Results 1  10
of
69
Doob’s inequality for noncommutative martingales
 J. reine angew. Math
"... Abstract. Let 1 ≤ p < ∞ and (xn)n∈N be a sequence of positive elements in a noncommutative Lp space and (En)n∈N be an increasing sequence of conditional expectations, then En(xn) ∥ ≤ cp xn∥ ..."
Abstract

Cited by 46 (27 self)
 Add to MetaCart
Abstract. Let 1 ≤ p < ∞ and (xn)n∈N be a sequence of positive elements in a noncommutative Lp space and (En)n∈N be an increasing sequence of conditional expectations, then En(xn) ∥ ≤ cp xn∥
An analytic approach to spectral flow in von Neumann algebras” preprint math.OA/0512454 on arXiv
"... Abstract. The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by BreuerFredholm operators in a semifinite von Neumann algebra. The la ..."
Abstract

Cited by 13 (7 self)
 Add to MetaCart
Abstract. The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by BreuerFredholm operators in a semifinite von Neumann algebra. The latter have continuous spectrum so that the notion of spectral flow turns out to be rather more difficult to deal with. However quite remarkably there is a uniform approach in which the proofs do not depend on discreteness of the spectrum of the operators in question. The first part of this paper gives a brief account of this theory extending and refining earlier results. It is then applied in the latter parts of the paper to a series of examples. One of the most powerful tools is an integral formula for spectral flow first analysed in the classical setting by Getzler and extended to BreuerFredholm operators by some of the current authors. This integral formula was known for Dirac operators in a variety of forms ever since the fundamental papers of Atiyah, Patodi and Singer. One of the purposes of this exposition is to make contact with this early work so that one can understand the recent developments in a proper historical context. In addition we show how to derive these spectral flow formulae in the setting of Dirac operators on (noncompact) covering spaces of a compact spin manifold using the adiabatic method. This answers a question of Mathai connecting Atiyah’s L 2index theorem to our analytic spectral flow. Finally we relate our work to that of Coburn, Douglas, Schaeffer and Singer on Toeplitz operators with almost periodic symbol. We generalise their work to cover the case of matrix valued almost periodic symbols on R N using some ideas of Shubin. This provides us with an opportunity to describe the deepest part of the theory namely the semifinite local index theorem in noncommutative geometry. This theorem, which gives a formula for spectral flow was recently proved by some of the present authors. It provides a farreaching generalisation of the original 1995 result of Connes and Moscovici. 1.
Invariant subspaces for operators in a general II1factor
, 2006
"... It is shown that to every operator T in a general von Neumann factor M of type II1 and to every Borel set B in the complex plane C, one can associate a maximal, closed, Tinvariant subspace, K = KT (B), affiliated with M, such that the Brown measure of T K is concentrated on B. Moreover, K is Thyp ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
It is shown that to every operator T in a general von Neumann factor M of type II1 and to every Borel set B in the complex plane C, one can associate a maximal, closed, Tinvariant subspace, K = KT (B), affiliated with M, such that the Brown measure of T K is concentrated on B. Moreover, K is Thyperinvariant, and the Brown measure of P K ⊥T  K ⊥ is concentrated on C \ B. In particular, if T ∈ M has a Brown measure which is not concentrated on a singleton, then there exists a nontrivial, closed, Thyperinvariant subspace. This paper substitutes and extends the unplublished manuscript [H1] by the first author, where similar results were proved under the assumption that M embeds in an ultrapower R ω of the hyperfinite II1factor R.
Sukochev Spectral flow and Dixmier traces, preprint math.OA/0205076
"... Abstract. We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semifinite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
Abstract. We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semifinite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd L (1,∞)summable BreuerFredholm module in terms of a Hochschild 1cycle. We explain how to derive a Wodzicki residue for pseudodifferential operators along the orbits of an ergodic R n action on a compact space X. Finally we give a short proof an index theorem of Lesch for generalised Toeplitz operators. 0 1 1.
Gundy’s decomposition for noncommutative martingales and applications
 Proc. London Math. Soc
"... Abstract. We provide an analogue of Gundy’s decomposition for L1bounded noncommutative martingales. An important difference from the classical case is that for any L1bounded noncommutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column na ..."
Abstract

Cited by 10 (7 self)
 Add to MetaCart
Abstract. We provide an analogue of Gundy’s decomposition for L1bounded noncommutative martingales. An important difference from the classical case is that for any L1bounded noncommutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of noncommutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak type (1, 1) boundedness for noncommutative martingale transforms and the noncommutative analogue of Burkholder’s weak type inequality for square functions. A sequence (xn)n≥1 in a normed space X is called 2colacunary if there exists a bounded linear map from the closed linear span of (xn)n≥1 to l2 taking each xn to the nth vector basis of l2. We prove (using our decomposition) that any relatively weakly compact martingale difference sequence in L1(M, τ) whose sequence of norms is bounded away from zero is 2colacunary, generalizing a result of Aldous and Fremlin to noncommutative L1spaces.
Dixmier traces as singular symmetric functionals and applications to measurable operators
 J. Funct. Analysis
"... We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the nonnormal Dixmier and ConnesDixmier traces (introduced by Dixmier and adapted to noncommutative geometry by Connes) living on ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the nonnormal Dixmier and ConnesDixmier traces (introduced by Dixmier and adapted to noncommutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the operator ideal L (1,∞) (which is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L (1,∞), i.e. those on which an arbitrary ConnesDixmier trace yields the same value. In the special case, when the operator ideal L (1,∞) is considered on a type I infinite factor, a bounded operator x belongs to L (1,∞) if and only if the sequence of singular numbers {sn(x)}n≥1 (in the descending order and counting the multiplicities) satisfies ‖x ‖ (1,∞):= 1 N supN≥1 Log(1+N) n=1 sn(x) < ∞. In this case, our characterization amounts to saying that a positive element x ∈ L (1,∞) is measurable if and only if limN→ ∞ 1 ∑N LogN n=1 sn(x) exists; (ii) the set of Dixmier traces and the set of ConnesDixmier traces are norming sets (up to equivalence) for the space L (1,∞) /L (1,∞) 0, where the space L (1,∞) 0 is the closure of all finite rank operators in L (1,∞) in the norm ‖. ‖ (1,∞).
Sukochev The Chern Character of Semifinite Spectral Triples
"... All authors were supported by grants from ARC (Australia) and NSERC (Canada), in addition ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
All authors were supported by grants from ARC (Australia) and NSERC (Canada), in addition
Embeddings of NonCommutative L_pSpaces into NonCommutative . . .
 GAFA GEOMETRIC AND FUNCTIONAL ANALYSIS
, 2000
"... It will be shown that for 1 < p < 2 the Schatten pclass is isometrically isomorphic to a subspace of the predual of a von Neumann algebra. Similar results hold for noncommutative Lp(N,τ)spaces defined by a finite trace on a finite von Neumann algebra. The embeddings rely on a suitable notion of p ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
It will be shown that for 1 < p < 2 the Schatten pclass is isometrically isomorphic to a subspace of the predual of a von Neumann algebra. Similar results hold for noncommutative Lp(N,τ)spaces defined by a finite trace on a finite von Neumann algebra. The embeddings rely on a suitable notion of pstable processes in the noncommutative setting.
Operator Valued Hardy Spaces
, 2003
"... We give a systematic study on the Hardy spaces of functions with values in the noncommutative L pspaces associated with a semifinite von Neumann algebra M. This is motivated by the works on matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), and on th ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
We give a systematic study on the Hardy spaces of functions with values in the noncommutative L pspaces associated with a semifinite von Neumann algebra M. This is motivated by the works on matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), and on the other hand, by the recent development on the noncommutative martingale inequalities. Our noncommutative Hardy spaces are defined by the noncommutative Lusin integral function. The main results of this paper include: (i) The analogue in our setting of the classical Fefferman duality theorem between H 1 and BMO. (ii) The atomic decomposition of our noncommutative H 1. (iii) The equivalence between the norms of the noncommutative Hardy spaces and of the noncommutative L pspaces (1 < p < ∞). (iv) The noncommutative HardyLittlewood maximal inequality. (v) A description of BMO as an intersection of two dyadic BMO. (vi) The interpolation results on these Hardy spaces. Plan: