Erling Størmer

Positive linear maps of operator algebras

Introduction.- 1 Generalities for positive maps.- 2 Jordan algebras and projection maps.- 3 Extremal positive maps.- 4 Choi matrices and dual functionals.- 5 Mapping cones.- 6 Dual cones.- 7 States and positive maps.- 8 Norms of positive maps.- Appendix: A.1 Topologies on B(H).- A.2 Tensor products.- A.3 An extension theorem.- Bibliography.- Index .

Symmetric states of infinite tensor products of C∗-algebras

Infinite tensor products of C*-algebras, and even more specially of the complex 2 x 2 matrices, have been of great importance in operator theory. For example, the perhaps most fruitful technique for constructing different types of factors, has been to take weak closures of infinite tensor products in different representations. In addition, some of the C*-algebras of main interest, those of the commutation and the anticommutation relations, are closely related to infinite tensor products of C*-algebras. Since we can also apply the theory, when all factors in the infinite tensor product are abelian, to measure theory on product spaces, we see that the theory of infinite tensor products of C*-algebra may have great potential importance. In the present paper we shall study the infinite tensor product VI * of a C*-algebra b with itself, viz. 2l = @ 23{ , where b, = 23, i = 1, 2,..., and then show how information on the C*-algebra ‘

Classification of Nuclear C*-Algebras. Entropy in Operator Algebras

l leads to both new and known results on the different subjects mentioned in the preceding paragraph. If G denotes the group of one-toone mappings of the positive integers onto themselves leaving all but a finite number of integers fixed, then G has a canonical representation as *-automorphisms of 2X, namely as those which permute the factors in the tensor product. Then ‘8 is asymptotically abelian with respect to G in the sense of [Id], h ence the theory of such C*-algebras, as developed in [Id] and [27], is applicable to ‘3. Our main concern will be with the G-invariant states of 2l. Such states are called sym-

A Gelfand-Neumark theorem for Jordan algebras

I. Classification of Nuclear, Simple C*-algebras.- II. A Survey of Noncommutative Dynamical Entropy.

Homogeneity of the state space of factors of type III1

Abstract Let A be a Jordan algebra over the reals which is a Banach space with respect to a norm satisfying the requirements: (i) ∥ a ° b ∥ ≤ ∥ a ∥ ∥ b ∥, (ii) ∥ a2 ∥ = ∥ a ∥2, (iii) ∥ a2 ∥ ≤ ∥ a2 + b2 ∥ for a, bϵA. It is shown that A possesses a unique norm closed Jordan ideal J such that A J has a faithful representation as a Jordan algebra of self-adjoint operators on a complex Hilbert space, while every “irreducible” representation of A not annihilating J is onto the exceptional Jordan algebra M38.

Jordan algebras of type I

Abstract A factor M is of type III 1 if and only if the action of its unitary group on its state space by inner automorphisms is topologically transitive in the norm topology.

DECOMPOSABLE POSITIVE MAPS ON C*-ALGEBRAS

Jordan, von Neumann, and Wigner [5] have classified all finite dimensional Jordan algebras over the reals. The present paper is an attempt to do the same in the infinite dimensional case. The following restriction will be imposed: we assume the Jordan alge- bras are weakly closed Jordan algebras of self-adjoint operators with minimal projections acting on a Hilbert space, i.e. are irreducible JW-algebras of type i.(1) The result is then quite analogous to that in [5], except we do not get hold of the Jordan algebra ~a s of that paper, as should be expected from the work of Albert [1]. We first classify all irreducible JW-algebras of type In, n>~3 (Theorem 3.9). These algebras are roughly all seif-adjoint operators on a Hilbert space over either the reals, the complexes, or the quaternions. Then all JW-factors of type

Equivalence of normal states on von Neumann algebras and the flow of weights

It is shown that a positive linear map of a C*-algebra A into B(H) is decomposable if and only if for all n G N whenever (x,j) and (*,,) belong to M„(A)+ then (<¡>(*,7)) belongs to M„(B(H))+ . A positive linear map <b of a C*-algebra A into B(H)—the bounded linear operators on a complex Hilbert space H— is said to be decomposable if there are a Hilbert space K, a bounded linear operator v of H into K, and a Jordan homomorphism it of A into B(K) such that <l>(x) = v*ir(x)v for all x G A. Such maps have been studied in [2, 3, 5, 7, 8, 9], and are the natural symmetrization of the completely positive ones, defined as those <b as above with m a homomorphism. If Mn(B) denotes the n X n matrices over a subspace £ of a C*-algebra and Mn(B)+ the positive part of Mn(B), the celebrated Stinespring theorem [4] states that a map <j>: A -> B(H) is completely positive if and only if for all n G N whenever (x¡¡) G Mn(A)+ then («H*,,)) G Mn(B(H))+. It is the purpose of the present note to provide an analogous characterization of decomposable maps. Theorem. Let A be a C*-algebra and <b a linear map of A into B(H). Then <¡> is decomposable if and only if for all n G N whenever (x,¡) and (Xj¡) belong to Mn(A)+ then (<t>(XlJ)) G Mn(B(H))+ . Proof. Suppose <j> is decomposable, so of the form v*rrv. If n is a homomorphism (resp. antihomomorphism) and (xi}) (resp. (xj¡)) belongs to Mn(A)+ then (<H*,,)) G Mn(B(H))+ . Since every Jordan homomorphism is the sum of a homomorphism and an antihomomorphism [6], if both (x¡¡) and (x„) belong to Mn(A)+ then (*(*„)) G Mn(B(H))+. Conversely suppose (x¡ ) and (xjt) G Mn(A)+ implies («K-*//)) e Mn(B(H))+ for all n G N. Since this property persists when <p is extended to the second dual of A we may assume A is unital and that A C B(L) for some Hilbert space L. Let t denote the transpose map on B(L) with respect to some orthonormal basis. Let Then Kis a self adjoint subspace of M2(B(L)) containing the identity. Define 0n on Mn(B(L)) by 8n((Xjj)) = (xj,). Then 6 is an antiautomorphism of order 2. Hence if Received by the editors September 25, 1981 and, in revised form, December 4, 1981. 1980 Mathematics Subject Classification. Primary 46L05; Secondary 46L50.

Cones of positive maps and their duality relations

Abstract Let M be a von Neumann algebra. Two positive normal functionals ϕ, ψ on M are called equivalent, ϕ∼ψ, if ψ is in the norm-closure of the orbit of ϕ under the action of inner automorphisms. Our main result is an isometric characterization of the quotient space M + ∗ ∼ : We construct a natural isometry [ϕ] a \ g4 of M + ∗ ∼ into the set of positive normal functionals on “the smooth flow of weights” of M, where the smooth flow of weights is realized as the center Z(N) of the crossed product N = M × σω R for some faithful normal semifinite weight ω on M. As an application we obtain that an automorphism α on a factor M with separable predual acts trivially on M + ∗ ∼ if and only if α acts trivially on the smooth flow of weights, i.e., the Connes-Takesaki modulus mod(α) of α vanishes. We also obtain a new proof of the diameter formula diam (S n (M)/≈)=2 1 − λ 1 + λ for the quotient of the state space of a factor of type IIIλ, 0 ⩽ λ ⩽ 1.

The crossed product of a UHF algebra by a shift

The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable, and k-entangled operators due to the Jamiolkowski–Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies.