ĂT) 1[HL) 0 L(T)") > (UH), MH) AL"T)), where [Z(⁄)n.Z(7)]n{[2()œ.Z(T)Ị! and ()n ¥'(T) are closed subsets of G(A#) and U(#), respectivelỵ Since a retract of a relative CW-pair is also a rela-
tive CW-pair [12, p 127], [14], the proof is completẹ
The proof of Theorem 4 is now completẹ
By using Lemma 5 and [14, p 402] we obtain the following:
CoROLLARY 6 #(#)/Ø()n (T7), Ø(⁄)/0Ø(#)n /(T)1n[2() n.2/(T1]E and (3)/2()n '(T) are homotopy equivalent to CW-complexes
THEOREM 7 Let T= A@®B@1, where A and B are disjoint operators on finite dimensional spaces, and A #0 Then the fibre bundle (3.1) does not have a
global cross section
Proof In the exact homotopy sequence (3.2) we have x„(2(2)) = 0 for all n and m(2()n.Z/'(T)) = Z™ (Proposition 3) for some m2 1 Therefore T„+ă()/()n f'(T)) = 1,(U(H) 1 '(T)) and, in particular,
TMH fU(A) 0 Z'(T)) ~ m(2(⁄)n Z'(7)) = ZO,
If there exists a global cross section for p, then the exact homotopy sequence
Trang 7splits [14, p 418], that 1s, 2(W(#)) ~ m;(2() n '(T)) ® m;(()/2()n fn xf'(T)) But Z cannot be a direct summand of 2,(%(#)) == 0; therefore p does
not have a global cross section
4 THE FINITE DIMENSIONAL CASE
In what follows, the notation U(n) = &(C") will be used Let A be an operator on C” Proceeding exactly as in Proposition |, we see that %(A) is homeomorphic with U(n)/U(1) 0 (4) (right cosets) and that
(4.1) U0) n z/(4)—> UÚ) ”> U0)JU(@w) n2!)
is a fibre bundlẹ (Indeed, this can also be proved by a direct argument.)
LEMMA 8 Let A= NO ~ N@1,,, where N is an irreducible operator in L(C%), n = qm Then the fibre bundle (4.1) does not have a global cross section unless q = 1, that is, unless A is a multiple of the identitỵ
Proof By Proposition 2, U(n)n (A) = {l,@ U: Ve U(m)} Hence, m(U(nn.Z{49)) ~ m(U0n)) ~ 2
The last terms of the exact homotopy sequence of the bundle we are consi- dering are
Z > Z =>z/(U(/U0) n '(4)—>0
We will show that /,(z) = gz for every z¢ Z The generator of 2,(U(n)) is the homotopy class of the map y, :{0, |] ~ U(), where y,(t) is the diagonal matrix diag(ẻ*", 1,1, ., 1) Since io y,(t) = diag(e’*", 1, ., le, 1,0 1, , Ă€?z 1, , , |), where ec? appears g times on the diagonal and the remaining ele- ments on the diagonal are equal to J, itis easy to show that jo y,, and qj, are homo- topic Therefore, i,[y,,] == đ[,}
Since 2,(U(#)) = Z is commutative, if there is a global! cross section, then the exact homotopy sequence will also split at z, But
n,(U(n)/U(n) n f'(A)) ~ 2 (U(n))/ker p., = a (UC) /im i, ~ Z/qZ = Z, Since Z, cannot be a direct summand of Z, there is no global cross section unless g:= 1
; (4) (s,) (s,)
Lemma 9 Let 4 = 4! @®Ao* @® @A,", 1 2 2, where the operators A, € £(C*) are irreducible and disjoint, and ¥) sik; == s Then the fibre bundle
(4.2) U(s) 0 Z⁄(4)—> UG) -> UG)JUG) ñ 3⁄44)
Trang 8Proof 7(U(s) 0 #(A) = Z@Z@ ®@ Z= Z™ (Proposition 3) and 7, (U(s)) = Z
The last terms of the exact homotopy sequence of the fibre bundle (4.2) are
= Z® ~ Z = m(UG)/UG)n '(4) — 0
Since n > 2, Z™ cannot be a direct summand of Z; therefore the exact homo- topy sequence cannot split This implies that the fibre bundle has no global cross
section B
Combining Lemmas 8 and 9, we obtain the following result
THEOREM 10 An operator acting on a finite dimensional space has a global unitary cross Section if and only if it is a multiple of the identitỵ
5 UNITARY CROSS SECTIONS IN THE CALKIN ALGEBRA
We need to introduce the following notation: % denotes the ideal of compact operators in Z(H), SL = L(A) KH is the Calkin algebra, (4#) denotes the unitary 8roup m Z and 2⁄;(.4Z) 1s the pathwise connected component of iin Uf), where T denotes the image of T ¢ L(#) under the canonical projection onto / Let #(T): = =: {V*FV: Vin Ut) In this section we will only consider operators of the form T::=A@®B@i1, AC L(C), Be LC’), for which t: U(x) ~ UT) is known to have a local unitary cross section [2] Also, notice that 7 is unitarily equivalent to (B@ 1)”
As before, we have a commutative diagram
%(#) ——> #()/#(#) n.'()
UP) a
where the map @ is continuous and bijectivẹ We will prove that if T is not a mul- tiple of the identity, then p does not have a global cross section, in which case t cannot have one either: if €: UT) — ⁄(.Z) is ănorm continuous) mapping such that to ¢ = Lag ¿Œ) =z 1, then @spsŠs@ =tofom = @ Since @ is injective, this implies that p o € og = 1 Therefore &o y is a cross section for p
THEOREM 11 Let T= B@1, where Be Y(C") The fibre bundle
%()n #'(Ð —> Ø(#) —>#(#)/2(#) n #'()
Trang 9Proof Consider the following morphism of bundles:
UA) 1 ol'(P) ——» Ut) 0 b'(T)
th £
(5.1) %(#)——————>x#{)
Py ?
%()/%4#)n /()—>()/(#)n fF)
where 1, %, i and /, are inclusion mappings, p and py are the canonical projections onto the classes of right cosets and :’’ is induced by the inclusions i and 7’ so that the diagram is commutativẹ Itis easy to see that i’’ is well defined and injectivẹ Since U,( xf) is the (pathwise connected) component of I in Ws), we have
ig | Ty(Up( I), 1) ~ 1, (U(x), 1) for n> 1
Observe that the path component of I in &(s/) N (7) is equal to the path component of | in Uf) 0 ó(T), whence
it (Ug) 1 8'(T), Ú) ~ m(@(#)n 7), 1) for n> l
Now consider the exact homotopy sequences of the bundles in (5.1) and the induced homomorphism i,, i, and iỵ We have the following commutative diagram: (Aol) 0 ob'(F), 1) 2 rey Uo), 1) P 1( g(t) [Ug ot) ñ 22G), p(Ñ) -> 1" i i i * ty * * ->m(()n 2), 1) ~> m(0(),Ï) —Š m(0(2)/2(2) n f'(P), ply) 2s (5.2)
Since i,, and i, are isomorphisms, the 5-lemma implies that ij’ is also an isomorphism Therefore if one of the exact sequences splits, the same holds for the other onẹ We will show that the upper sequence in (5.2) does not split unless T isa multiple of 1 This implies that the exact homotopy sequence of p : U(x) > 2⁄(Z)/2(Z)n
Trang 10Let r: GH) ¬ U(H#) and h: GH) xX [0, 1] + GH) be defined by r(Vf)== V(V*V)~1!2 for Vin GH)
and
AV, D2 VI — t(V*V)-1E + rlỊ for Vìn Ø(#),0 <t< 1,
respectivelỵ It is easy to see (by using the spectral theorem for compact hermitan operators) that ĂGM)N(L +X) X (0,1) <— #⁄)n(L-+-Z) and r((#)n
ă +#)) < U(H#)N( +) Therefore r and A give maps of pairs r:(#(#).2(#)n(1 — )) > AH), MH) ACU + ) and
h: (GH) x [0, 1], #(Z)n( + ) x [0, l) ¬ (@(), ##)n(1 ‹- #))
with ĂV,0) =r(V) and ;(, l) = V Thus, &@(#) is a deformation retract of GH), UA) AU + #) is a deformation retract of GS/)N (1 + #) and #(#)‡ ;{Z)0n (1 ++ 4H) is a deformation retract of #(Z)/2(#)n(1 +2) As a con- sequence,
n((#)/#(#)n( + #)) > m(0(#)/2(#) ñ(L +2), n>0
Since #()/Z()n(1 +2) is homotopy equivalent to the space of Fredhoim operators of index zero [10], which is homotopy equivalent to BU (the classifying space of vector bundles with group U(oo) = lim U()) [1], [10], we obtain
0 #n=2m — 1
x(44(22))=,()/#(4#)n (L+2))~>x,(BU)= | | + IlnH=2m, m 2 Ì
[15, p 215]
Case Ị Let T= B@1, Be L(C") If B=A®@I]1,, where 4EL(C’) is irreducible, then B@ 1 ~ A@1, @ 1 is unitarily equivalent to A @ 1 Therefore we can assume that B = A is irreduciblẹ By using a faithful unital *-representation of the finite dimensional C*-algebra C*(T), and Proposition 2, one can show that %(Z)n.Z⁄Œ) = {(„@ U)”:Ue()} Therefore
Ty( No(s) 0 f(T), 1) = m(®()1#()n( +2), n >1
Let [y] be a generator of z;(2()/2()n(1 + )) >~ 7:
Trang 11If y(s, t) = (U(s, 2), then p(s, 1) = (Us, 1)” = U, @ U(s, t))~ Thus, the inclusion ;¿;: 2¿(.Z) Nn 5!) — 2⁄g(.Z) induces the homomorphism ‘“‘multipli- cation by 2°: rạy: z;(2¿(.Z) 0 á(T), 1)¬n,(2,(), 1), lgy(Z) == nz for every z in mă%()n xf'(T), 1) It follows that in the exact homotopy sequence
có >n((#)n 2, D -%n(Cø), 1) 2S tro Uo ol) Ugl st) 0 of'(T), p(t) +0
~ ~
(5.3)
we have z;(,()/2.() n 'Œ), pÑÑ)) ~ ZjnZ = Z,
This implies that for 7 > | the exact homotopy sequence cannot split because Z, cannot be a direct summand of Z Therefore py does not have a global cross section unless n == 1, that is, unless T is a roultiple of 1,
„é%
Case 2 Assume that 8= @ B,, m > 2, where the operators 8; (acting
i=l
on finite dimensional spaces) are irreducible and disjoint Let %@(1) be the pathwise connected component of 1 in Uy(f) 1 F xvài If V is in HT), then by using a faithful unital “representation of C*(7) and Proposition 3, we can show that ~ mona m V= @ V;, where the V; ’s are Fredholm operators such that ` ind V; = 0 Let i=l ¿=1 a: [0,1] > %(F) be a path from I to V Then by applying to ăr) the same argu- ~ m ~ ment as to V, we obtain ăt) = @ a,(t) Since «,(0) = 1,;, Vi; = #,(1) isin %8, @1)ˆ7 i=]
for each ft, i= 1,2, .,m Therefore
Trang 12The results in this section have an interesting consequencẹ D Deckard and L Ạ Fialkow [2] showed that if t has a local cross section ọ, then it is possible to choose ¢ in such a way that whenever two operators T,, J, in (7) have a com- pact difference T,—T., then o(T,)—(T,) is also compact
If T # 41, then the analogous statement is false for global cross sections Namely, if t has a global cross section g and T # /1, then it is impossible to choose y in such a way that compact differences are preserved, because then there would be a global cross section in the Calkin algebra, in contradiction with Theorem 11
PART IỊ
6 A CHARACTERIZATION OF THE SIMILARITY ORBIT AS A HOMOGENEOUS SPACE
THEOREM 12 (L.ẠFialkow—D.ẠHerrero [6]) An operator T in Ở() has @ local similarity cross section if and only if it is similar to a nice Jordan operator Furthermore, in this case the continuous bijection » that makes the dia- gram
GH) > GH GH) 0 '(T) s
ST)
commutative is actually a homeomorphism, and a local cross section (~, #) can be constructed so that if T,, T, are in S(T) and T, — T,€%, then @(T)) — p(T.) €% Remark [f T is similar to a nice Jordan operator, Theorem 12 implies that GH GH) 0 f(T) is pathwise connected Moreover, since G4) (T) is a closed subgroup of 4(#), the existence of a local cross section for p implies that
WH) 0 sh'(T) > WH) A> GHG H) 0 f(T)
is a fibre bundle (14, p 57] (is the inclusion map)
Also, via y, the existence of a global cross section for s is equivalent to the existence of a global cross section for p
7 CHARACTERIZATION OF 9(%) 0 #(T)
Let T be similar to a nice Jordan operator J Then T= W-JW for some W in G(#) and
Trang 13„ Therefore AT) = (J), so it is enough to characterize o/'(J) Let # = = , ỞQ= ° dạ 2 )ẹZ() Gn? <.Z(,), L<m<ng< <m, 1<a;< 00, for cach j=l, "2, ,k) Each #; can be decomposed as a direct sum #7; = @) of qn with r,s=1 = 8 2? (with dim #} = a,) in such a way that the matrix (Q; i 1
respect to this decomposition satisfies Q;,-,.41 = le for l1 <r<7m;— landQ,;;= 0 for any other palr Œ, s)
Let 4c Z() and let (44;;,,:) be the matrix of 4 with respect to the decom-
k tr :
position #H = @ (3 #3) (Aij,rs': #5 — 27) Straightforward computations show
j=1 Vist /
that the commutant ’(Q) is the set of all the operators A in #(#) which satisfy the following conditions:
a) If j 2 i, then Aijis = Aij,2,s41 —= = Aij,njt1 5.n,; for 1 + Ì — Hị S < s <n; and Ajj,-s = 0 for allr > s+ n,; — nj
b) If j < i, then Aijjis = Aijest1 = 1 = Aij,njs tlas,n, for l<s< Hy and
Ai, =0 for all r > s, and
C) Ajjis:27 => 2/} is an arbitrary bounded linear mapping
Let Ajj,s = Aij,1s and let ĂA) = © A0) J1
The above notation is used in the following:
k
Lemma 13 7f O=@ an? then
j=1
GH) 1\ of'(Q) = {4 = (Aij.rs) € B'(Q): ĂA) = © A02 is inert
Lemma 14 G(#) 0 '(Q) is homotopy equivalent to a product xX {WC!):1 <j <k, a; < oo} Proof By Lemma 13, (97) 0 '(Q) is the set of matrices of operators (Ajj, +s) k in (@) with ĂA) = ® 4 invertiblẹ Define ù:#(2)n.Z'(@) x [0, 1] > #(⁄)n.Z'(@) by (A, t) = ĂA) + t(A — ĂA)) Clearly, ACA, 1) is in GH) 1 4/(Q) for cach / in [0,1] and /# is continuous
Since ĂA, 1) = A and ĂA, 0) = ĂA) for all A in GY(9/)N '(Q), we see that the set of invertible matrices of the form 4(A) is homotopy equivalent to
GH) N (Q)
Trang 14Next, observe that the mapping d:x{Ø(C?):1 <7 < k} ¬ {Ă4):A in Ø()n Z'(0)} (if a; = 00, then Ci must be understood as #) defined by k k d(Ay, , Ay) = @ AV? = ® A; @ In, fol fa ; is a homotopy equivalencẹ Since G(#) is contractible [11], we are donẹ Z CoRoLLaRy 15 m,(Ø()n (0) = ®{z,((C?)):1 < j < k, ø; < co} In particular, m((2) n f'(Q)) = LH,
where B == card{j: a; < co}, and GH) 1 f'(Q) is pathwise connected
Proof This follows from the fact that m(C”) = Z YA Proposition 16 Let J =: ® [/;1; + Q;] be a nice Jordan operator (2¡,
deal
mM
Jaye sey hyq are distinct complex numbers) with QO; = ® q
j=l
< <k;,,, tm; and %;; == 00 for at most one value of j for each i, 1 <i <n Then GH) 0 f'(J) is homotopy equivalent to @ product
(a; :)
, leky<kp<
Os
X{HC"):a;;< 00, 1<j<m, 1 <i <n}
Proof The fact that /(A,1,;+ @j) -:⁄Z(Ø) and o(f41,+ 0) no(Ayl, + O,) = Ø for ? # h imply [8] that
GH) 0 ) ~ x ،,)n (0) i=l
Now the result follows from Lemma 14 f3
§ GLOBAL SIMILARITY CROSS SECTIONS
THEOREM 17 Let T be similar to a nice Jordan operator The fibre bundle
Trang 15where G(T) = GH) 0 f(T), has a global cross section if and only if T is similar to avery nice Jordan operator, that is, T is similar to an operator of the form
m
@A; + qk, @ 1), where 41, 49, ,2, are distinct complex numbers and q,is the
J1 |
kxk Jordan cell in Y(C*)
Proof Assume that 7 is similar to a very nice Jordan operator By Proposi- tion 16, G(T) ts homotopy equivalent to the product of m copies of G(s/); therefore it is contractiblẹ By substituting G(#/) for (2) in the proof of Theorem 4, we obtain that (#)/F(T) is weakly homotopy equivalent to a contractible spacẹ By Lemma 5, (G(#), G(T)) is a relative CW-complex, whence (#)/G(T) is a CW-com- plex Therefore 4(#)/G(T) is actually contractible [14] As in the proof of Theorem 4, it follows that p has a global cross section
i=1
lf T is similar to a nice Jordan operator J = ® (2 + ® 4 ?) with some ui; < 00 (ịẹ, J is not very nice), then Corollary 16 states that t(GT)) = Z®,
”
where B =: Yj card {j:a,; < oo} > |
fic
If we substitute G97) for &(H#) and B for m in the proof of Theorem 7, we obtain that p does not have a global cross section
In the finite dimensional case, we have the following result: THEOREM 18 Let Té Y(C") The fibre bundle
GT) > GC) 4 GCYGT),
where G(T) == GC") N &'(T), has a global cross section if and only if T is a multiple of the identitỵ —
Proof Let us assume that 7 is similar to J = ® Ez + ° qq 2 where
¡=1
Ay, Ag, A, are distinct complex numbers and 1 < kj < cài <kj,, TẾ there are two or more terms of the form qả then 2,(G(T)) = 7,(G(J)) has at least ii
two direct summands isomorphic with Z Therefore the exact homotopy sequence
— m((1)) ~> m(Ø(C9) ~ Z2 x.(2(C9/2()) ¬ 0
cannot split This implies that p does not have a global cross section
Now assume that 7is similar to Al,,, + ¢% Let G(g”) denote the intersection
Trang 16Let [y,,] be the generator of 2,(U(m)) = 2,(@(C”)) defined in Lemma 8, and let d: GC”) — Gqk”) be the mapping defined in Lemma 14 Since d(y,,(t)) = == diag(y,,(4),- -:¥m(t)) is homotopic with (y,.4(f), we have ide[ynl == &[y„„Ì A repetition of the argument in the proof of Lemma 8 (with g replaced by k) shows that p does not have a global cross section unless k = 1 f 2
9 SIMILARITY CROSS SECTIONS IN THE CALKIN ALGEBRA
With the same notation as in Section 5, let Y(.2/) denote the group of inver- tible elements of »/ and let Yo(.27) be the pathwise connected component of Tin GA) Let
S(T) = {W-1TW :W #(#)}
and let s: (2) > S(T) be defined by s(\W) =W-TW, We Ga)
The mapping s: G(s) + Y(T) has a local cross section if and only if T is similar to a compact perturbation of a nice Jordan operator [6] As in Section 5, we have a commutative diagram
Gl) ——> Ø()I(#)n #'()
S(T)
where @ is continuous and bijectivẹ
When T = 21, s and p obviously have global cross sections (¢(.%) nN st'(T)= =: G(x), and F(T) = {T}) We will prove that p does not have a global cross sec- tion if T is not a multiple of the identitỵ It will follow that s does not have one either, as in Section 5
Consider the fibre bundle
GA) 1 oh (TP) > Gol sb) 73 Gol Sf Go sf) 0 f(T)
An argument completely analogous to the one employed in Section 5 shows that p has a global cross section only if the exact homotopy sequence of
Po GA) > Go( Gol) 0 Œ),
.~3 m(9,()n s/'(Ÿ), D —3 mÁ9(), D => —> m(9(⁄)/9() n s'), p()— :
Trang 17Since Z;(.Z) is Isomorphic with #()/#()n (Í + #), we have that
n,(9g(Z)) = m,(#()/9()n (1 + #)) ~ m,(BU)= 2 n na n1
[1Š, p 215] and zx;(#s¿()) = 0
Let GT) be the component of I in Y(o/)n (7) For the computation of 2,(Go() Nn '(T), 1) = 2,(GT), 1) (n > 1) we need the following:
LEMMA 19 Let œ6: ý(2Zf) ¬ BH = (W)JX be the canonical projection of L(H) onto @ If JE L(A) is a nice Jordan operator, then ø⁄'(J)= n(#"(J)) and
GT) = (GH) 0 (J)
Proposition 20 Let Te sf be similar to m(AL + gO), k >2 Then py: G(x) > — #,(./)/ZặZ) n A “(T) does not have a global cross section
_ Proof By Lemmas 13 and 19, if W belongs to $(T) then it can be represented as a matrix VV, Vẹ Vy Vy, Ve , W= 0 * VV, Vy
with V, € Go((9,)) As in Lemma 14, it follows that 9(T) is homotopy equivalent to #;((2)); the map W —> V, isa homotopy equivalence and its homotopy inverse is the map V, > Ve, This implies that the inclusion map ip: Go(7) n / “(T) > Gy( A) induces the homomorphism “multiplication by k’’ in the second homotopy group The same argument as in Case 1 of Theorem 11 leads to the conclusion that po does not have a global cross section unless k = 1 2
PROPOSITION 21 Let Te & be similar to J, where J = @ (Â,j1,-E 4) isa
f=1
Trang 18Proof Let J =: Jy ® Je ® Jy, where Jy == Aly +g, Jg = gle + gi and 1 3 ⁄¿-= @ (21; + 4”) Then ¿=-8 ‘ GT) = {We 9() n ø!Ở) :Ÿ = Ÿ¿ @ W¿ @ W,, W,e9J,), j =: 1,2,3} (see Lemma 19 and the proof of Proposition 20) Thus, wo 3 ~ > 1z(ặ) n Z'{T), 1) ~ @ m;(9°%(J,)) ~ ZOZOx( HJ) de
Since the latter group cannot be a direct summand of 1,(G(s7)) ~ Z, it follows that the exact homotopy sequence
= m(%(#)n 3⁄4), 1) > eGo), 1) -> ToGo Golf) 0 "1, pol) >
—0>0—
cannot split Therefore py does not have a global cross section Ñ The results of this section can be summarized in the following
THEOREM 22 p: G(x) > #()/2(Z)n ø⁄'(7) has a giobal cross section if and only if T = 21 for some ^À€ C
As a consequence, we deduce that if an operator T has a global similarity cross section o: (7) ~ GH), then @ cannot preserve compact differences unless 7’ is a multiple of the identitỵ
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