Second order arithmetic means in operator ideals

Equality of the second order arithmetic means of two principal ideals does not imply equality of their first order arithmetic means (second order equality cancellation). We provide fairly broad sufficient conditions on one of the principal ideals for this implication to hold true. We present also sufficient conditions for second order inclusion cancellations. These conditions are formulated in terms of the growth properties of the ratio of regularity sequence associated to the sequence of s-number of a generator of the principal ideal. These results are then extended to general ideals.Comment: 19 pages. To appear in Operators and Matrice

Traces on operator ideals and arithmetic means

This article - a part of a multipaper project investigating arithmetic mean ideals - investigates the codimension of commutator spaces [I, B(H)] of operator ideals on a separable Hilbert space, i.e., ``How many traces can an ideal support?" We conjecture that the codimension can be only zero, one, or infinity. Using the arithmetic mean (am) operations on ideals introduced by Dykema, Figiel, Weiss, and Wodzicki, and the analogous am operations at infinity that we develop in this article, the conjecture is proven for all ideals not contained in the largest am-infinity stable ideal and not containing the smallest am-stable ideal. It is also proven for all soft-edged ideals (i.e., I= IK(H)) and all soft-complemented ideals (i.e., I= I/K(H)), which include many classical operator ideals. In the process, we prove that an ideal of trace class operators supports a unique trace (up to scalar multiples) if and only if it is am-infinity stable and that, for a principal ideal, am-infinity stability is equivalent to regularity at infinity of the sequence of s-numbers of the generator. Furthermore, we apply trace extension methods to two problems on elementary operators studied by V. Shulman and to Fuglede-Putnam type problems of the second author.Comment: 41 pages, 1 figur

Soft ideals and arithmetic mean ideals

This article investigates the soft-interior and the soft-cover of operator ideals. These operations, and especially the first one, have been widely used before, but making their role explicit and analyzing their interplay with the arithmetic mean operations is essential for the study of the multiplicity of traces (see arXiv:0707.3169v1 [math.FA]). Many classical ideals are "soft", i.e., coincide with their soft interior or with their soft cover, and many ideal constructions yield soft ideals. Arithmetic mean (am) operations were proven to be intrinsic to the theory of operator ideals by the work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators and arithmetic mean operations at infinity were studied in arXiv:0707.3169v1 [math.FA]. Here we focus on the commutation relations between these operations and soft operations. In the process we characterize the am-interior and the am-infinity interior of an ideal.Comment: 21 page

An infinite dimensional Schur-Horn theorem and majorization theory with applications to operator ideals

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.Comment: 44 pages Changes in 2nd version: 1. We discuss overlaps with a 1966 paper by Gohberg and Markus. 2. Majorization characterizes diagonals in the partial isometry orbit. We added results on diagonals in the unitary orbi

Sums of equivalent sequences of positive operators in von Neumann factors

Let A be a positive operator in an infinite sigma-finite von Neumann factor M and let B_j be a sequence of positive elements in M. We give sufficient conditions for decomposing A into a sum of elements C_j equivalent to B_j for all j ( C equivalent to B in M means that C=XX* and B=X*X for some X in M) and when C_j are unitarily equivalent to B_j for all j. This extends recent work of Bourin and Lee for the case of B_j= B for all j and M=B(H) and answers affirmatively their conjecture. For the case when B_j= B for all j we provide necessary conditions, which in the type III case are also sufficient

A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur-Horn majorization theorem

The work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals, and we explored their role in a multipaper project which we survey in this article. We start by presenting the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we explore their connections with commutator spaces, traces, elementary operators, lattice and sublattice structure of ideals, arithmetic mean ideal cancellation properties of first and second order, and softness properties - a term that we introduced but a notion ubiquitous in the literature on operator ideals. Arithmetic mean closure of ideals leads us to investigate majorization for infinite sequences and this in turn leads us to an infinite Schur-Horn majorization theorem which extends theorems by A. Neumann, by Arveson and Kadison, and by Antezana, Massey, Ruiz and Stojanoff. We also list ten open questions that we encountered in the development of this material.Comment: 33 page

Operator valued frames

We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends to higher multiplicity (e.g., multiframes) their dilation approach. We prove several results for operator-valued frames concerning their parametrization, duality, disjointeness, complementarity, and composition and the relationship between the two types of similarity (left and right) of such frames. We then apply these notions to prove that the collection of multiframe generators for the action of a discrete group on a Hilbert space is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. The proof is obtained by parametrizing this collection by a class of partial isometries in a larger von Neumann algebra. In the multiplicity one case this class reduces to the unitary class which is path-connected in norm, but in the infinite multiplicity case this class is path connected only in the strong operator topology and the proof depends on properties of tensor product slice maps.Comment: 37 page

Operator valued frames on C*-modules

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.Comment: 15 pages, to appear in Contemporary Mathematics. Updated reference list and introduction, corrected typo

Strict comparison of positive elements in multiplier algebras

Main result: If a C*-algebra is simple, $\sigma$-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier also has strict comparison of positive elements by traces. The same results holds if "finitely many extremal traces" is replaced by "quasicontinuous scale". A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary $\sigma$-unital C*-algebra can be approximated by a bi-diagonal series. An application of strict comparison: If the algebra is a simple separable stable $\sigma$-unital with real rank zero, stable rank one, strict comparison of positive elements by traces, then whether a positive element is a linear combination of projections depends on the trace values of its range projection.Comment: 30 page

The minimal ideal in multiplier algebras

Let $\mathcal A$ be a simple, $\sigma$-unital, non-unital, non-elementary C*-algebra and let $I_{min}$ be the intersection of all the ideals of $\mathcal M(\mathcal A)$ that properly contain $\mathcal A$. $I_{min}$ coincides with the ideal defined by Lin (Simple C*-algebras with continuous scales and simple corona algebras. 112, (1991) Proc. Amer.Math. Soc) in terms of approximate units of $\mathcal A$ and $I_{min}/\mathcal A$ is purely infinite and simple. If $\mathcal A$ is separable, or if $\mathcal A$ has the (SP) property and its dimension semigroup $D(\mathcal A)$ of Murray-von Neumann equivalence classes of projections of $\mathcal A$ is order separable, or if $\mathcal A$ has strict comparison of positive elements by traces, then $\mathcal A\ne I_{min}$. If the tracial simplex $\mathcal T(\mathcal A)$ is nonempty, let $I_{con}$ be the closure of the linear span of the elements $A\in\mathcal M(\mathcal A)_+$ such that the evaluation map $\hat A(\tau)=\tau(A)$ is continuous. If $\mathcal A$ has strict comparison of positive element by traces then $I_{min}= I_{con}$. Furthermore, $I_{min}$ too has strict comparison of positive elements in the sense that if $A, B\in (I_{min})_+$, $B\not \in\mathcal A$ and $d_\tau(A)< d_\tau(B)$ for all $\tau\in \mathcal T(\mathcal A)$ for which $d_\tau(B)< \infty$, then $A\preceq B$. However if $\mathcal A$ does not have strict comparison of positive elements by traces then $I_{min}\ne I_{con}$ can occur: a counterexample is provided by Villadsen's AH algebras without slow dimension growth. If the dimension growth is flat, $I_{con}$ is the largest proper ideal of $\mathcal M(\mathcal A)$