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Владикавказский математический журнал Январь-март, 2004, Том 6, Выпуск 1
UDC 517.98
ON IDEAL OF COMPACT OPERATORS IN REAL FACTORS
A. A. Rakhimov, A. A. Katz, R. Dadakhodjaev
Dedicated to the memory of Professor Yuri A. Abramovich
In the present paper the real ideals of relatively compact operators of W-algebras are considered.
A description (up to isomorphism) of real two-sided ideal of relatively compact operators of the complex
W*-factors is given.
1. Introduction
It is well known that the set of relatively completely continuous operators in a Hilbert space forms a two-sided uniformly closed ideal. Analogous classes of such operators for operator algebras, in particular, for the von Neumann algebras, has been studies in classical papers of M. Sonis, V. Ovchinnikov, H. Halpern and V. Kaftal. Let now B(H) be the algebra of all bounded linear operators on a complex Hilbert space H. A weakly closed *-subalgebra M with identity element 1 in B(H) is called a PF*-algebra. Let P(M) be the set of all projections of M, I be the ideal of all operators with the finite range projection relatively to M, J = I be the ideal of compact operators relatively to M. It is known [2], that I and J are proper iff M is infinite; and that J is the maximal two-sided ideal of M without infinite projections. The compact operators relative to M were defined by Sonis [6] (in the case of the algebras with Segal measure, i. e. for finite PF*-algebras) as the operators which send bounded sets into relatively compact sets. In the paper [4] it has been introduced and considered an analogous notion of finiteness and compactness in purely infinite PF*-algebras. In the present paper we will introduce and consider the ideal of compact operators relative to a real W*-algebra. Similarly to the complex case, it has been given a description and classification (up to isomorphism) of the real two-sided ideal of the relatively compact operators.
2. Preliminary Information
A real *-subalgebra R with 1 in B(H) is called a real W*-a,lgebra if it is closed in the weak operator topology and R П iR = {0}. A real PF*-algebra R is called a real factor if its center Z(R) contains only elements of the form {Al}, A € K. We say that a real PF*-algebra R is of the type Ifln, loo, Hi, Hqo, or Шд (0 ^ A ^ 1) if the enveloping W*-algebra U(R) = R + iR has the corresponding type in the ordinary classification of PF*-algebras [1].
A linear mapping a with a(x*) = a(x)* of the algebra R into itself is called an *-auto-morphism if a(xy) = a(x)a(y); an *-antiautomorphism if a(xy) = a(y)a(x); involutive if a2(x) = a(a(x)) = x, for all x,y € R.
© 2004 Rakhimov A. A., Katz A. A., Dadakhodjaev R.
A trace on a (complex or real) PF*-algebra N is an additive and positively homogenus function r on the set N+ of positive elements of N with values in [0,+00], satisfying the following condition: t(uxu*) = t(x), for an arbitrary unitary u and x in N.
The trace r is said to be finite if r(l) < +00; semifinite if given any x € N+ there is a nonzero y € N+, y ^ x with r(y) < +00.
Let R C B(H) be a real PF*-algebra, M = R + iR be the enveloping W*-algebra for R. Let r be a semifinite trace on R. Subspace K of H with KqR, i. e. Pk € R, is called finite relative to r if t(Pk) < 00, where Pk projection of H on K-, com,pact relative to r if K is an approximate of the bounded sets from relatively finite subspaces.
Real operator x of H (i. e. x € R) is called real com,pact relative to r if it is the operator mapping bounded sets into relatively compact sets.
3. Compact Operators in Semifinite Real Factor
Let I(R) be the set of all relatively compact operators of R.
Theorem 1. Let R be a semifinite real factor. Then I(R) is a unique (non zero) uniformly closed two-sided ideal of R.
<1 Let I(R) be a uniformly closed two-sided ideal in R. By Proposition 3 of [5] it contains all bounded operators with the finite metric range. Because I(R) is uniformly closed, from Proposition 2 of [5] it follows that <Joo(Ri t) C I(R), where <Joo(Ri t) is the set of all relatively completely continuous operators in R. Let us prove that I(R) C CTqo (R, t). Let x € J. Because x* € J and Re(®), Im(®) € J, without the loss of generality we can say that x is Hermitian. Let now {ca} be the family of the spectral projections for the operator x, and let the interval A = (a,/3) be without zero. Then P& /', — /',, (r I(R), and by Proposition 3 of [5] the metric range of Pa is finite. Because A/,./'-,. —x uniformly, from Proposition 2 of [5] is follows that x € o"oo(-R5t)- The theorem is proven. >
Theorem 2. Let R be a semifinite real factor, U = R + iR is its enveloping factor. Let I(U) be a unique (nonzero) uniformly closed two-sided ideal ofU. Then
I(U) = I(R) + iI(R).
<1 Since I(R) is a uniformly closed two-sided ideal, then I(R) + il(R) is also a uniformly-closed two-sided ideal. In fact, let {cn = an + ibn} be a Cauchy sequence in I(R) + iI(R), i. e. \\cn ~ cmII 0 as n,m —00. Then \\(an — am) + i(bn — bm)\\ —0 as n,m —00. Using the lemma 1.1.3 (iii) from fl] we have
max{||a„ - arn ||, \\bn - brn\\} sg ||(a„ - arn) + i(bn - bm) ||,
therefore, \\an — am\\ 0 and \\bn — bm\\ 0 as n,m —00. Thus, {an} and {bn} are Cauchy sequences in I(R), hence they converge to a and b in I(R) respectively. Thus, cn = a„ + ibn —a + ib in I(R) + il(R), which is uniformly closed. Now, if x = a + ib € U, y = c + id € I(R) + il(R), then xy = (ac - bd) + i(ad + be) € I(R) + il(R). Similarly, yx € I(R) + iI(R). Therefore, I(R) +iI(R) is a uniformly closed two-sided ideal of U and we have proved that I(R) + il(R) C I(U).
Now, since for x € I(U) we have x = a + ib, a,b € R, let I(U) = A + iB, for some A,BcR. But, for a € A, b € B we have ab, ba € I(U). Therefore, ab, ba € A, whence A = B, i. e. I(U) = A + iA. Then I(R) c A as A, I(R) C R. Let {an} be a Cauchy sequence in A C I(U). Since I(U) is uniformly closed, {an} converges to a € /(i/). But, R is also
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A. A. Rakhimov, A. A. Katz, R. Dadakhodjaev
uniformly closed, therefore, a € R. Then a £ A Now, let x € A, y £ i?. Since I(U) is a two-sided ideal of U, xy, yx € /(?/), i. e. yx £ A as xy^yx € R. Therefore, is a uniformly-closed two-sided ideal of i? with /(i?) C A. Then by Theorem 1 we have A = I(R). This completes the proof. >
4. Real ideals of compact operators of factors of type IIIa (A ^ 1)
Let us recall [3] the notion of the crossed product of a PF*-algebra by a locally compact topological group by its *-automorphism. Let N be a (complex or real) PF*-algebra in B(H), 7 : G —Aut(M) be a group homomorphism such that each map g —jg is strongly-continuous. Let L2{G,H) be the Hilbert space of all ii-valued square integrable functions on G. We consider an *-algebra U C B(L2(G,H)) generated by operators of the form ?r7(a) (a € M) and u(g) (g € G), where
(n1(a)0(h)=^1(a)m, Hg)0(h)=a9~1h), £ = £(h) € L2(G,H), g,heG.
The algebra U is called crossed product of M by G, and denoted by W*(M, G) (or Mx7G). Moreover, there exists a canonical embedding tt7 : M —tt7(M) C U. Each element x € U has the form: x = where x(-) is a M-valued function on G.
Let 0 be an *-automorphism of N. For the action {0n} of the group Z on N we denoted by W*(0,N) (or N xe Z) the crossed product of N by 0.
Now, let R be a factor of type IIIa (A ^ 1). Then ([7]), either there exist a real factor F of type IIqo and an automorphism 0 of F such that R is isomorphic to the crossed product F X0 Z or there exist a complex factor N of a type 11^ and an antiautomorphism a of N such that R is isomorphic to ((N © Nop) xff Z,/3), where is the opposite W^*-algebra for N, [3(x,y) = (y,x), for all x,y € N.
In the first case let I(R) be the norm closure of span{® € R+ : E(x) € 1(F)}, where E : R —F is a unique faithful normal conditional expectation. In the second case let I(R) be the norm closure of span{® € R+ : E(x) € I(N © Nop)}, where E is a unique faithful normal conditional expectation from M to N © №p.
If we now apply Theorem 1 and use the scheme of proof of Theorem 6.2 from [4] then we prove a real analogue of the theorem of Halpern-Kaftal:
Theorem 3. In each case I(R) is a unique (non zero) uniformly closed two-sided ideal of R.
Similarly to Theorem 2 we can prove the following:
Theorem 4. Let M be an injective factor of type III\, 0 < A < 1, R and Q be non isomorphic real factors with the some enveloping factor M, i. e. R + iR = Q + iQ = M. If I(M) is a (nonzero) uniformly closed two-sided ideal of M then
I(M) = I(R) + il(R), I(M) = I(Q) + il(Q),
where I(R) and I(Q) are non isomorphic unique uniformly closed two-sided ideals of R and Q respectively.
5. Main Result
Let M be a factor, a be an involutive *-antiautomorphism of M. Then (flj) the set R = {x € M : a(x) = x*} is a real factor and the enveloping PF*-algebra U(R) of R coincides with M, and conversely, given an arbitrary real factor R there exists a unique involutive
*-antiautomorphism cxr of the PF*-algebra IJ(R) such that R = {x € IJ(R) : a(x) = x*}. Moreover, two real factors R\ and R2 are *-isomorphic if and only if the enveloping factors U(Ri) and U(Ri) are *-isomorphic and the involutive *-antiautomorphisms otRl and cxr2 are conjugate, i. e. otRl = 9 ■ oir2 ■ for some *-automorphism 0.
It is known (flj) that
in a factor of type I„, n even, there exists a unique conjugacy class of involutive ^anti-automorphisms;
in a factor of type I„, n odd or n = 00, there exist exactly two conjugacy classes of involutive *-antiautomorphisms;
in an injective factor of type Hi, II^, or IIIi there exists a unique conjugacy class of involutive *-antiautomorphisms;
in an injective factor of type IIIa, 0 < A < 1, there exist exactly two conjugacy classes of involutive *-antiautomorphisms.
Hence, and from Theorems 1 and 3 we obtain the following result:
Theorem 5. Let M be a factor. Then the following assertions are true:
(1) if M has type In, n even, then in M there exist two (non zero) uniformly closed two-sided real ideals up to isomorphisms;
(2) if M has type In, n odd or n = 00, then in M there exist three (non zero) uniformly closed two-sided real ideals up to isomorphisms;
(3) if M is an injective factor of type II\ or type II0Q then in M there exist two (non zero) uniformly closed two-sided real ideals up to isomorphisms;
(4) if M is an injective factor of type 111\ (0 < A < 1) then in M there exist three (non zero) uniformly closed two-sided real ideals up to isomorphisms.
References
1. Ajupov Sh. A., Rakhimov A. A., Usmanov Sh. M. Jordan real and Li structures in operator algebras.— Dordrecht: Kluwer, 2001.-225 p.
2. Breuer M. Fredholm theories in von Neumann algebras. I // Math. Ann.—1968.—V. 178.—P. 243-254.
3. Connes A. line classification des facteurs de type III // Ann. Sc. Ec. Norm. Sup.—1973.—V 6.—P. 133252.
4. Halpern H., Kaftal V. Compact operators in type IIIa and type IIIo factors // Math. Ann.—1986.— V. 273.—P. 251-270.
5. Rakhimov A. A., Katz A. A., Dadakhodjaev R. The ideal of compact operators in real factors of types I and II // Mat. Tr.-2002.-V. 5, № l.-P. 129-134. [Russian]
6. Sonis M. G. On a class of operators in von Neumann algebras with Segal measures // Math. USSR Sb.—1971.—V. 13.-P. 344-359.
7. Stacey P. J. Real structure in er-finite factors of type IIIa, where 0 < A < 1 // Proc. London Math. Soc. 3.—1983.—V. 47.—P. 275-284.
Received by the editors 17 July 2003.
Dr. Abdugafur A. Rakhimov,
Department of Mathematics, Karadeniz Technical University,
Trabzon 61080, Turkey
E-mail: rakhimov@ktu.edu.tr
Dr. Alexander A. Katz,
Department of Mathematics and Computer Science,
St. John's University, 300 Howard Ave., Staten Island, NY 10301, USA
E-mail: katza@stjohns.edu
Dr. Rashithon Dadakhodjaev,
Department of Mathematics, National University of Uzbekistan,
Vuz Gorodok, Tashkent 700000, Uzbekistan
E-mail: Eashidhon@yandex.ru
