Real ideals of compact operators of complex factors

Real ideals of compact operators for (complex) factors are investigated. A description (up to isomorphisms) of real two-sided ideals of relatively compact operators of a complex W*-factors is given. A relative weak (RW)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_r$$\end{document} convergence in a real Hilbert space is introduced. The classical Hilbert characterization of compactness of operators is extended to the compact operators in semifinite real W*-algebras.


Introduction
In the present paper we investigate the real ideals of compact operators for complex factors and give a description (up to isomorphisms) of real two-sided ideal of relatively compact operators of the complex W*-factors. A concept of relative weak (RW) r convergence in a real Hilbert space is introduced. The classical Hilbert characterization of compactness of operators is extended to the compact operators in semifinite real W*-algebras.

Preliminaries
Let B(H ) be the algebra of all bounded linear operators on a complex separable Hilbert space H . A weakly closed *-subalgebra A containing the identity operator 1 I in B(H ) is called a W * -algebra. A real *-subalgebra R ⊂ B(H ) is called a real W * -algebra if it is closed in the weak operator topology, 1 I ∈ R and R ∩i R = {0}. A real W * -algebra R is called a real factor if its center Z (R) consists of the elements {λ1 I, λ ∈ R}, where R is the field of all real numbers. We say that a real W * -algebra R is of the type I f in , I ∞ , II 1 , II ∞ , or III λ , (0 ≤ λ ≤ 1) if the enveloping W * -algebra A(R) has the corresponding type in the ordinary classification of W * -algebras. A linear mapping α of an algebra into itself with α(x * ) = α(x) * is called an *-automorphism if α(x y) = α(x)α(y); it is called an involutive *-antiautomorphism if α(x y) = α(y)α(x) and α 2 (x) = x. If α is an involutive *-antiautomorphism of a W * -algebra M, we denote by (M, α) the real W * -algebra generated by α, i.e. (M, α) = {x ∈ M : α(x) = x * }. Conversely, every real W*-algebra R is of the form (M, α), where M is the complex envelope of R and α is an involutive *-antiautomorphism of M (see [1,2,5,9]). Therefore we shall identify from now on the real von Neumann algebra R with the pair (M, α).
A trace on a (complex or real) W*-algebra N is a linear function τ on the set N + of positive elements of N with values in [0, +∞], satisfying τ (uxu * ) = τ (x), for an arbitrary unitary u and for any x in N .

Real and complex ideals of W*-algebras
It is easy to see that the subspace I c is equal to I + i I , therefore a real subspace I is a real ideal if and only if I · M ⊂ I + i I .
Since each complex subspace of M is a real subspace, any complex ideal is automatically a real ideal of M. Let I be a real ideal of M. If there exists a real W*-subalgebra R of M with R + i R = M, such that I ⊂ R, then I is called a pure real ideal of M. In this case, it is obvious that we have I · R ⊂ I . Note that, the reverse is not true, i.e. from I · R ⊂ I it does not follow I ⊂ R. But a complex subspace J = I + i I always is a complex ideal of M. On the other hand if I ⊂ R is a real subspace of M and I + i I is a complex ideal, then I is a pure real ideal, i.e. we obtain I · R ⊂ I .
Let, now I and Q be pure real ideals of M. In general, the set I + i Q is not a (complex) subspace. More precisely the set I + i Q is a complex subspace if and only if I = Q. Therefore we consider the smallest complex subspace J of M, containing I and Q. Obviously J is equal to (I + Q) + i(I + Q). Thus, if I and Q are real ideals, then J = (I + Q) + i(I + Q) is a complex ideal.

Ideals of compact operators
Let (M, α) be a real factor and let τ be an α-invariant semifinite trace on M. A subspace K ⊂ H is called τ -finite (or finite relative to τ ), if τ (P K ) < +∞, where P K is the canonical projection of H on K with P K ∈ M. Now, let K be a subset of H . A subset K is called τ -compact (or compact relative to τ ), if K is approximated in the norm · H by a bounded sequence of τ -finite subspaces.
A real operator A on H (i.e. A ∈ (M, α)) is called real compact relative to τ if it is an operator mapping bounded sets into relatively compact sets. We denote by I (respectively, by J ) the set of all relatively compact operators of (M, α) (respectively, of M). Let us recall the following result. Now, let us recall [4] the notion of the crossed product of a W*-algebra M by a locally compact topological group G by its *-automorphism. Let γ : G → Aut (M) be a group homomorphism such that the map g → γ g is strongly continuous. Let L 2 (G, H ) be the Hilbert space of all H -valued square integrable functions on G. We consider a *-algebra N ⊂ B(L 2 (G, H )), generated by operators of the form: π γ (a) and u(g), where a ∈ M, g ∈ G, and Similarly one can define the notion of crossed product for real W*-algebras (see [1,[12][13][14]). Let M be a factor of type III λ (λ = 1) and let α be an involutive *-antiautomorphism of M. Then by [12] (see also [1]), either -there exist a factor N of type II ∞ and an α-invariant automorphism θ of N such that (M, α) is isomorphic to the (real) crossed product (N , α) × θ Z or -there exist a factor N of type II ∞ and an antiautomorphism σ of N such that (M, α) Let's consider the first case. Let (M, α) be isomorphic to the (real) crossed product [12][13][14]). Let E : M → N be the unique α-invariant faithful normal conditional expectation (see [15,16]). Let us state an auxiliary lemma whose proof immediately follows from the linearity and α-invariance of E.
It is easy to see that any two-sided ideal is hereditary.

Lemma 3 The following is valid
x ≤ y, f or some y ∈ I + }.
From Lemma 3, in particular, it follows, that a projection from (M, α) is finite if and only if it majorized by some finite projection of (N , α).
Let I 1 be the norm closure of I. If we apply Proposition 1, Lemma 3 and the scheme of the proofs of Propositions 4.1, 4.5 and Theorem 4.3 [6], then we can prove the following real analogue of Halpern-Kaftal's theorem.

Theorem 2 Let M be a factor of type III λ (λ = 1) and let α be an involutive *-antiautomorphism of M. If the real factor (M, α) is isomorphic to the (real) crossed product (N , α) × θ Z, then I 1 is a unique (up to an inner automorphism) smallest hereditary real C*-subalgebra of (M, α), containing the ideal I , and it is a two-sided module over (N , α).
Let us consider the second case. Let (M, α) be isomorphic to ((N ⊕ N op ) × σ Z, β), where N is a II ∞ -factor, N op is the opposite W*-algebra for N and β(x, y) = (y, x), for all x, y ∈ N .
Recall that, a factor N is generated by the fixed point algebra of the one parameter group {σ ψ t : t ∈ R} of modular automorphisms, associated with some α-invariant faithful normal semifinite weight ψ. More precisely, the W*-subalgebra M ψ = {x ∈ M : σ ψ t (x) = x, t ∈ R} contains a central projection p such that N is isomorphic to factor pM ψ . In this case the real W*-algebra (M ψ , α) is isomorphic to (N ⊕ N op , β) (for more details see [1,[12][13][14]).
Let E : (M, α) → (N ⊕ N op , β) be a faithful normal conditional expectation (see [15,16]) and let J be the unique (nonzero) uniformly closed two-sided ideal of the semifinite (complex) factor N (see Theorem 1). Similarly to the first case, we denote by I 2 the norm closure of span{x ∈ (M, α) : Applying the same reasonings, as in the first case, and the scheme of proofs of Propositions 4.1, 4.5 and Theorem 4.3 [6], we obtain one more real analogue of Halpern-Kaftal's theorem.

Theorem 3 Let M be a factor of type III λ (λ = 1) and let α be an involutive *-antiautomorphism of M. If the real factor (M, α) is isomorphic to
, then I 2 is the unique (up to inner automorphism) smallest hereditary real C*-subalgebra of (M, α), containing the ideal I and is a two-sided module over (N , α). Here I is the unique (nonzero) uniformly closed two-sided ideal of semifinite real factor (N , α).
Thus, summarizing all above, in the injective case, we can describe all (nonzero) uniformly closed two-sided real ideals of semifinite and pure infinite complex factors. Recall that [1,5], if M is an injective factor of type II, then there exists a unique conjugacy class of involutive *-antiautomorphisms in M; therefore there exists a unique (up to isomorphisms) real subfactor of M, generating M. If M is an injective factor of type III λ (0 < λ < 1), then in M there exist exactly two conjugacy classes of involutive *-antiautomorphism; therefore there exist two (up to isomorphisms) real subfactors of M, generating M. Hence we obtain the following result. Theorem 4 Let M be a factor. Then the following assertions are true: 1. If M is an injective factor of type II 1 or type II ∞ , then there exist (up to isomorphisms) two (nonzero) uniformly closed two-sided real ideals in M. One of them is the complex ideal J , the other is the pure real ideal I . 2. If M is an injective factor of type III λ (0 < λ < 1), then there exist (up to isomorphisms) three (nonzero) uniformly closed two-sided real ideals in M. One of them is the complex ideal J , the two others are the pure real ideals I 1 and I 2 .

Relative weak convergence in semifinite real W*-algebras
In this section, we study the relative weak (RW) convergence in a real Hilbert space. We first recall that the elements of the two-sided closed ideal I generated by the projections which are finite relative to a real W*-algebra (M, α) are called compact operators of (M, α).
Let Note that a weakly convergent sequence is necessarily bounded, but following the Example 2 of [7], it is easy to construct an example, in which a unbounded sequence satisfies the second condition of Definition 1.
Example Let H and K be infinite-dimensional separable real Hilbert spaces with orthonormal bases {η n }, {γ n } respectively. We put R = B(H ) ⊗ R1 I K and ξ n = 2n i=n η i ⊗ γ i . Since ξ n 2 = 2n i=n η i 2 γ i 2 = n, it is an unbounded sequence. Let P be a finite protection of R. Then there is a finite projection P 0 in B(H ) such that P = P 0 ⊗ 1 I K . Without loss of generality we may assume that it is one-dimensional, i.e., that P 0 =< ·, ξ > ξ, for an element ξ ∈ H with ξ = 1. Then Since the series ∞ k=1 | < η k , ξ > | 2 converges, the sequence 2n k=n | < η k , ξ > | 2 converges to 0, when n → ∞. Hence Pξ n → 0. Therefore Pξ n S −→ Pξ = 0, but the sequence {ξ n } is unbounded. Moreover it is easy to show that the sequence {η n ⊗γ } (RW ) r converges to 0, but it does not converge strongly to 0, and the sequence {ξ ⊗γ n } converges weakly to 0, but does not (RW ) r converge to 0.
It is easy to see that the following assertions are true.
Proof The proof of implication "⇒" is obvious. Let's prove the implication "⇐". Assume, that there is some projection p ∈ M with pξ n → 0. Since the projections p and α( p) are equivalent we have α( p)ξ n → 0 (see [1]). It is easy to see that a = p + α( p) ∈ (M, α) and a ≥ p (because α( p) ≥ 0), therefore aξ n → 0. Then exists a spectral projection e ∈ (M, α) of an element a with eξ n → 0. It is a contradiction with the assumption.
The following theorem is a generalization of Hibert's characterization of the compact operators.

Theorem 5 (Theorem 7, [7]) An element A is compact in M iff it maps (RW) converging sequences into strongly converging ones.
We have the following real analogue of the above characterization.

Theorem 6 An element A is compact in (M, α) iff it maps (RW) r converging sequences into strongly converging ones.
Proof Firstly, let us prove the sufficiency. Let I be the two-sided closed (pure real) ideal generated by the projections which are finite relative to a real W*-algebra (M, α) and put J = I + i I . As it was noticed above J is the two-sided closed (complex) ideal, generated by the projections which are finite relative to a W*-algebra M. If we put Since I 1 ⊂ (M, α), we obtain I 1 ⊂ I . Therefore A ∈ I , i.e. A is compact relatively to (M, α) Now we shall prove the necessity. It suffices to show that I ⊂ I 1 . Suppose that K ∈ I and ξ n (RW ) r −→ ξ . Without loss of generality we can assume that ξ = θ . We repeat step by step the scheme of proof of Theorem 1.3 [8], to obtain the real analogue of the generalized Rellich condition for K , i.e., for every λ > 0 there is a projection P of (M, α) such that K P ≤ λ and 1 I − P is finite. From ξ n (RW ) r −→ 0, by definition, we obtain (1 I − P)ξ n S −→ 0, and hence K (1 I − P)ξ n S −→ 0. Since K − K (1 I − P) = K P ≤ λ and ξ n is bounded by definition, we have K ξ n S −→ 0. Thus, we have shown that for any K ∈ I the condition ξ n (RW ) r −→ ξ implies K ξ n S −→ K ξ . Hence K ∈ I 1 , and therefore I 1 ⊂ I . Theorem 6 shows that Hibert's characterization of the compact operators remains valid in semifinite real W*-algebras.