arXiv subjects

Erik Christensen

Publications and source records attributed to Erik Christensen.

At least 19 recordsLinked to original sources

arXiv2025

The Schur multiplier norm and its dual norm

We present a formula for the Schur multiplier norm of a complex self-adjoint matrix, and a formula for the norm, which is dual to the Schur multiplier norm, of a self-adjoint matrix. For a complex self-adjoint $n \times n $ matrix $X$ we show that its Schur multiplier norm is determined by $$ \|X\|_S = \min \{\, \|\mathrm{diag}(P)\|_\infty \, :\, - P \leq X \leq P \, \}.$$ The dual space of $( M_n(\bc), \|.\|_S)$ is $(M_n(\bc), \|.\|_{cbB}).$ For $X=X^*:$ $$ \|X\|_{cbB} = \min \{ \, \mathrm{Tr}_n\big(\Delta(\lambda)\big)\, :\, \lambda \in \br^n, \, - \Delta(\lambda) \leq X \leq \Delta(\lambda)\,\}. $$

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arXiv2025

Column bounded matrices and Grothendieck's inequalities

It follows from Grothendieck's little inequality that to any complex (m x n) matrix X of column norm at most 1, and an 0 <e <1, there exist a natural number q, an (m x q) matrix C with $(1-e)^2 \leq CC^* \leq (4/π) (1 + e)^2$ and an (q x n ) matrix Z with entries in the complex torus such that X= q$^{-(1/2)}$(CZ). Both of Grothendieck's complex inequalities follow from this factorization result.

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arXiv2024

Extremal Schur Multipliers

The Schur product of two complex m x n matrices is their entry wise product. We show that an extremal element X in the convex set of m x n complex matrices of Schur multiplier norm at most 1 satisfies the inequality rank(X) =< (m +n)^(1/2) . For positive n x n matrices with unit diagonal, we give a characterization of the extremal elements, and show that such a matrix satisfies rank(X) =< n^(1/2).

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arXiv2024

Jordan norms for multilinear maps on C*-algebras and Grothendieck's inequalities

There exists a generalization of the concept, completely bounded norm for multilinear maps on C*-algebras. We will use the word, Jordan norm, for this norm. The Jordan norm of a multilinear map is obtained via factorizations of the map, such that bounded operators and Jordan homomorphisms form a long product, as in the case of a completely bounded multilinear map. We show that any bounded bilinear form on a pair of C*-algebras is Jordan bounded and its Jordan norm is at most twice the norm.

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arXiv2023

Some points of view on Grothendieck's inequalities

Haagerup's proof of the non commutative little Grothendieck inequality raises some questions on the commutative little inequality, and it offers a new result on scalar matrices with non negative entries. The theory of completely bounded maps implies that the commutative Grothendieck inequality follows from the little commutative inequality, and that this passage may be given a geometric form as a relation between a pair of compact convex sets of positive matrices, which, in turn, characterizes the little constant in the complex case.

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arXiv2023

Unique Matrix Factorizations associated to Bilinear Forms and Schur Multipliers

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m x n matrices. The theory of operator spaces provides a set up which describes 4 norm optimal factorizations of Grothendieck's sort. It is shown that 3 of the optimal factorizations are uniquely determined and the remaining one is unique in some cases.

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arXiv2023

Norm optimal factorizations of scalar and block matrices

For an $m \times n$ complex matrix $X$ of rank $r$ with Schur multiplier $S_X$ we show that there exist an $ r \times m $ complex matrix $L$ and an $ r\times n $ complex matrix $R$ such that $X = L^*R$ and $\|S_X\|\, =\, \|\mathrm{diag} (L^*L) \|^{\frac{1}{2}} \| \mathrm{diag} (R^*R) \| ^{\frac{1}{2}},$ and the norm condition is optimal. Let the completely bounded norm of the bilinear form $B_X$ induced by $X$ on $(\mathbb{C}^m, \|.\|_\infty) \times (\mathbb{C}^n, \|.\|_\infty)$ be denoted $\|B_X\|_{cb},$ then $X$ has a factorization $ X = Δ(η)^* C Δ(ξ)$ with $η$ in $\mathbb{C}^m,$ $ξ$ in $\mathbb{C}^n$ such that the outer factors are diagonal operators with $\|ξ\|_2 = \|η\|_2=1 $ and $C$ has operator norm equal to $\|B_X\|_{cb},$ and the norm condition is optimal. A generalization to operator valued Schur block multipliers is presented too.

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arXiv2023

Bilinear forms, Schur multipliers, complete boundedness and duality

Grothendieck's inequalities for operators and bilinear forms imply some factorization results for complex m x n matrices. Based on the theory of operator spaces and completely bounded mappings we present norm optimal versions of these results and two norm optimal factorization results related to the Schur product. We show that the spaces of respectively bilinear forms and Schur multipliers are conjugate duals to each other with respect to their completely bounded norms.

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arXiv2021

Minimal Stinespring Representations of Operator Valued Multilinear Maps

A completely positive linear map $φ$ from a C*-algebra $A$ into $B(H)$ has a Stinespring representation as $φ(a) = X^*π(a)X,$ where $π$ is a *-representation of $A$ on a Hilbert space $K$ and $X$ is a bounded operator from $H$ to $K. $ Completely bounded multilinear operators on C*-algebras as well as some densely defined multilinear operators in Connes' non commutative geometry also have Stinespring representations of the form $$ Φ(a_1, \dots, a_k ) = X_0π_1(a_1)X_1 \dots π_k(a_k)X_k$$ such that each $a_i$ is in a *-algebra $A_i$ and $X_0, \dots X_k $ are densely defined closed operators between the Hilbert spaces. We show that for both completely bounded maps and for the geometrical maps, a natural minimality assumption implies that two such Stinespring representations have unitarily equivalent *-representations in the decomposition.

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arXiv2021

C*-dynamical rapid decay

Some well known results by Haagerup, Jolissaint and de la Harpe may be extended to the setting of a reduced crossed product of a C*-algebra A by a discrete group $G.$ We show that for many discrete groups, which include Gromov's hyperbolic groups and finitely generated discrete groups of polynomial growth, an inequality of the form $$\|X\| \leq C \sqrt{\sum_{g \in G} (1+|g|)^4 \|X_g\|^2 } $$ holds for any finitely supported operator $X$ in the reduced crossed product.

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arXiv2020

Unbounded expectations to some von Neumann algebras

For any injective von Neumann algebra R and any discrete, countable group G, which acts by *-automorphisms on R, we construct an idempotent mapping of an ultra-weakly dense subspace of B(H) onto the reducerd crossed product von Neumann algebra, such that it is R-bimodular and satisfies some nice relations with respect to positivity. In the case of an amenable group our unbounded expectation turns into a usual conditional expectation of norm 1.

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arXiv2019

Decompositions of Schur block products

Given two m x n matrices A = (a_{ij}) and B=(b_{ij}) with entries in B(H), the Schur block product is the m x n matrix A \square B := (a_{ij}b_{ij}). There exists an m x n contraction matrix S = (s_{ij}), such that A \square B = diag(AA*)^(1/2) S diag(B*B)^(1/2). This decomposition is also valid for the block Schur tensor product. It is shown, via the theory of random matrices, that the set of contractions S, which may appear in such a decomposition, is a very thin subset of the unit ball of M_n(B(H)).

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arXiv2019

The Hadamard product in a crossed product C*-algebra

We show that for a C*-algebra A and a discrete group G with an action of G on A, the reduced crossed product C*-algebra possesses a natural generalization of the convolution product, which we suggest should be named the Hadamard product. We show that this product has a natural Stinespring representation and we lift some known results on block Schur products to this setting, but we also show that the block Schur product is a special case of the Hadamard product in a crossed product algebra.

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arXiv2018

On the complete boundedness of the Schur block product

We give a Stinespring representation of the Schur block product, say (*), on pairs of square matrices with entries in a C*-algebra as a completely bounded bilinear operator of the form: A:=(a_{ij}), B:= (b_{ij}): A (*) B := (a_{ij}b_{ij}) = V* pi(A) F pi(B) V, such that V is an isometry, pi is a *-representation and F is a self-adjoint unitary. This implies an inequality due to Livshits and two apparently new ones on diagonals of matrices. ||A (*) B|| \leq ||A||_r ||B||_c operator, row and column norm; - diag(A*A) \leq A* (*) A \leq diag(A*A), and for all vectors f, g: | |^2 \leq < diag(AA*) g, g> .

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arXiv2016

Structural properties of close II$_1$ factors

We show that a number of key structural properties transfer between sufficiently close II$_1$ factors, including solidity, strong solidity, uniqueness of Cartan masas and property $Γ$. We also examine II$_1$ factors close to tensor product factors, showing that such factors also factorise as a tensor product in a fashion close to the original.

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arXiv2015

Commutator inequalities via Schur products

For a self-adjoint unbounded operator D on a Hilbert space H, a bounded operator y on H and some complex Borel functions g(t) we establish inequalities of the type ||[g(D),y]|| \leq A|||y|| + B||[D,y]|| + ...+ X|[D, [D,...[D, y]...]]||. The proofs take place in a space of infinite matrices with operator entries, and in this setting it is possible to approximate the matrix associated to [g(D), y] by the Schur product of a matrix approximating [D,y] and a scalar matrix. A classical inequality of Bennett on the norm of Schur products may then be applied to obtain the results.

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arXiv2015

Higher Weak Derivatives and Reflexive Algebras of Operators

Let D be a self-adjoint operator on a Hilbert space H and x a bounded operator on H. We say that x is n-times weakly D-differentiable, if for any pair of vectors a, b from H the function < exp(itD)x exp(-itD) a, b> is n-times differentiable. We give several characterizations of this property, among which one is original. The results are used to show, that for a von Neumann algebra M on H, the sub-algebra of n-times weakly D-differentiable operators has a representation as a reflexive algebra of operators on a bigger Hilbert space.

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arXiv2015

On weakly D-differentiable operators

For an unbounded self-adjoint operator D on a Hilbert space H and a bounded operator a on H we say that a is weakly D-differentiable if for any pair of vectors x, y in H the function is differentiable at t =0. We find several conditions which are all equivalent to weak D-differentiability.

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