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Register a new userAn abstract characterization of unital operator spaces
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In this article, we give an abstract characterization of the ``identity of an operator space $V$ by looking at a quantity $n_{cb}(V,u)$ which is defined in analogue to a well-known quantity in Banach space theory. More precisely, we show that there exists a complete isometry from $V$ to some $mathcal{L}(H)$ sending $u$ to ${rm id}_H$ if and only if $n_{cb}(V,u) =1$. We will use it to give an abstract characterization of operator systems. Moreover, we will show that if $V$ is a unital operator space and $W$ is a proper complete $M$-ideal, then $V/W$ is also a unital operator space. As a consequece, the quotient of an operator system by a proper complete $M$-ideal is again an operator system. In the appendix, we will also give an abstract characterisation of ``non-unital operator systems using an idea arose from the definition of $n_{cb}(V,u)$.
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Let $mathcal{M}$ be a semifinite von Neumann algebra. We equip the associated noncommutative $L_p$-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for $1<p<infty$ let $$L_{p,p}(mathcal{M})=big(L_{infty}(mathcal{M}),,L_{1}(mathcal{M})big)_{frac1p,,p}$$ be equipped with the operator space structure via real interpolation as defined by the second named author ({em J. Funct. Anal}. 139 (1996), 500--539). We show that $L_{p,p}(mathcal{M})=L_{p}(mathcal{M})$ completely isomorphically if and only if $mathcal{M}$ is finite dimensional. This solves in the negative the three problems left open in the quoted work of the second author. We also show that for $1<p<infty$ and $1le qleinfty$ with $p eq q$ $$big(L_{infty}(mathcal{M};ell_q),,L_{1}(mathcal{M};ell_q)big)_{frac1p,,p}=L_p(mathcal{M}; ell_q)$$ with equivalent norms, i.e., at the Banach space level if and only if $mathcal{M}$ is isomorphic, as a Banach space, to a commutative von Neumann algebra. Our third result concerns the following inequality: $$ big|big(sum_ix_i^qbig)^{frac1q}big|_{L_p(mathcal{M})}lebig|big(sum_ix_i^rbig)^{frac1r}big|_{L_p(mathcal{M})} $$ for any finite sequence $(x_i)subset L_p^+(mathcal{M})$, where $0<r<q<infty$ and $0<pleinfty$. If $mathcal{M}$ is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if $pge r$.
Let $mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal trace $tau$. Let $E(mathcal{M},tau) $ be a symmetric operator space affiliated with $ mathcal{M} $, whose norm is order continuous and is not proportional to the Hilbertian norm $left|cdotright|_2$ on $L_2(mathcal{M},tau)$. We obtain general description of all bounded hermitian operators on $E(mathcal{M},tau)$. This is the first time that the description of hermitian operators on asymmetric operator space (even for a noncommutative $L_p$-space) is obtained in the setting of general (non-hyperfinite) von Neumann algebras. As an application, we resolve a long-standing open problem concerning the description of isometries raised in the 1980s, which generalizes and unifies numerous earlier results.
In a recent work, Moslehian and Rajic have shown that the Gruss inequality holds for unital n-positive linear maps $phi:mathcal A rightarrow B(H)$, where $mathcal A$ is a unital C*-algebra and H is a Hilbert space, if $n ge 3$. They also demonstrate that the inequality fails to hold, in general, if $n = 1$ and question whether the inequality holds if $n=2$. In this article, we provide an affirmative answer to this question.
We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Aglers theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices.
The main result of this paper is the extension of the Schur-Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequences x and y that converge to 0, there exists a compact operator A with eigenvalue list y and diagonal sequence x if and only if y majorizes x (sum_{j=1}^n x_j le sum_{j=1}^n y_j for all n) if and only if x = Qy for some orthostochastic matrix Q. The similar result requiring equality of the infinite series in the case that the sequences x and y are summable is an extension of a recent theorem by Arveson and Kadison. Our proof depends on the construction and analysis of an infinite product of T-transform matrices. Further results on majorization for infinite sequences providing intermediate sequences generalize known results from the finite case. Majorization properties and invariance under various classes of stochastic matrices are then used to characterize arithmetic mean closed operator ideals.
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