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Register a new userNotes on real interpolation of operator $L_p$-spaces
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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 $pge r$.
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We consider the reduction of problems on general noncommutative $L_p$-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncommutative $L_p$-space by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products by a locally compact abelian group or to their noncommutative $L_p$-spaces. We present applications of these results to the theory of noncommutative martingale inequalities by reducing most recent general noncommutative martingale/ergodic inequalities to those in the tracial case.
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 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 $n_{cb}(V,u)$.
We consider the Fourier-Stietljes algebra B(G) of a locally compact group G. We show that operator amenablility of B(G) implies that a certain semitolpological compactification of G admits only finitely many idempotents. In the case that G is connected, we show that operator amenability of B(G) entails that $G$ is compact.
Frames on Hilbert C*-modules have been defined for unital C*-algebras by Frank and Larson and operator valued frames on a Hilbert space have been studied in arXiv.0707.3272v1.[math.FA]. Goal of the present paper is to introduce operator valued frames on a Hilbert C*-module for a sigma-unital C*-algebra. Theorem 1.4 reformulates the definition given by Frank and Larson in terms of a series of rank-one operators converging in the strict topology. Theorem 2.2. shows that the frame transform and the frame projection of an operator valued frame are limits in the strict topology of a series of elements in the multiplier algebra and hence belong to it. Theorem 3.3 shows that two operator valued frames are right similar if and only if they share the same frame projection. Theorem 3.4 establishes a one to one correspondence between Murray-von Neumann equivalence classes of projections in the multiplier algebra and right similarity equivalence classes of operator valued frames and provides a parametrization of all Parseval operator-valued frames on a given Hilbert C*-module. Left similarity is then defined and Proposition 3.9 establishes when two left unitarily equivalent frames are also right unitarily equivalent.
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