Lugovaya, G. D.

Sherstnev, A. N.

English

Institute of Mathematics AS CR, v. v. i.

Mathematica SlovacaG. D. Lugovaya; A. N. SherstnevOn the extension problem for unbounded measures on projectionsMathematica Slovaca, Vol. 50 (2000), No. 4, 473--481Persistent URL: http://dml.cz/dmlcz/131771Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 2000Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMathematica Slovaca ©2000 , . l L c . c n /<^r\r\r\\ M A A-/~> AOI Mathematical Institute Math. SlOVaCa, 50 (2000), NO. 4, 473-481 Slovak Academy of Sciences ON THE EXTENSION PROBLEM FOR UNBOUNDED MEASURES ON PROJECTIONS G. D . LUGOVAYA — A . N . SHERSTNEV (Communicated by Anatolij Dvurecenskij ) ABSTRACT. We construct an unbounded semifinite finitely additive measure on projections in a von Neumann algebra with no direct summands of type I2 not extending to a weight. Thus we give a negative solution to the problem set in [SHERSTNEV, A. N.: On certain problems in the theory of unbounded measures on projections, Atti Sem. Mat. Fis. Univ. Modena 42 (1994), 357-366] 1. Introduction The famous problem of linearity in non-commutative measure theory deals with extending orthoadditive measures on projections in a von Neumann algebra to linear functionals. The problem has been set up and solved for factors and unitarily-invariant measures by F. Murray and J. von Neumann in their classi-cal works on rings of operators. The full extension program for unitary-invariant measures to integrals has been carried out by I. S e g a l [1]. For general or-thoadditive bounded measures the above problem has been solved exhaustively due to the efforts of several mathematicians during 1957-1983 ([2]-[5]). The problem was positively solved for all types of von Neumann algebras with no di-rect summand of type I2. In 1984, the authors [6] obtained the positive solution to the problem for unbounded a-orthoadditive measures on the algebra of all bounded linear operators on a Hilbert space, and then this result was extended to the case of semifinite von Neumann algebras by M. M a t v e i c h u k [7]. The aim of this paper is to present a negative solution to the problem for the case of unbounded finitely additive measures. In the next section, the main result of the paper is stated. In Section 3, we introduce the notion of an orthoideal of orthoprojections in a von Neumann alge-2000 M a t h e m a t i c s Sub jec t C l a s s i f i c a t i o n : Primary 46L10, 46L51. Key words : von Neumann algebra, measure on projections, orthoideal. This research was supported by the RFFI (Russia), Grant No. 98-01-00103 and URFI, Grant No. 1617. 473 G. D. LUGOVAYA — A. N. SHERSTNEV bra. This notion plays a key role in constructing the measure. The construction is described in Section 4. In Section 5, all necessary proofs are adduced. The results of this paper were announced in [8], [9]. 2. The main result Let M be a von Neumann algebra. We shall denote by M p r and M + the classes of all orthoprojections and positive operators (respectively) in M . We de-note by B(H) the algebra of all bounded linear operators on a Hilbert space H. For x G B(H) we denote by %(x) the range of x. Let I ( = IH) be the identity operator on H. 1. DEFINITION. A mapping m\ M p r —> [0, +oo] is said to be a, finitely additive measure if m(p + q) = m(p) + m(q) ( p , q G M p r , pq = 0). A measure m is said to be unbounded if m(I) = +oo. We call m semifinite if there exists a net (pa)aeA C Mp r with pa /* I and m(pa) < +oo for every a G A. Nonsemifinite measures are rather difficult to view even if dimiJ < +oo, and we shall not examine them here. Recall that a weight on a von Neumann algebra M is a map (p\ M+ -> [0, +oo] such that ip(x + y) = <p(x) + ip(y), ip(\x) = \<p(x) (x,yeM+, A > 0 . 0 - O O E E O ) . When ip(I) < +oo, a weight ip extends to a positive linear functional ip G M* by linearity (we use the same letter cp for this extension). The fundamental result of C h r i s t e n s e n -Y e a d o n [3], [4] lies in the fact that every bounded finitely additive measure on the orthoprojections in a von Neumann algebra with no direct summand of type I2 extends to a positive linear functional. The main result of this paper is: 2. THEOREM. There exists a von Neumann algebra M with no direct sum-mands of type I2 and an unbounded semifinite finitely additive measure m: M p r —r [0, +oo] not extending to a weight. Thus we give the negative answer to a question raised by one of the authors in 1992 ([10; Problem 2.7]). The proof of the theorem consists in constructing an appropriate von Neu-mann algebra M and a two-valued measure m: Mpr -> {0. +00} with the re-quired properties. 474 ON THE EXTENSION PROBLEM FOR UNBOUNDED MEASURES ON PROJECTIONS 3. Orthoideals of projections 3. DEFINITION. A set J C M p r is said to be an orthoideal of projections in a von Neumann algebra M if the following properties are satisfied: (i) P, g G J, pq = 0 implies p + q G J; (ii) p < q (pe M p r , q G J) implies p G J. 4. If 5 is a subset of M p r , then among all orthoideals containing 5 , there exists the smallest. This is the intersection of all orthoideals containing S. It is natural to call it the orthoideal generated by S. We shall denote the orthoideal by J(S). This orthoideal can be described simply when S = {p, q] is a pair of projections in M p r . Consider the following increasing sequence of sets (Jn) C Mp r , which is defined by induction: J0 = {P, q, 0} , Jn = {reMpr: (3s,teJn_1)(st = 0 & r <s + t)} ( n G N ) . oo Then J(p,q)= U J„-n=0 Denote by M f p r the set of all finite-dimensional orthoprojections in M . 5. PROPOSITION. Let p,q£ M p r and S = {p, q] U M f p r . Then J(S) = {r + k: re J(P, q), k G M f p r , rk = 0} . P r o o f . It is obvious that projections of the form r + k, where r G J(p, g), fc G Affpr and rfc = 0 are contained in the orthoideal generated by 5 . Thus it is sufficient to verify that {r + k : re J(I>, q), k G M f p r , rk =-. 0} is itself an orthoideal. The condition (i) in Definition 3 is obviously satisfied. Now, let r G J(p, q), k e M f p r , rk = 0, and s G M p r be such that s < r + k. We show that s has the form s = r1 + k± (r1 G J(p, g), k1 G Mf p r ) . Put rx = 5 A r . Then rx G J(p, g) because rx < r . It remains to be proved that kx — s — r1 G Affpr. We may assume (and we do this) that k1Ak = 0. Suppose on the contrary dimk1 = +oo and let {e^ e 2 , . . . } be an orthonormal sequence in TZ(kx). Consider the decompositions ej = fj+9j (3 € N ) : w h e r C fj = rej . gj = fc^. . Note that fj:gj ^ 9 (j e N). (Indeed, if f. = 0, then ê  G / ^ ( ^ A fc) = {0}; if g- = 0, then ê . G ^(A^ A r) = {0}). Let n = dirnk. Then there are numbers M+l A p . . . , A n + 1 not all equal to zero with ]T \.g — 0. Hence j=i n+1 n+1 n+1 n+1 E v.=E v,+E v.-=E Vi€ *(*i A *•>=w • j - 1 j = l j = l j = l 475 G. D. LUGOVAYA — A. N. SHERSTNEV This contradicts the linear independence of the system {e1,e2,...}. D 6. Remark. If J ( c M p r ) is an orthoideal, then a two-valued finitely additive measure on M p r is naturally associated with J by ГПj(r) = | 0 ííreJ, +00 if r € MP r \ J. 4. The construction 7. Let K be an infinite-dimensional separable Hilbert space and T > 0 a com-pact operator acting on K with Ker T = {6} (0 is the null vector in K). We will denote the projection-valued measure on the Borel algebra 03(E) constructed by the resolution of the identity for T by e(-) = eT(-). Let Af = {T}'f be the commutative von Neumann algebra in K generated by T . Put H = K®K©K and introduce the von Neumann algebra M = {[u^] : ut- G N} acting on H (here, [ui •] are 3 x 3-matrices). This algebra has no direct summands of type I2. For the construction of the required orthoideal we fix twro orthoprojections, p = TAn ATT TATT and = ^ 2 Г 2 7 A 2 Г 0 2 Д 2 Г T 0 4 T Д 2 Г Г 0 (for V e B(K) we set Av = (IK + V+V)-1). Consider the orthoideal J = r G J(p, q), k G M f p r , rfc = 0} introduced in Propo^ tion 5. The 771; {r-h k : measure rrij canonically associated with the orthoideal due to Remark 6 is semifmite. (In fact, let un = (^, +00) . By the spectral theorem for the compact operator T , e(un) G Af**T and e(un) /* IK (since K e r T = {9}). Therefore. [ > K ) < y Z 1 ^ a n d m j ([e(a; n )* 0 .]) = 0 (n € N).) 8. The proof of Theorem 2. We will showT that the measure ntj constructed in Paragraph 7 does not extend to a weight. We should prove two key lemmas (this will be done in Section 5): 9. LEMMA. Let E be a Hilbert space and let r. s . £ be orthoprojections in E. Then the following conditions are equivalent. (i) 9t(r) c<R(s)+m(t); (ii) £/iere exists C > 0 sttcJi £/m£ r < C(s -f-1). 10. LEMMA. T/ie orthoprojection r = 1к 0 0 0 0 0 L 0 0 0 belongs to Aíp r \ J. 476 ON THE EXTENSION PROBLEM FOR UNBOUNDED MEASURES ON PROJECTIONS Now, we complete the proof of Theorem 2. Suppose that <p: M+ -» [0, +00] is a weight such that </?|jVfpr = mj. Note that the orthoprojections r, p, q satisfy the condition (i) in Lemma 9: r{U 9, h} = {/, 0,6} = {2/, 2T/, 6} + {-/, - 2 T / , 9} = p{2/, 2T/, 0} + , { - / , - 2 T / , 0} (f,g,heK). Hence there exists C > 0 such that r < C(P+q) . Since p,q G J and r G ilfPr\ J (see Lemma 10), we obtain the contradiction +00 = m(r) = cp(r) < Ccp(p + q) = C(<p(p) + <p(g)) = C(m(p) + m(q)) = 0, which proves the theorem. 5. Proofs In this section, we give the proofs of Lemmas 9 and 10. 11. LEMMA. Let E be a Hilbert space and xj G B(E)+ (l<j<n). Then n({x1 + ... + xnfi2)=n(x\'2) + ... + n(xn^). (Here, the right-hand side is regarded as the algebraic sum of linear subspaces.) P r o o f . Put a = xx-\ h xn. Then x- < a and [9; I, §1, Lemma 2] give xyi = djai/2 = ai/2d* ( i < j < „ ) , where dj are operators in B(E). Hence n(x\/2) + • • • + Tl(xl/2) C Tl(a^2). Conversely, observe that b = ^ d*d- is the range projection of a. In fact, we have a = a1//2ba1/2 and d-h = 0 (1 < j < n), and for all f,g e E, he Kera, {(Y,d*dyi2fyi29+h) = {(Y.d*dyi2fyi2g) =,(a^ba^f,g) = (af,g) = (a1/2f,ai/2g) ^{aWfytg + h). For every vector of the form a1/2/ we now have a"2f = tt'Hf = X>1/2W = j:xy2{djf)en(x\<2) + ... + n(x^), and the lemma follows. • 477 G. D. LUGOVAYA — A. N. SHERSTNEV 12. The proof of Lemma 9. Let us first assume that the condition (ii) is satisfied. From the inequality r < C(s+t) it follows that there exists an operator x G B(E), ||x|| < y/C such that r = x(s + t)1'2 = (s + tf^x*. Therefore 7̂ (r) - n((s + t)1/2x*) c n((s +1)1/2) = n(s) + n(t) by Lemma 11. Conversely, let the condition (i) holds and F = TZ(s) + n(t). The operator (s+t)1/2 \p is bounded and injective. Hence there exists a closed inverse operator (s + t)'1/2: 1Z(s) + n(t) -•> F. Since H(r) C TZ(s) + n(t), the closed operator A = (s + t)~1/2r is defined everywhere on E. Therefore it is bounded. Putting C = \\A\\2 we have r = (s + t)x>2A = (s + t)xl2AA*(s + t)1'2 < C(s + t), and the Lemma follows. 13. The proof of Lemma 10. We start with the observation that it is sufficient to prove the "reduced" version of this lemma. To be precise, let "IK o o] 7T = o IK o (e Mp r ) . Consider the reduced von Neumann algebra Mn = L o o oj I nx\n(7r) ' x ^ ^ f a c t m S o n t n e Hilbert space H^ = K © K, A, Pn = •T TA ATT T TATT ҡp\н A2T 2TA 2T 2A2TT 4TA2TT щн It should be noted that pn = Pr<T\ (respectively qn = PT^2T^) is the ortho-projection onto the graph of T (respectively 2T). Let Jn be the orthoideal generated by S = {p^^Q^} U M^ p r . Our next goal is to prove the following lemma. 14. LEMMA. The orthoprojection rn = iк o o o belongs to M£r \ JK . We will slightly simplify the problem by restricting ourselves to considera-tion of the ideal J(pn, qn) generated by the two-element set {pn. qn} (see Para-graph 4). Namely, we will prove: 15. PROPOSITION. 7/ r = rll rl2 r21 r22 eJ(p7T,q7T), then Ker(IK-rn) = {6}. Now, since every orthoprojection r' G J^ is of the form r' = r -J- k, T G J(p7T,q7r), dimk < +oo, it follows from Proposition 15 that dimKer(IK — r'n) = d i m K e r ( / x — r n — kn) < +oo, whereas d imKer ( / K - (r7r)11) = dimKerO = +oo , which proves Lemma 14. We begin with some general facts concerning the matrix structure of ortho-projections onto the graphs of operators. 478 ON THE EXTENSION PROBLEM FOR UNBOUNDED MEASURES ON PROJECTIONS 16. PROPOSITION. Let T be a closed densely defined operator acting on a Hilbert space K, E = K®K. Then r ( T ) x = {{-T*/,/} : / G D(T*)}, and the orthoprojection in E onto the graph T(T) has the form PГ(T) — Д (TA)* T Д T(TA)* where A = (I + T*T) - 1 Moreover, if r is a subprojection of T p m > ^ i e n r c a n be represented in the form r = ^a T(T_1)* > where the operator a satisfies 0 < a = a2 + (Ta)*TcY. P r o o f . The operator matrix of an orthoprojection in B(E) has the form 7 ^ . In addition a 2 + 7 7 * = a, P2 + 7 * 7 = P> cry+ 7/3---7. If that is the matrix of Pr/T\, then a 7 7* ß a 7 Г ß ePг(т) (feк) ePг(т) (feк) 7* = Ta, 7 = (Ta)* = aT* , (3 = T(Ta)* = T • aT*, If / € D(T*T), then [»JJT%.] [/,] = [$£%,] = [/,] implies a = (I + T'T)'1. П 17. The proof of Proposit ion 15. In the notation of Paragraph 4, we have oo J(p7r,q1z) = (J Jn(Pn,qn), where 71 = 0 J0(p7r,qv) = {pv,qn,0}, J„_o7r,q„) = {reM?: (3s,t e Jn_l(pn,q7T))(st = 0 k r < s+ t)} ( n e N ) . We will prove the following assertion: the projections in Jn (pn, q__) are subpro-jections of projections onto graphs of compact operators which are functions of T . In addition, Proposition 15 holds for the corresponding null-space. 479 G. D. LUGOVAYA — A. N. SHERSTNEV Let us make the matrix structure of such subprojections more precise. Given a nonnegative compact operator V which is a function of T, we have, according to Proposition 16, r = J"" y11 y . We shall seek r n in the form r n = xAv , where x G JV is an unknown operator. As Ar is commutative, xAv = Avx x*Д, and xAv = r n ri\VK + V2>11 = XAyAy^Ay = X ^ X = X X = X Thus, there exists u G 23(M) such that r n = e(uj)Ay. Hence Г = Д ve(w) VAVЄ(OJ) VAve(ш) V2Ave(ш) (*) We will prove the above assertion by induction on n. It can easily be seen that it is true for every projection in J0(p„.>O- Suppose it holds for the projections in Jk(pv,qv) (0 < k < n). Consider s,t £ Jn(p^,q^) such that st = 0. Write s and t in the matrix form: Ave(u) VAve{w) _VAve(u) V2Ave(u) Awe(5) WAwe(5) [WAwe(S) W2Awe(S) Since s and t are orthogonal, we conclude that e(uj)e(5) = e(u D 5) = 0 (as V,W > 0, by the assumption of induction). Without loss of generality, we can suppose that u fl 5 = 0. Then an easy computation shows that the projection r = s + t G «/n+i(p7r,?7r) 1s a subprojection of the projection onto the graph of the non-negative compact operator X = Ve(u) + We(5). By the inductive assumption, it follows that Kei(IK-ru) = {fen(e(u;Ud)) = {f£7l(e(u;US)) = {feTl(e(uUS)) The proof is complete. Ave(uj)f + Awe(ô)f = f} Ave(u;)f = e(u;)f, Awe(ð)f = e(ð)f} e(u;)f = e(ð)f = } = { } . R E F E R E N C E S [1] SEGAL, I . : A non-commutative extension of abstract integration, Ann. Math . 5 7 (1053), 401-457. [2] GLEASON, A. M. : Measures on the closed subspaces of a Hilbert space, J. Math. Лlech. 6 (1957), 885-894. [3] C H R I S T E N S E N , E . : Measures on projections and physгcal states, Coram. Math. Phуs. 86 (1982), 113-115. 480 ON THE EXTENSЮN PROBLEM FOR UNBOUNDED MEASURES ON PROJECTIONS [4] Y E A D O N , F . : Finiteiy additive measures on projections in finite W* -algebras. Bull. Lon-don Math. Soc. 16 (1984), 145-150. [5] M A T V E J C H U K , M. S.: Description of finite measures in semifìnite algebras (Russian), Funktsional. Anal. i Pгilozhen. 15 (1981), 41-53. [English translat ion: Functional Anal. Appl. 15 (1982), 187-197]. [6] LUGOVAYA, G. D — S H E R S T N E V , A. N . : On the linearity problem for unbounded mea-sures on projections, Funktsional. Anal. (Uľyanovsk) no. 23 (1984), 76-81. (Russian) [7] M A T V E J C H U K , M. S.: The extension of an unbounded measure to a weight, Izv. Vyssh. Uchebn. Zaved. Mat . 4 (1987), 47-51. (Russian) [8] LUGOVAYA, G. D . — S H E R S T N E V , A. N . : On a problem of the non-commutative mea-sure theory. In: Algebra and Analysis (Abstracts of the Conf. Dedicated to the 100-al B. M. Gagaeff), Kaz. M a t h . S o c , Kazan, 1997, p. 139. (Russian) [9] LUGOVAYA, G. D . — S H E R S T N E V , A. N . : On the extension problem for unbounded measures on projections to weights, Dokl. Akad. Nauk 3 6 5 (1999), 165-166. (Russian) [10] SHERSTNEV, A. N . : On certain problems in the theory of unbounded measures on pro-jections, Atti Sem. Mat . Fis. Univ. Modena 4 2 (1994), 357-366. [11] DIXMIER, J . : Les algèbres ďopérateurs dans ľespace Hilbertien (algèbres de von Neu-mann). In: Cahiers scientifiques. Fasc. XXV, 2e ed. revue et augmentee (Gauthier-Villars, ed.), Paris, 1969. (French) Received February 23, 1999 Department of Math. and Mechanics Kazan State University Ul. Kremlevskaya 18 Kazan, 420008 RUS SI A E-mail: Anatolij.Sherstnev@ksu.ru 481 

On the extension problem for unbounded measures on projections

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On the extension problem for unbounded measures on projections

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Institute of Mathematics AS CR, v. v. i.