Haagerup Norm Research Articles

Haagerup tensor product of C⁎-ternary rings

The paper investigates Haagerup tensor product of C⁎-ternary rings, more particularly of TROs. We prove that for a TRO M and a C⁎-algebra B, the embedding ζ of M⁎⁎⊗hB⁎⁎ into (M⊗hB)⁎⁎ is an isometry, ⊗h being the Haagerup tensor product. Next, we study the maximal tensor product ⊗tmax for C⁎-trings and compare Haagerup norm with the tmax norm. It is shown that for unital C⁎-algebras A and B, A⊗hB is a C⁎-ternary ring if and only if A=C or B=C. The natural contractive map ϵ:M⊗hB→M⊗tminB is shown to be a monomorphism, ⊗tmin being the injective tensor product. Finally, ϵ-ideals of B(H,K)⊗hB(L) have been characterized.

Elementary operators on Hilbert modules over prime C⁎-algebras

Let X be a right Hilbert module over a C⁎-algebra A equipped with the canonical operator space structure. We define an elementary operator on X as a map ϕ:X→X for which there exists a finite number of elements ui in the C⁎-algebra B(X) of adjointable operators on X and vi in the multiplier algebra M(A) of A such that ϕ(x)=∑iuixvi for x∈X. If X=A this notion agrees with the standard notion of an elementary operator on A. In this paper we extend Mathieu's theorem for elementary operators on prime C⁎-algebras by showing that the completely bounded norm of each elementary operator on a non-zero Hilbert A-module X agrees with the Haagerup norm of its corresponding tensor in B(X)⊗M(A) if and only if A is a prime C⁎-algebra.

On closed Lie ideals of certain tensor products of ‐algebras

AbstractFor a simple ‐algebra A and any other ‐algebra B, it is proved that every closed ideal of is a product ideal if either A is exact or B is nuclear. Closed commutator of a closed ideal in a Banach algebra whose every closed ideal possesses a quasi‐central approximate identity is described in terms of the commutator of the Banach algebra. If α is either the Haagerup norm, the operator space projective norm or the ‐minimal norm, then this allows us to identify all closed Lie ideals of , where A and B are simple, unital ‐algebras with one of them admitting no tracial functionals, and to deduce that every non‐central closed Lie ideal of contains the product ideal . Closed Lie ideals of are also determined, A being any simple unital ‐algebra with at most one tracial state and X any compact Hausdorff space. And, it is shown that closed Lie ideals of are precisely the product ideals, where A is any unital ‐algebra and α any completely positive uniform tensor norm.

Numerical radius Haagerup norm and square factorization through Hilbert spaces

We study a factorization of bounded linear maps from an operator space A to its dual space A * . It is shown that $T: A \rightarrow A*$ factors through a pair of column Hilbert space ℋ c and its dual space if and only if T is a bounded linear form on A ⊗ A by the canonical identification equipped with a numerical radius type Haagerup norm. As a consequence, we characterize a bounded linear map from a Banach space to its dual space, which factors through a pair of Hilbert spaces.

Compact Schur multipliers

Compact Schur multipliers on the algebra B(t) of all bounded linear operators on an infinite-dimensional separable complex Hilbert space 7will be identified as the elements of the Haagerup tensor product co 9h Co (the completion of co (&co in the Haagerup norm). Other ideals of Schur multipliers related to compact operators will also be characterized.

Equivalence of norms on operator space tensor products of $C^\ast $-algebras

The Haagerup norm 11 Ilh on the tensor product A X B of two C*-algebras A and B is shown to be Banach space equivalent to eith.er the Banach space projective norm 11 11? or the operator space projective norm 11 . IIA if and only if either A or B is finite dimensional or A and B are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective nortn are equivalent on A X B if and only if A or B is subhomogeneous.

The Haagerup Norm on the Tensor Product of Operator Modules

It is proved that the (analogy of the) Haagerup norm on the tenser product of submodules of B ( H ) over a von Neumann algebra T ⊆ B ( H ) is injective. If R ⊆ S ⊆ B ( H ) are von Neumann algebras with S injective and T = R ′ ∩ S , then the natural map from S ⊗ T S S equipped with the Haagerup norm to CB( R , S ) (the space of all completely bounded maps from R to S ) is shown to be an isometry, and from this we deduce the result of Chattejee and Smith that the natural map from the central Haagerup tenser product R ⊗ C R to CB( R , R ) is an isometry for each von Neumann algebra R . It is also shown that for an elementary operator on a prime C*-algebra with zero socle or on a continuous von Neumann algebra the norm is equal to the completely bounded norm.

Self-duality for the Haagerup tensor product and Hilbert space factorizations

D. Blecher and V. Paulsen showed that the Haagerup tensor product V ⊗ h W for operator spaces V and W preserves inclusions. It is proved to also preserve complete quotient maps, and to be self-dual in the sense that it induces the Haagerup norm on the algebraic tensor product V ∗ ⊗ W ∗ . The full operator dual space (V ⊗ h W) ∗ is computed. It coincides with the natural operator space \\ ̃ gG 2(V, W ∗) of maps ϑ: V → W ∗ which have completely bounded factorizations through Hilbert spaces (with vectors identified with row matrices). More generally, one has the natural complete isometry \\ ̃ gG 2(V ⊗ h W, X) ≊ \\ ̃ gG 2(V, \\ ̃ gG 2(W, X)) . Given Hilbert spaces H and K with vectors regarded as column matrices, it is shown that one may identify the operator spaces B( H, K) and CB( H, K).

Tensor products of operator spaces

In this paper we lay the foundations for a systematic study of tensor products of subspaces of C ∗-algebras. To accomplish this, various notions of duality are introduced and employed. Elementary proofs of the complete injectivity of the Haagerup norm, and of the extension theorem for completely bounded maps, are given. Pisier's gamma norms are examined and found to be special cases of the Haagerup norm. We identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense. Indeed most of the elementary theory of Banach space tensor norms generalizes to the category of operator spaces.

The maximal C*-norm and the Haagerup norm

AbstractThe difference between the maximal C*-norm ‖ ‖max and the Haagerup norm ‖ ‖h on the tensor product space of C*-aIgebras is studied. Let A and B be C*-algebras. It is shown that ‖ ‖max is equivalent to ‖ ‖h on A ⊗ B if and only if A or B is finite-dimensional.

Geometry of the tensor product of C*-algebras

When and ℬ are C*-algebras their algebraic tensor product ⊗ ℬ is a *-algebra in a natural way. Until recently, work on tensor products of C*-algebras has concentrated on norms α which make the completion ⊗α ℬ into a C*-algebra. The crucial role played by the Haagerup norm in the theory of operator spaces and completely bounded maps has produced some interest in more general norms (see [8; 12]). In this paper we investigate geometrical properties of algebra norms on ⊗ ℬ. By an ‘algebra norm’ we mean a norm which is sub-multiplicative: α(u.v) ≤ ≤ α(u).α(v).