On property of injectivity for real W*-algebras and JW-algebras

In this paper injective real W*-algebras are investigated. It is shown that injectivity is equivalent to the property of E (extension property). It is proven that a real W*-algebra is injective iff its hermitian part is injective, and it is equivalent to, that the enveloping W*-algebra is also injective. Moreover, it is shown that if the second dual space of a real C*-algebra is injective, then the real C*-algebra is nuclear.


Introduction
The theory of operator algebras, acting on a Hilbert space was initiated in thirties by papers of Murray and von Neumann. In the papers they have studied the structure of algebras which were later called von Neumann algebras or W*-algebras. They are weakly closed complex *-algebras of operators on a Hilbert space. At present the theory of von Neumann algebras is a deeply developed theory with various applications.
In the middle of sixtieth D.Topping and E.Stormer have initiated the study of Jordan (non associative and real) analogues of von Neumann algebras-so called JW-algebras, i.e. real linear spaces of self-adjoint operators on a complex Hilbert space, which contain the identity operator 1, closed with respect to the Jordan (i.e. symmetrised) product x • y = (x y+ yx)/2 and closed in the weak operator topology. The structure of B A. A. Rakhimov rakhimov2002@yahoo.com 1 Moscow State University in Tashkent Named After M.V.Lomonosov, Tashkent, Uzbekistan 2 Institute of Mathematics Named After V.I.Romanovsky, Tashkent, Uzbekistan these algebras has happened to be close to the structure of von Neumann algebras and it was possible to apply ideas and methods similar to von Neumann algebras theory in the study of JW-algebras. Thus D. Topping has classified JW-algebras into those of type I, II 1 , II ∞ , III, later E. Stormer and Sh. Ayupov considered the problem on connections between the type of a JW-algebra and the type of its enveloping W*-algebra. Moreover, E. Stormer gave a complete study of type I JW-algebras and has also proved that any reversible JW-algebra A (in particular of type II and III) is isomorphic to the direct sum A c ⊕ A r , where the JW-algebra A c is the self-adjoint part U(A c ) s of its enveloping W*-algebra U(A c ), whence the JW-algebra A r coincides with the self-adjoint part R(A r ) s of the real enveloping W*-algebra R(A r ) such that R(A r ) i R(A r ) = {0} (so called real W*-algebra). In this connection the study of real W*-algebras was carried out parallel to the theory of JW-algebras.
Thus the structure theory of real W*-algebras is relatively new, though in many aspects it is similar to the classical case of complex W*-algebras (von Neumann algebras). In this paper we introduce the notions of injectivity and nuclearity for real W*-algebras (in general for real C*-algebras) and study their relations with property of E (the extension property).

Preliminaries
Let B(H ) be the algebra of all bounded linear operators, acting on a complex Hilbert space H . Recall that a weakly closed *-subalgebra A ⊂ B(H ) with the identity 1 is called a W*-algebra. A real *-subalgebra R ⊂ B(H ) with 1 is called a real W*algebra, if it is weakly closed and R ∩ i R = {0}. A JW-algebra is a real subspace of the space of self-adjoint operators on a real or complex Hilbert space, closed under the operator Jordan product a • b = (ab + ba)/2 and closed in the weak operator topology.
Let A be a Banach *-algebra over the field C. The algebra A is called a C*-algebra, if aa * = a 2 for any a ∈ A. A real Banach *-algebra R is called a real C*-algebra, if aa * = a 2 and an element 1 + aa * is invertible for any a ∈ R. It is easy to see that R is a real C*-algebra if and only if a norm on R can be extended onto the complexification A = R + i R of the algebra R so that algebra A is a C*-algebra (see [9, 5.1.1]). Denote by M n (A) algebra of all n × n matrices over A which is also a C*-algebra. Recall that a continuous linear map ϕ between two C*-algebras A and B is called completely positive, if for any n ≥ 1, the natural map ϕ n from the C*-algebra A ⊗ M n to the C*-algebra B ⊗ M n , defining by is positive, where M n is the C*-algebra of n × n matrices over C. We say that a W*algebra A is called injective if the following condition is held: for every C*-algebra B, for every self-adjoint linear subspace S of B, containing the identity 1, and for every completely positive linear map ϕ : S → A, there is a completely positive linear map ϕ : B → A such that ϕ| S = ϕ [1,10]. All notions above are defined similarly for real C*-and W*-algebras. Recall [10], that a W*-algebra A ⊂ B(H ) is injective if and only if it has the property of E (the extension property), i.e. there exists a projection P : B(H ) → A such that P = 1, P(1) = 1. In this case the map P is completely positive (see [10]).

Injective real W*-algebras
In the following lemma we will show that the map P between two real W*-algebras is completely positive.
In order to show the converse of Theorem 1 we will similarly prove the real analogues of Stinespring's theorem and some lemmas about completely positive maps in [1].

Theorem 2 (The real analogy of Stinespring's theorem.) Let R be a real C*-algebra with identity and H r be a real Hilbert space. Then every completely positive linear
where π is a representation of R on some real Hilbert space K r and V is a bounded operator from H r to K r .
Proof Consider the vector space R ⊗ H r as tensor product of two the vector spaces R and H r . We define a bilinear form < ·, · > on R ⊗ H r as follows The fact that ϕ is completely positive guarantees that < ·, · > is positive semi-definite. For each x ∈ R, define a linear transformation π 0 (x) on R ⊗ H r by π 0 (x) : x j ⊗ξ j → x x j ⊗ξ j . π 0 is an algebraic homomorphism for S is a linear subspace of R⊗ H r , invariant under π 0 (x) for every x ∈ R and < ·, · > determines a positive definite inner product on the quotient (R ⊗ H r )/S in the usual way: < u + S, v + S >=< u, v >. Letting K r be the real Hilbert space completion of the quotient, the preceding paragraph implies that there is a unique representation π of R on K r such that π( Let S be a subspace of a real C*-algebra R, and H r be a real Hilbert space. B(S, H r ) (resp. B(R, H r )) will denote the vector space of all bounded linear maps of S (resp. R) into B(H r ). Note that B(S, H r ) is a real Banach space in the obvious norm. We shall endow B(S, H r ) with a certain topology, relative to which it becomes the dual space of another real Banach space. For Let C P(S, H r ) (resp. C P(R, H r )) denote the set of all completely positive linear maps of S (resp. R) into B(H r ). Each is a subset of B (S, H r ) and B(R, H r ), respectively, and thus inherits a BW-topology from the larger space as above. In addition, it is apparent that both C P(S, H r ) and C P(R, H r ) are convex cones, and the set C P(R, H r )| S of all restrictions of maps in C P(R, H r ) to S is a subcone of C P(S, H r ).

Lemma 2 C P(R, H r )| S is a closed cone in B(S, H r ), relative to the BW-topology.
Proof We claim first that ϕ = ϕ| S , for every ϕ ∈ C P(R, H r ). Choose π and V , as in Theorem 2, such that ϕ(x) = V * π(x)V , x ∈ R. Then ϕ ≤ V * · V = V * V = ϕ(1) ; since 1 ∈ S it follows that ϕ ≤ ϕ| S . The opposite inequality is trivial.
Nextly, observe that C P(R, H r ) is a BW-closed subset of B(R, H r ); indeed, since C P(R, H r ) is convex, then by definition, it is closed iff C P(R, H r ) ∩ B l (R, H r ) is (relatively) closed for every l > 0. But, if ϕ ν is a bounded net in C P(R, H r ) such that ϕ ν → ϕ ∈ B(R, H r ) (BW), then ϕ ν (x) → ϕ(x) in the weak operator topology for every x ∈ R, and this makes it plain that ϕ must also be completely positive. By Remark 1 1), it follows that for every l > 0, C P(R, H r ) ∩ B l (R, H r ) is BW-compact. The first paragraph of the proof shows that restriction map ϕ → ϕ| S carries C P(R, H r ) ∩ B l (R, H r ) onto C P(R, H r )| S ∩ B l (S, H r ), and by Remark 1 2), the restriction is BW-continuous; we conclude that C P(R, H r )| S ∩ B l (S, H r ) is compact and closed. Since C P(R, H r )| S is convex, it follows from the definition of the BW-topology that this set is closed.

Lemma 3 If f is an arbitrary BW-continuous linear functional with
The lemma is proven analogically to Lemmas 1.2.5 and 1.2.6 in [1].

Proposition 1 The set C P(S, H r ) is a subcone of C P(R, H r )| S .
The proof is completed by Lemmas 2 and 3 and a standard separation theorem. Hence, we obtain:

Theorem 3 B(H r ) is an injective real W*-algebra.
Proof Let S be a self-adjoint linear subspace of a real C*-algebra R, which contains the identity 1 of R, and let ϕ : S → B(H r ) be an arbitrary completely positive linear map. Then ϕ ∈ C P(S, H r ). By Corollary 1, ϕ ∈ C P(R, H r )| S , i.e. the map ϕ has a completely positive extension on R. Hence, we have that B(H r ) is injective.
And now, we will show the converse of Theorem 1.

Theorem 4 Let R ⊂ B(H ) be a real W*-algebra. If the algebra R has the property of E, then it is injective.
Proof Let there be a projection P : B(H ) → R such that P = 1, P(1) = 1 and let ϕ : S → R be a completely positive map, where S is a self-adjoint subspace of a real C*-algebra B, which contains the identity 1 of B. Since B(H r ) is injective, the map ϕ : S → R ⊂ B(H r ) can be extended to a completely positive map ϕ on B, i.e. there is a map ϕ : B → B(H r ) such that ϕ| S = ϕ. Consider a map It is easy to see the map ϕ 1 is completely positive and ϕ 1 | S = ϕ. Hence, R is injective.
From Theorems 1 and 4 we have the following result.

Corollary 2 Let R ⊂ B(H ) be a real W*-algebra. The algebra R is injective, if and only if it has the property of E.
Recall that if R is a real W*-algebra, then R s = {x ∈ R : x = x * } is a JW-algebra by Jordan product x • y = (x y + yx)/2. A JW-algebra (or a JC-algebra) R s is called injective if for any C*-algebras B ⊂ C and any morphism ϕ : B s → R s there is a morphism ϕ : C s → R s such that ϕ| B S = ϕ.
The following result has been proven in [8,Theorem 1.2]. But, there is a gap in the second part of the proof of Theorem 1.2 in [8]. In the following theorem, using Theorem 3 we will complete the second part of the proof of Theorem 1.2 in [8]. Then, using the first part of Theorem 1.2 in [8] and Theorem 5 we can formulate the following corollary.

Theorem 6 Let R ⊂ B(H ) be a real W*-algebra. The algebra R has the property of E if and only if the JW-algebra R s has the property of E.
Proof Suppose, that R has the property of E, i.e. there exists a projection P : B(H ) → R with P = 1, P(1) = 1. We define a map P 1 : B(H ) → R s , as It is easy to see that the map P 1 is a projection with P 1 = 1, P 1 (1) = 1. Hence, R s has the property of E.
Conversely, let P : B(H ) → R s is a projection with P = 1, P(1) = 1. Then by [6,Theorem 3.2] there exists a projection P from B(H ) onto R + i R such that P = 1, P(1) = 1. And now, we consider the map where E 2 (x +iy) = x. It is clear that P 1 is a projection such that P 1 = 1, P 1 (1) = 1.

Theorem 7 A real W*-algebra R has the property of E if and only if the enveloping W*-algebra R + i R of R has the property of E.
Proof Let R has the property of E, i.e. there exists a projection P : B(H ) → R with P = 1, P(1) = 1. Then by [6,Theorem 3.2] there exists a projection P from B(H ) onto R + i R such that P = 1, P(1) = 1; therefore, R + i R has the property of E.
Conversely, let P is a projection P from B(H ) onto R + i R such that P = 1, [3]). It is easy to see that E is a projection from R + i R onto R. Indeed, let y = E(x), then we have i.e. y ∈ R. It is clear, that E = 1, E(1) = 1. Then, the map P = E • P is a projection from B(H ) onto R such that P = 1, P(1) = 1, i.e. R has the property of E.
This result is true for injectivity, too.

Theorem 8 A real W*-algebra R is injective if and only if the enveloping W*-algebra R + i R of R is injective.
Proof Let R be injective. Then, by Theorem 1, R has the property of E and by Theorem 7, R +i R has the property of E. By [10,Theorem], R +i R is injective. Conversely, let R + i R be injective. By [10,Theorem], R + i R has the property of E. From Theorem 7, it follows that R has the property of E. Then, by Theorem 4, R is injective.
From theorems above we have the following result.

Corollary 4 1) A JW-algebra R s has the property of E if and only if the enveloping W*-algebra R + i R of R has the property of E. 2) JW-algebra R s is injective if and only if the enveloping W*-algebra R + i R of R is injective.
Remark 2 Thus, for the *-algebras R s , R, R + i R the notion of injectivity and the property of E coincide. Now, we will consider the dual space.

Theorem 9
If R is a real C*-algebra, then the second dual R * * of R is a real W*algebra.
Proof It is known that the complexification A = R + i R of R is a C*-algebra. By [9, Proposition 1.1.4] we have A * * = R * * + i R * * . By [12, Theorem 1.17.2] A * * is a W*-algebra, therefore R * * is a real W*-algebra.
Let A and B be C*-algebras, A ⊗ B be their tensor product over the the field C. A norm · γ on the *-algebra A ⊗ B is called a C*-norm, if x x * γ = x 2 γ for all x ∈ A ⊗ B. Every C*-norm on A ⊗ B is crossnorm, i.e. it satisfies the condition a ⊗ b γ = a b for all a ∈ A, b ∈ B. If · γ is C*-norm, then the closure of the *-algebra A ⊗ B, by the norm · γ is a C*-algebra which we denote by A ⊗ γ B. There are two following crossnorms: It can be easily shown, that for any C*-norm · γ it is held that: u min ≤ · γ ≤ u max . It is known [12] that if A and B are W*-algebras, then there is a unique central projection Z in (A ⊗ min B) * * such that A ⊗ min B is identified with a weak-* dense C*-subalgebra of (A ⊗ min B) * * Z . The W*-algebra (A ⊗ min B) * * Z is called the W*-tensor product of A and B, and is denoted by A⊗B.
Theorem 10 Let R and Q be real W*-algebras, U(R) = R +i R and U(Q) = Q +i Q be their enveloping W*-algebras, respectively. Then Proof Let α R and α Q be canonical *-antiautomorphisms of U(R) and U(Q), generating R and Q, respectively, i.e.
it is held: i.e. (α R ⊗α Q ) 2 is an identity map on U(R)⊗U(Q). Consider a real W*-algebra Then it can be easily shown that F ∩ i F = {0} and U(R)⊗U(Q) = F + i F. If x ∈ R and y ∈ Q, then α R ⊗α Q (x ⊗ y) = α R (x) ⊗ α Q (y) = x * ⊗ y * = (x ⊗ y) * , therefore, Remark 3 Recall that the Jordan analogue of Theorem 10, is proven in [7], i.e. for the JW algebras R and Q it is shown that U(R⊗Q) = U(R)⊗U(Q).
Definition 1 A C*-algebra A is called nuclear, if for any C*-algebra B the norms · min and · max on A ⊗ B (all of C*-norms on A ⊗ B) coincide, i.e. A is called nuclear, if there is the unique C*-norm on A ⊗ B.
The notion of the nuclearity is defined similarly for real C*-algebras. From Theorem 10 we can have the following result.

Proposition 2 A real C*-algebra R is nuclear, if the C*-algebra R + i R is nuclear.
Proof Let R be a real C*-algebra. Then Q + i Q is a C*-algebra and by Theorem 9 Since there is the unique C*-norm on Remark 4 In the paper [Lemma 6.2 (2), [4]] using the complexification of the tensor product of two real C * -algebras it was stated in passing that a real C * -algebra R is nuclear iff the enveloping C * -algebra M = R + i R is nuclear. However, generally speaking, it is not clear that R + i R is nuclear when R is nuclear. Indeed, it is known, that any real W * -algebra is generated by the involutive *-antiautomorphism of enveloping W * -algebra (see [3]). Since there is a W * -algebra N which does not have a *-antiautomorphism algebra M⊗N can also not have a *-antiautomorphism. Therefore, C * -algebra M⊗N can not be expressed as complexification of a real C *algebra.
Therefore, we will formulate the following problem.

Problem 1
Is the C*-algebra R + i R nuclear, whether R is a nuclear real C*-algebra?
Let R be a real C*-algebra. It is known, that the (complex) second dual space (R + i R) * * is a W*-algebra. Then, by (R + i R) * * = R * * + i R * * the (real) second dual space R * * is a real W*-algebra. There is the following connection, relative to injectivity and nuclearity between a real W*-algebra and its second dual space.

Theorem 11
Let R be a real C*-algebra. If the real W*-algebra R * * has the property of E, then R is nuclear.
Proof Since A * * = R * * + i R * * , by Theorem 7 A * * also has the property of E. So, by [10,Theorem] A * * is injective. Then, by [5,Theorem 3], A is nuclear. By Proposition 2 R is also nuclear.
The converse is not obvious, therefore we will formulate the last problem of the paper.
Problem 2 Does the real W*-algebra R * * have the property of E, whether R is a nuclear real C*-algebra?
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