In this paper we generalize Brown's spectral distribution measure to a large class of unbounded operators affiliated with a finite von Neumann algebra. Moreover, we compute the Brown measure of all unbounded R-diagonal operators in this class. As a particular case, we determine the Brown measure of z=xy^{-1}, where (x,y) is a circular system in the sense of Voiculescu, and we prove that for all positive integers n, z^n is in L^p(M) iff 0<p< 2/(n+1).
It is shown that the *-algebra of all (closed densely defined linear) operators affiliated with a finite type I von Neumann algebra admits a unique center-valued trace, which turns out to be, in a sense, normal. It is also demonstrated that for no other von Neumann algebras similar constructions can be performed.
This paper is devoted to derivations on the algebra $S(M)$ of all measurable operators affiliated with a finite von Neumann algebra $M.$ We prove that if $M$ is a finite von Neumann algebra with a faithful normal semi-finite trace $τ$, equipped with the locally measure topology $t,$ then every $t$-continuous derivation $D:S(M)\rightarrow S(M)$ is inner. A similar result is valid for derivation on the algebra $S(M,τ)$ of $τ$-measurable operators equipped with the measure topology $t_τ$.
Let $M$ be a type I von Neumann algebra with the center $Z,$ and let $LS(M)$ be the algebra of all locally measurable operators affiliated with $M.$ We prove that every $Z$-linear derivation on $LS(M)$ is inner. In particular all $Z$-linear derivations on the algebras of measurable and respectively totally measurable operators are spatial and implemented by elements from $LS(M).$
Spectral theory and functional calculus for unbounded self-adjoint operators on a Hilbert space are usually treated through von Neumann's Cayley transform. Based on ideas of Woronowicz, we redevelop this theory from the point of view of multiplier algebras and the so-called bounded transform (which establishes a bijective correspondence between closed operators and pure contractions). This also leads to a simple account of the affiliation relation between von Neumann algebras and self-adjoint operators.
Let $M$ be a type I von Neumann algebra with the center $Z,$ a faithful normal semi-finite trace $τ.$ Let $L(M, τ)$ be the algebra of all $τ$-measurable operators affiliated with $M$ and let $S_0(M, τ)$ be the subalgebra in $L(M, τ)$ consisting of all operators $x$ such that given any $ε>0$ there is a projection $p\in\mathcal{P}(M)$ with $τ(p^{\perp})<\infty, xp\in M$ and $\|xp\|<ε.$ We prove that any $Z$-linear derivation of $S_0(M, τ)$ is spatial and generated by an element from $L(M, τ).$
We study L^2-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes. We give a definition of L^2-cohomology and show how the study of the first L^2-Betti number can be related with the study of derivations with values in a bi-module of affiliated operators. We show several results about the possibility of extending derivations from sub-algebras and about uniqueness of such extensions. Along the way, we prove some results about the dimension function of modules over rings of affiliated operators which are of independent interest.
The dimension of any module over an algebra of affiliated operators ${\mathcal U}$ of a finite von Neumann algebra ${\mathcal A}$ is defined using a trace on ${\mathcal A}.$ All zero-dimensional ${\mathcal U}$-modules constitute the torsion class of torsion theory $(\mathrm{\bf T},\mathrm{\bf P})$. We show that every finitely generated ${\mathcal U}$-module splits as the direct sum of torsion and torsion-free part. Moreover, we prove that the theory $(\mathrm{\bf T},\mathrm{\bf P})$ coincides with the theory of bounded and unbounded modules and also with the Lambek and Goldie torsion theories.…
In this paper we suggest an approach for constructing an L1-type space for a positive selfadjoint operator affiliated with von Neumann algebra. For such operator we intro- duce a seminorm, and prove that it is a norm if and only if the operator is injective. For this norm we construct an L1 -type space as the complition of the space of hermitian ultraweakly continuous linear functionals on von Neumann algebra, and represent L1- type space as a space of continuous linear functionals on the space of special sesquilinear forms. Also, we prove that L1 -type space is isometrically isomorphic to the…
Results of Haagerup and Schultz (2009) about existence of invariant subspaces that decompose the Brown measure are extended to a large class of unbounded operators affiliated to a tracial von Neumann algebra. These subspaces are used to decompose an arbitrary operator in this class into the sum of a normal operator and a spectrally negligible operator. This latter result is used to prove that, on a bimodule over a tracial von Neumann algebra that is closed with respect to logarithmic submajorization, every trace is spectral, in the sense that the trace value on an operator depends only on the …
In the present paper we introduce a certain class of non commutative Orlicz spaces, associated with arbitrary faithful normal locally-finite weights on a semi-finite von Neumann algebra $M.$ We describe the dual spaces for such Orlicz spaces and, in the case of regular weights, we show that they can be realized as linear subspaces of the algebra of $LS(M)$ of locally measurable operators affiliated with $M.$
Let $P_E$ be the set of all values of a spectral measure $E$ and $A(P_E)$ be the smallest von Neumann algebra containing $P_E$. We give a simple description of all sets of generators of $A(P_E)$ in terms of the integrals with respect to $E$. The treatment covers not only the case of generators belonging to $A(P_E)$, but also the case of (possibly unbounded) generators affiliated with this algebra.
The theory of direct integral decompositions of both bounded and unbounded operators is further developed; in particular, results about spectral projections, functional calculus and affiliation to von Neumann algebras are proved. For operators belonging to or affiliated to a tracial von Neumann algebra that is a direct integral von Neumann algebra, the Brown measure is shown to be given by the corresponding integral of Brown measures.
This paper is devoted to derivations on the algebra $S_0(M, τ)$ of all $τ$-compact operators affiliated with a von Neumann algebra $M$ and a faithful normal semi-finite trace $τ.$ The main result asserts that every $t_τ$-continuous derivation $D:S_0(M, τ)\rightarrow S_0(M, τ)$ is spatial and implemented by a $τ$-measurable operator affiliated with $M$, where $t_τ$ denotes the measure topology on $S_0(M, τ)$. We also show the automatic $t_τ$-continuity of all derivations on $S_0(M, τ)$ for properly infinite von Neumann algebras $M$. Thus in the properly infinite case the condition of $t_τ$-cont…
We prove that each local Lie $n$-derivation is a Lie $n$-derivation under mild assumptions on the unital algebras with a nontrivial idempotent. As applications, we obtain descriptions of local Lie $n$-derivations on generalized matrix algebras, triangular algebras, nest algebras, von Neumann algebras, and the algebras of locally measurable operators affiliated with a von Neumann algebra.
This paper introduces the notion of exponential arcs in Hilbert space and of exponential arcs connecting vector states on a sigma-finite von Neumann algebra in its standard representation. Results from Tomita-Takesaki theory form an essential ingredient. Starting point is a non-commutative Radon-Nikodym theorem that involves positive operators affiliated with the commutant algebra. It is shown that exponential arcs are differentiable and that parts of an exponential arc are again exponential arcs. Special cases of probability theory and of quantum probability are used to illustrate the approac…
This paper is devoted to local derivations on subalgebras on the algebra $S(M, τ)$ of all $τ$-measurable operators affiliated with a von Neumann algebra $M$ without abelian summands and with a faithful normal semi-finite trace $τ.$ We prove that if $\mathcal{A}$ is a solid $\ast$-subalgebra in $S(M, τ)$ such that $p\in \mathcal{A}$ for all projection $p\in M$ with finite trace, then every local derivation on the algebra $\mathcal{A}$ is a derivation. This result is new even in the case standard subalgebras on the algebra $B(H)$ of all bounded linear operators on a Hilbert space $H.$ We also ap…
In this paper we present a generalization of the Radon-Nikodym theorem proved by Pedersen and Takesaki. Given a normal, semifinite and faithful (n.s.f.) weight $ϕ$ on a von Neumann algebra M and a strictly positive operator $δ$, affiliated with M and satisfying a certain relative invariance property with respect to the modular automorphism group $σ^ϕ$ of $ϕ$, with a strictly positive operator as the invariance factor, we construct the n.s.f. weight $ϕ(δ^{1/2} . δ^{1/2})$. All the n.s.f. weights on M whose modular automorphisms commute with $σ^ϕ$ are of this form, the invariance factor being af…
We prove that norm continuous derivations from a von Neumann algebra into the algebra of operators affiliated with its tensor square are automatically continuous for both the strong operator topology and the measure topology. Furthermore, we prove that the first continuous L^2-Betti number scales quadratically when passing to corner algebras and derive an upper bound given by Shen's generator invariant. This, in turn, yields vanishing of the first continuous L^2-Betti number for II_1 factors with property (T), for finitely generated factors with non-trivial fundamental group and for factors wi…
Given a von Neumann algebra $M$ we consider the central extension $E(M)$ of $M.$ For type I von Neumann algebras $E(M)$ coincides with the algebra $LS(M)$ of all locally measurable operators affiliated with $M.$ In this case we show that an arbitrary automorphism $T$ of $E(M)$ can be decomposed as $T=T_a\circ T_ϕ,$ where $T_a(x)=axa^{-1}$ is an inner automorphism implemented by an element $a\in E(M),$ and $T_ϕ$ is a special automorphism generated by an automorphism $ϕ$ of the center of $E(M).$ In particular if $M$ is of type I$_\infty$ then every band preserving automorphism of $E(M)$ is inner…
The paper is devoted to local derivations on the algebra $S(\mathcal{M},τ)$ of $τ$-measurable operators affiliated with a von Neumann algebra $\mathcal{M}$ and a faithful normal semi-finite trace $τ.$ We prove that every local derivation on $S(\mathcal{M},τ)$ which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the condition of c…
We extend the Fuglede-Putnam theorem from the algebra $B(H)$ of all bounded operators on the Hilbert space $H$ to the algebra of all locally measurable operators affiliated with a von Neumann algebra.
We prove that a finite von Neumann algebra ${\mathcal A}$ is semisimple if the algebra of affiliated operators ${\mathcal U}$ of ${\mathcal A}$ is semisimple. When ${\mathcal A}$ is not semisimple, we give the upper and lower bounds for the global dimensions of ${\mathcal A}$ and ${\mathcal U}.$ This last result requires the use of the Continuum Hypothesis.
We show that the identity is the sum of two commutators in the algebra of all operators affiliated with a von Neumann algebra of type II$_1$, settling a question, in the negative, that had puzzled a number of us.
As the reversed version of usual symmetric norms, we introduce the notion of symmetric anti-norms $\|\cdot\|_!$ defined on the positive operators affiliated with a finite von Neumann algebra with a finite normal trace. Related to symmetric anti-norms, we develop majorization theory and superadditivity inequalities of the form $\|ψ(A+B)\|_!\ge\|ψ(A)\|_!+\|ψ(B)\|_!$ for a wide class of functions $ψ$.
We prove that every derivation acting on a von Neumann algebra $\mathcal{M}$ with values in a quasi-normed bimodule of locally measurable operators affiliated with $\mathcal{M}$ is necessarily inner.
Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $τ,$ let $S_0(M, τ)$ be the algebra of all $τ$-compact operators affiliated with $M.$ We give a complete description of all derivations on the algebra $S_0(M, τ).$ In particular, we prove that if $M$ is of type I$_\infty$ then every derivation on $S_0(M, τ)$ is spatial.
The present paper presents a survey of some recent results devoted to derivations, local derivations and 2-local derivations on various algebras of measurable operators affiliated with von Neumann algebras. We give a complete description of derivation on these algebras, except the case where the von Neumann algebra is of type II$_1$. In the latter case the result is obtained under an extra condition of measure continuity of derivations. Local and 2-local derivations on the above algebras are also considered. We give sufficient conditions on a von Neumann algebra $M$, under which every local or…
We prove that a von Neumann algebra $M$ is abelian if and only if the square of every derivation on the algebra $S(M)$ of measurable operators, affiliated with $M$, is a local derivation. We also show that for general associative unital algebras this is not true.
The paper is devoted to 2-local derivations on the algebra $LS(M)$ of all locally measurable operators affiliated with a type I$_\infty$ von Neumann algebra $M.$ We prove that every 2-local derivation on $LS(M)$ is a derivation.
It is established that every derivation continuous with respect to the local measure topology acting on the *-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$ is necessary inner. If $\mathcal{M}$ is a properly infinite von Neumann algebra, then every derivation on $LS(\mathcal{M})$ is inner. In addition, it is proved that any derivation on $\mathcal{M}$ with values in Banach $\mathcal{M}$-bimodule of locally measurable operators is inner.
For a semi-finite von Neumann algebra $\mathcal A$, we study the case of equality in Young's inequality of s-numbers for a pair of $τ$-measurable operators $a,b$, and we prove that equality is only possible if $|a|^p=|b|^q$. We also extend the result to unbounded operators affiliated with $\mathcal A$, and relate this problem with other symmetric norm Young inequalities.
TU Dresden（德累斯顿工业大学）；Institute of Geometry, TU Dresden（德累斯顿工业大学几何研究所）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）；University of Chinese Academy of Sciences（中国科学院大学）
The analytic von Neumann regular closure $R(Γ)$ of a complex group algebra $\CΓ$ was introduced by Linnell and Schick. This ring is the smallest $*$-regular subring in the algebra of affiliated operators $U(Γ)$ containing $\CΓ$. We prove that all the algebraic von Neumann regular closures corresponding to sofic representations of an amenable group are isomorphic to $R(Γ)$. This result can be viewed as a structural generalization of Lück's Approximation Theorem. \noindent The main tool of the proof which might be of independent interest is that an amenable group algebra $KΓ$ over any field $K$ …
We generalise the local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra \A of a general semifinite von Neumann algebra. In this setting it gives a formula for spectral flow along a path joining an unbounded self adjoint Breuer-Fredholm operator, affiliated to the von Neumann algebra, to a unitarily equivalent operator. Our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is `almost' a (b,B)-cocycle in the cyclic cohomology of \A.
In this paper we extend the Pietsch correspondence for ideals of compact operators and traces on them to the semifinite setting. We prove that a shift-monotone space $E(\Z)$ of sequences indexed by $\Z$ defines a Calkin space $E(\cM,τ)$ of $τ$-measurable operators affiliated with a semifinite von Neumann algebra $\cM$ equipped with a faithful normal semifinite trace $τ$. Furthermore, we show that shift-invariant functionals on $E(\Z)$ generate symmetric functionals on $E(\cM,τ)$. In the special case, when the algebra $\cM$ is atomless or atomic with atoms of equal trace, the converse also hold…
In the paper, we prove an analogue of the Kato-Rosenblum theorem in a semifinite von Neumann algebra. Let $\mathcal{M}$ be a countably decomposable, properly infinite, semifinite von Neumann algebra acting on a Hilbert space $\mathcal{H}$ and let $τ$ be a faithful normal semifinite tracial weight of $\mathcal M$. Suppose that $H$ and $H_1$ are self-adjoint operators affiliated with $\mathcal{M}$. We show that if $H-H_1$ is in $\mathcal{M}\cap L^{1}\left(\mathcal{M},τ\right)$, then the ${norm}$ absolutely continuous parts of $H$ and $H_1$ are unitarily equivalent. This implies that the real par…
We introduce a new topology, weaker than the gap topology, on the space of selfadjoint operators affiliated to a semifinite von Neumann algebra. We define the real-valued spectral flow for a continuous path of selfadjoint Breuer-Fredholm operators in terms of a generalization of the winding number. We compare our definition with Phillips' analytical definition and derive integral formulas for the spectral flow for certain paths of unbounded operators with common domain, generalizing those of Carey--Phillips. Furthermore we prove the homotopy invariance of the real-valued index. As an example w…
The joint Brown measure and joint Haagerup--Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are investigated, and several natural properties are proved for these. It is shown that the support of the joint Brown measure is contained in the Taylor joint spectrum of the tuple, and also in the ostensibly smaller left Harte spectrum. A simultaneous upper triangularization result for finite commuting tuples is proved and the joint Brown measure and joint Haagerup--Schultz projections are shown to be have well under the Arens multi…
Let $\mathcal{M}$ be a type II$_1$ von Neumann factor and let $S(\mathcal{M})$ be the associated Murray-von Neumann algebra of all measurable operators affiliated to $\mathcal{M}.$ We extend a result of Kadison and Liu \cite{KL} by showing that any derivation from $S(\mathcal{M})$ into an $\mathcal{M}$-bimodule $\mathcal{B}\subsetneq S(\mathcal{M})$ is trivial. In the special case, when $\mathcal{M}$ is the hyperfinite type II$_1-$factor $\mathcal{R}$, we introduce the algebra $AD(\mathcal{R})$, a noncommutative analogue of the algebra of all almost everywhere approximately differentiable func…
The present paper is devoted to study of ring isomorphisms of $\ast$-subalgebras of Murray--von Neumann factors. Let $\cM,$ $\cN$ be von Neumann factors of type II$_1,$ and let $S(\cM),$ $S(\cN)$ be the $\ast$-algebras of all measurable operators affiliated with $\cM$ and $ \cN,$ respectively. Suppose that $\cA\subset S(\cM),$ $\cB\subset S(\cN)$ are their $\ast$-subalgebras such that $\cM\subset \cA,$ $\cN\subset \cB.$ We prove that for every ring isomorphism $Φ: \cA \to \cB$ there exist a positive invertible element $a \in \cB$ with $a^{-1}\in \cB$ and a real $\ast$-isomorphism $Ψ: \cM \to \…
We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group $U(\mathcal{H})$ in a Hilbert space $\mathcal{H}$ with $U(\mathcal{H})$ equipped with the strong operator topology. More precisely, for any strongly closed subgroup $G$ of the unitary group $U(\mathfrak{M})$ in a finite von Neumann algebra $\mathfrak{M}$, we show that the set of all generators of strongly continuous one-parameter subgroups of $G$ forms a complete topological Lie algebra with respect to the strong resolvent topology. We also characterize the algebra …
We prove that if $\mathcal M$ is a properly infinite von Neumann algebra and $LS(\mathcal M)$ is the local measurable operator algebra affiliated with $\mathcal M$, then every Jordan derivation from $LS(\mathcal M)$ into itself is continuous with respect to the local measure topology $t(\mathcal M)$. We construct an extension of a Jordan derivation from $\mathcal M$ into $LS(\mathcal M)$ up to a Jordan derivation from $LS(\mathcal M)$ into itself. Moreover, we prove that if $\mathcal M$ is a properly von Neumann algebra and $\mathcal A$ is a subalgebra of $LS(\mathcal M)$ such that $\mathcal M…
We consider the Aluthge transform $|T|^{1/2}U|T|^{1/2}$ of a Hilbert space operator $T$, where $T=U|T|$ is the polar decomposition of $T$. We prove that the map that sends $T$ to its Aluthge transform is continuous with respect to the norm topology and with respect to the $*$--SOT topology on bounded sets. We consider the special case in a tracial von Neumann algebra when $U$ implements an automorphism of the von Neumann algebra generated by the positive part $|T|$ of $T$, and we prove that the iterated Aluthge transform converges to a normal operator whose Brown measure agrees with that of $T…
Given a von Neumann algebra $M$ denote by $S(M)$ and $LS(M)$ respectively the algebras of all measurable and locally measurable operators affiliated with $M.$ For a faithful normal semi-finite trace $τ$ on $M$ let $S(M, τ)$ (resp. $S_0(M, τ)$) be the algebra of all $τ$-measurable (resp. $τ$-compact) operators from $S(M).$ We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra $M.$ In particular, we prove that if $M$ is of type I$_\infty$ then every derivation on $LS(M)$ (resp. $S(M)$ and $S(M,τ)$) is inner, and each deriva…
This paper is concerned with derivations in algebras of (unbounded) operators affiliated with a von Neumann algebra $\mathcal{M}$. Let $\mathcal{% A}$ be one of the algebras of measurable operators, locally measurable operators or, $τ$-measurable operators. We present a complete description of von Neumann algebras $\mathcal{M}$ of type $I$ in terms of their central projections such that every derivation in $\mathcal{A}$ is inner. It is also shown that every derivation in the algebra $LS(\mathcal{M})$ of all locally measurable operators with respect to a properly infinite von Neumann algebra $\…
Let \(\mathcal{A}\) be a unital Banach algebra such that any Jordan derivation from \(\mathcal{A}\) into any \(\mathcal{A}\)-bimodule \(\mathcal{M}\) is a derivation. We prove that any 2-local derivation from the algebra $M_n(\mathcal{A})$ into $M_n(\mathcal{M})$ $(n\geq 3)$ is a derivation. We apply this result to show that any 2-local derivation on the algebra of locally measurable operators affiliated with a von Neumann algebra without direct abelian summands is a derivation.
We prove that any derivation of the *-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated with a properly infinite von Neumann algebra $\mathcal{M}$ is continuous with respect to the local measure topology $t(\mathcal{M})$. Building an extension of a derivation $δ:\mathcal{M}\longrightarrow LS(\mathcal{M})$ up to a derivation from $LS(\mathcal{M})$ into $LS(\mathcal{M})$, it is further established that any derivation from $\mathcal{M}$ into $LS(\mathcal{M})$ is $t(\mathcal{M})$-continuous.
In \cite{bik1}, A. M. Bikchentaev conjectured that for positive $τ-$measurable operators $a$ and $b$ affiliated with an arbitrary semifinite von Neumann algebra $\mathcal M$, the operator $b^{1/2}ab^{1/2}$ is submajorized by the operator $ab$ in the sense of Hardy-Littlewood. We prove this conjecture in full generality and present a number of applications to fully symmetric operator ideals, Golden-Thompson inequality and (singular) traces.
We consider the locally measure topology $t(\mathcal{M})$ on the *-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$. We prove that $t(\mathcal{M})$ coincides with the $(o)$-topology on $LS_h(\mathcal{M})=\{T\in LS(\mathcal{M}): T^*=T\}$ if and only if the algebra $\mathcal{M}$ is $σ$-finite and a finite algebra. We study relationships between the topology $t(\mathcal{M})$ and various topologies generated by faithful normal semifinite traces on $\mathcal{M}$.
In classical matrix theory, there exist useful extremal characterizations of eigenvalues and their sums for Hermitian matrices (due to Ky Fan, Courant-Fischer-Weyl and Wielandt) and some consequences such as the majorization assertion in Lidskii's theorem. In this paper, we extend these results to the context of self adjoint elements of finite von Neumann algebras, and their distribution and quantile functions. This work was motivated by a lemma in a paper by Voiculescu and Bercovici, that described such an extremal characterization of the distribution of a self-adjoint operator affiliated to …
An open question, raised independently by several authors, asks if a closed amenable subalgebra of ${\mathcal B}({\mathcal H})$ must be similar to an amenable C*-algebra; the question remains open even for singly-generated algebras. In this article we show that any closed, commutative, operator amenable subalgebra of a finite von Neumann algebra ${\mathcal M}$ is similar to a commutative C*-subalgebra of ${\mathcal M}$, with the similarity implemented by an element of ${\mathcal M}$. Our proof makes use of the algebra of measurable operators affiliated to ${\mathcal M}$.
We compute spectra and Brown measures of some non self-adjoint operators in $(M_2(\cc), {1/2}Tr)*(M_2(\cc), {1/2}Tr)$, the reduced free product von Neumann algebra of $M_2(\cc)$ with $M_2(\cc)$. Examples include $AB$ and $A+B$, where A and B are matrices in $(M_2(\cc), {1/2}Tr)*1$ and $1*(M_2(\cc), {1/2}Tr)$, respectively. We prove that AB is an R-diagonal operator (in the sense of Nica and Speicher \cite{N-S1}) if and only if Tr(A)=Tr(B)=0. We show that if X=AB or X=A+B and A,B are not scalar matrices, then the Brown measure of X is not concentrated on a single point. By a theorem of Haagerup…
We relate the spectral flow to the index for paths of selfadjoint Breuer-Fredholm operators affiliated to a semifinite von Neumann algebra, generalizing results of Robbin-Salamon and Pushnitski. Then we prove the vanishing of the von Neumann spectral flow for the tangential signature operator of a foliated manifold when the metric is varied. We conclude that the tangential signature of a foliated manifold with boundary does not depend on the metric. In the Appendix we reconsider integral formulas for the spectral flow of paths of bounded operators.
We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman-Solomyak representation of the spectral shift function of M.G. Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow.
