A general approach to the well-behaved unbounded *-representations of a *-algebra X is proposed. Let B be a normed *-algebra equipped with a left action |> of X on B such that (x |> a)^+ b=a^+(x^+ |> b) for a,b\in B and x\in X. Then the pair (X,B) is called a compatible pair. For any continuous non-degenerate *-representation ρof B there exists a closed *-representation ρ' of X such that ρ'(x)ρ(b)=ρ(x |> b), where x\in X and b\in B. The *-representations ρ' are called the well-behaved *-representations associated with the compatible pair (X,B). A number of examples are developed in detail.
Induced representations of $\ast$-algebras by unbounded operators in Hilbert space are investigated. Conditional expectations of a $\ast$-algebra $\cA$ onto a unital $\ast$-subalgebra $\cB$ are introduced and used to define inner products on the corresponding induced modules. The main part of the paper is concerned with group graded $\ast$-algebras $\cA=\oplus_{g\in G}\cA_g$ for which the *-subalgebra $\cB:=\cA_e$ is commutative. Then the canonical projection $p:\cA\to\cB$ is a conditional expectation and there is a partial action of the group $G$ on the set $\cBp$ of all characters of $\cB$ w…
The paper contains results on the structure of Jordan maps and several kinds of triple maps on standard algebras of unbounded operators in Hilbert spaces. These results are unbounded counterparts to results on algebras of bounded operators obtained by Lu, Molnár and others ([3,4,6 -- 10]).
Applying a result of abstract ring theory we get that bijective additive mappings on standard algebras of unbounded operators preserving zero products are multiples of ring isomorphisms. The structure of additive bijective mappings on certain classes of standard algebras of unbounded operators preserving orthogonality in both directions is also investigated. The results are quite similar to those for algebras of bounded operators
In this paper we study decompositions and classification problems for unbounded derivations in Bunce-Deddens-Toeplitz and Bunce-Deddens algebras. We also look at implementations of these derivations on associated GNS Hilbert spaces.
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.
A derivation $δ$ on a $C^*$-algebra has kernel stabilization if for all $n\in \mathbb{N}$, $\ker δ^n=\ker δ.$ Our main result shows that a weakly-defined derivation studied recently by E. Christensen has kernel stabilization. As corollaries, we (1) show that a family of $*$-derivations on $C^*$-algebras studied by Bratteli and Robinson has kernel stabilization and (2) provide sufficient conditions for when operators satisfying the Heisenberg Commutation Relation must both be unbounded.
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).
In this paper we discuss some physical applications of topological *-algebras of unbounded operators. Our first example is a simple system of free bosons. Then we analyze different models which are related to this one. We also discuss the time evolution of two interacting models of matter and bosons. We show that for all these systems it is possible to build up a common framework where the thermodynamical limit of the algebraic dynamics can be conveniently studied and obtained.
For any injective von Neumann algebra R and any discrete, countable group G, which acts by *-automorphisms on R, we construct an idempotent mapping of an ultra-weakly dense subspace of B(H) onto the reducerd crossed product von Neumann algebra, such that it is R-bimodular and satisfies some nice relations with respect to positivity. In the case of an amenable group our unbounded expectation turns into a usual conditional expectation of norm 1.
We introduce a class of densely defined, unbounded, 2-Hochschild cocycles ([PT]) on finite von Neumann algebras $M$. Our cocycles admit a coboundary, determined by an unbounded operator on the standard Hilbert space associated to the von Neumann algebra $M$. For the cocycles associated to the $Γ$-equivariant deformation ([Ra]) of the upper halfplane $(Γ=PSL_2(\mathbb Z))$, the "imaginary" part of the coboundary operator is a cohomological obstruction - in the sense that it can not be removed by a "large class" of closable derivations, with non-trivial real part, that have a joint core domain, …
We define a C*-hull for a *-algebra, given a notion of integrability for its representations on Hilbert modules. We establish a local-global principle which, in many cases, characterises integrable representations on Hilbert modules through the integrable representations on Hilbert spaces. The induction theorem constructs a C*-hull for a certain class of integrable representations of a graded *-algebra, given a C*-hull for its unit fibre.
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 the study of unbounded derivations on Banach quasi *-algebras with a particular emphasis to the case when they are infinitesimal generators of one parameter automorphisms groups. Both of them, derivations and automorphisms are considered in a weak sense; i.e., with the use of a certain families of bounded sesquilinear forms. Conditions for a weak *-derivation to be the generator of a *-automorphisms group are given.
Evans-Hudson flows are constructed for a class of quantum dynamical semigroups with unbounded generator on UHF algebras, which appeared in \cite{Ma}. It is shown that these flows are unital and covariant. Ergodicity of the flows for the semigroups associated with partial states is also discussed.
In the present paper derivations and *-automorphisms of algebras of unbounded operators over the ring of measurable functions are investigated and it is shown that all L^0-linear derivations and L^{0}-linear *-automorphisms are inner. Moreover, it is proved that each L^0-linear automorphism of the algebra of all linear operators on a bo-dense submodule of a Kaplansky-Hilbert module over the ring of measurable functions is spatial.
We discuss some results concerning fixed point equations in the setting of topological *-algebras of unbounded operators. In particular, an existence result is obtained for what we have called {\em weak $τ$ strict contractions}, and some continuity properties of these maps are discussed. We also discuss possible applications of our procedure to quantum mechanical systems.
We show how the fine structure in shift-tail equivalence, appearing in the noncommutative geometry of Cuntz-Krieger algebras developed by the first two authors, has an analogue in a wide range of other Cuntz-Pimsner algebras. To illustrate this structure, and where it appears, we produce an unbounded representative of the defining extension of the Cuntz-Pimsner algebra constructed from a finitely generated projective bi-Hilbertian module, extending work by the third author with Robertson and Sims. As an application, our construction yields new spectral triples for Cuntz- and Cuntz-Krieger alge…
The concept of logarithmic representation of infinitesimal generators is introduced, and it is applied to clarify the algebraic structure of bounded and unbounded infinitesimal generators. In particular, by means of the logarithmic representation, the bounded components can be extracted from generally-unbounded infinitesimal generators. In conclusion the concept of module over a Banach algebra is proposed as the generalization of Banach algebra. As an application to mathematical physics, the rigorous formulation of rotation group, which consists of unbounded operators being written by differen…
While Jordan algebras are commutative, their non-associativity makes it so that the Jordan product operators do not necessarily commute. When the product operators of two elements commute, the elements are said to operator commute. In some Jordan algebras operator commutation can be badly behaved, for instance having elements $a$ and $b$ operator commute, while $a^2$ and $b$ do not operator commute. In this paper we study JB-algebras, real Jordan algebras which are also Banach spaces in a compatible manner, of which C*-algebras are examples. We show that elements $a$ and $b$ in a JB-algebra op…
We introduce a notion of Morita equivalence for non-selfadjoint operator algebras equipped with a completely isometric involution (operator *-algebras). We then show that the unbounded Kasparov product by a Morita equivalence bimodule induces an isomorphism between equivalence classes of twisted spectral triples over Morita equivalent operator *-algebras. This leads to a tentative definition of unbounded bivariant K-theory and we prove that this bivariant theory is related to Kasparov's bivariant K-theory via the Baaj-Julg bounded transform. Moreover, the unbounded Kasparov product provides a …
Let $ \mathcal D$ be a dense linear manifold in a Hilbert space $\mathcal H$ and let $L^+(\mathcal D)$ be the *-algebra of all linear operators $A$ such that $A \mathcal D \subset \mathcal D, A^* \mathcal D \subset \mathcal D$. Denote by $F(\mathcal D)$ the *-ideal of $L^+(\mathcal D)$ consisting of all finite-rank operators. A characterization of the structure of additive mappings of $F(\mathcal D)$ preserving operators of rank one or projections of rank one is given. The corresponding results for algebras of bounded operators on Banach spaces were given by P. uSemrl.
The induction and reduction precesses of an O*-vector space $\M$ obtained by means of a projection taken, respectively, in $\M$ itself or in its weak bounded commutant $\M'_\w$ are studied. In the case where $\M$ is a partial GW*-algebra, sufficient conditions are given for the induced and the reduced spaces to be partial GW*-algebras again.
The Kubo-Ando theory deals with connections for positive bounded operators. On the other hand, in various analysis related to von Neumann algebras it is impossible to avoid unbounded operators. In this article we try to extend a notion of connections to cover various classes of positive unbounded operators (or unbounded objects such as positive forms and weights) appearing naturally in the setting of von Neumann algebras, and we must keep all the expected properties maintained. This generalization is carried out for the following classes: (i) positive $τ$-measurable operators (affiliated with …
We prove that if A is a σ-unital exact C*-algebra of real rank zero, then every state on K_0(A) is induced by a 2-quasitrace on A. This yields a generalisation of Rainone's work on pure infiniteness and stable finiteness of crossed products to the non-unital case. It also applies to k-graph algebras associated to row-finite k-graphs with no sources. We show that for any k-graph whose C*-algebra is unital and simple, either every twisted C*-algebra associated to that k-graph is stably finite, or every twisted C*-algebra associated to that k-graph is purely infinite. Finally we provide sufficien…
We study bounded and unbounded representations of the $*$-algebra $Q_{n,λ}(*)$ generated by $n$ idempotents whose sum equals $λe$ ($λ\in{\mathbb C}$, $e$ is the identity).
It is known that the generator of a strictly continuous one parameter unitary group in the multiplier algebra of a $C^{*}$-algebra is affiliated to that $C^{*}$-algebra . We show that under natural non degeneracy conditions, this self adjoint unbounded operator lies indeed in the (unbounded) multiplier algebra of the Pedersen's ideal of the $C^{*}$-algebra.
Let $E$ and $F$ be Hilbert $C^*$-modules over a $C^*$-algebra $\CAlg{A}$. New classes of (possibly unbounded) operators $t:E\to F$ are introduced and investigated. Instead of the density of the domain $\Def(t)$ we only assume that $t$ is essentially defined, that is, $\Def(t)^\bot=\{0\}$. Then $t$ has a well-defined adjoint. We call an essentially defined operator $t$ graph regular if its graph $\Graph(t)$ is orthogonally complemented in $E\oplus F$ and orthogonally closed if $\Graph(t)^{\bot\bot}=\Graph(t)$. A theory of these operators is developed. Various characterizations of graph regular …
In this paper the concept of unbounded Fredholm operators on Hilbert C*- modules over an arbitrary C*-algebra is discussed and the Atkinson theorem is generalized for bounded and unbounded Feredholm operators on Hilbert C*-modules over C*-algebras of compact operators. In the framework of Hilbert C*-modules over C*-algebras of compact operators, the index of an unbounded Fredholm operator and the index of its bounded transform are the same.
In this article, we consider algebras $\mathcal{A}$ of non-formal pseudodifferential operators over $S^1$ which contain $C^\infty(S^1),$ understood as multiplication operators. We apply a construction of Chern-Weil type forms in order to get $2k-$closed cocycles. For $k=1,$ we obtain a cocycle on the algebra of (maybe non classical) pseudodifferential operators with the same cohomology class as the Schwinger cocycle on the algebra of Classical pseudodifferential operators, previously extended and studied by the author on algebras of the same type.
We consider Hilsum's notion of bordism as an equivalence relation on unbounded $KK$-cycles and study the equivalence classes. Upon fixing two $C^*$-algebras, and a $*$-subalgebra dense in the first $C^*$-algebra, a $\mathbb{Z}/2\mathbb{Z}$-graded abelian group is obtained; it maps to the Kasparov $KK$-group of the two $C^*$-algebras via the bounded transform. We study properties of this map both in general and in specific examples. In particular, it is an isomorphism if the first $C^*$-algebra is the complex numbers (i.e., for $K$-theory) and is a split surjection if the first $C^*$-algebra is…
Anar A. Dosiev in [Local operator spaces, unbounded operators and multinormed $C^*$-algebras, J. Funct. Anal. 255 (2008), 1724-1760], obtained a Stinespring's theorem for local completely positive maps (in short: local CP-maps) on locally $C^{\ast}$-algebras. In this article a suitable notion of minimality for this construction has been identified so as to ensure uniqueness up to unitary equivalence for the associated representation. Using this a Radon-Nikodym type theorem for local completely positive maps has been proved. Further, a Stinespring's theorem for unbounded operator valued local c…
In this paper, we introduce the notion of multiplier of a Hilbert algebra. The space of bounded multipliers is a semifinite von Neumann algebra isomorphic to the left von Neumann algebra of the Hilbert algebra, as expected. However, in the unbounded setting, the space of multipliers has the structure of a *-algebra with nice properties concerning commutant and affiliation: it is a pre-GW*-algebra. And this correspondence between Hilbert algebras and its multipliers is functorial. Then, we can endow the Hilbert algebra with a nice topology constructed from unbounded multipliers. As we can see f…
It is shown how to construct *-homomorphic quantum stochastic Feller cocycles for certain unbounded generators, and so obtain dilations of strongly continuous quantum dynamical semigroups on C* algebras; this generalises the construction of a classical Feller process and semigroup from a given generator. The construction is possible provided the generator satisfies an invariance property for some dense subalgebra A_0 of the C* algebra A and obeys the necessary structure relations; the iterates of the generator, when applied to a generating set for A_0, must satisfy a growth condition. Furtherm…
The aim of this article is to describe a class of *-algebras that allows to treat well-behaved algebras of unbounded operators independently of a representation. To this end, Archimedean ordered *-algebras (*-algebras whose real linear subspace of Hermitian elements are an Archimedean ordered vector space with rather weak compatibilities with the algebraic structure) are examined. The order induces a translation-invariant uniform metric which comes from a C*-norm in the bounded case. It will then be shown that uniformly complete Archimedean ordered *-algebras have good order properties (like e…
Bockstein operations and AD algebras with unbounded torsion in $K_1$
School of Mathematics and Statistics, Northeast Normal University（东北师范大学数学与统计学院）；School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）；Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学高等研究院）
The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert C*-module is isomorphic to a direct summand in the standard module of square summable sequences in the base C*-algebra. In this paper, this result will be generalized by incorporating a densely defined derivation on the base C*-algebra. This leads to a differentiable version of the Kasparov absorption theorem. The extra compatibility assumptions needed are minimal: It will only be required that there exists a sequence of generators with mutual inner products in the domain of the derivation. The diffe…
Given a C*-algebra A with a semicontinuous semifinite trace tau acting on the Hilbert space H, we define the family R of bounded Riemann measurable elements w.r.t. tau as a suitable closure, a la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions, and show that R is a C*-algebra, and tau extends to a semicontinuous semifinite trace on R. Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A'' and can be approximated in measure by operators in R, in analogy with unbounded Riemann integr…
Using unbounded Hilbert space representations basic results on the transition probability of positive linear functionals $f$ and $g$ on a unital *-algebra are obtained. The main assumption is the essential self-adjointness of GNS representations $π_f$ and $π_g$. Applications to functionals given by density matrices and by integrals and to vector functionals on the Weyl algebra are given.
We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach algebras as defined by Runde, but also all group algebras, and all discrete (weakly cancellative) semigroup algebras. Such algebras also behave in a similar way to C$^*$- and W$^*$-algebras; we show that interpolation space techniques can be used in the place of GNS type arguments. We define a notion of injectivity for dual Banach algebras, and show that this i…
We introduce unbounded strongly irreducible operators and transitive operators. These operators are related to a certain class of indecomposable Hilbert representations of quivers on infinite-dimensional Hilbert spaces. We regard the theory of Hilbert representations of quivers is a generalization of the theory of unbounded operators. A non-zero Hilbert representation of a quiver is said to be transitive if the endomorphism algebra is trivial. If a Hilbert representation of a quiver is transitive, then it is indecomposable. But the converse is not true. Let $Γ$ be a quiver whose underlying und…
Consider the $\mathcal{B}$-valued probability space $(\mathcal{A}, E, \mathcal{B})$, where $\mathcal{A}$ is a tracial von Neumann algebra. We extend the theory of operator valued free probability to the algebra of affiliated operators $\tilde{\mathcal{A}}$. For a random variable $X \in \tilde{\mathcal{A}}^{sa}$ we study the Cauchy transform $G_{X}$ and show that the operator algebra $(\mathcal{B} \cup \{X\})"$ can be recovered from this function. In the case where $\mathcal{B}$ is finite dimensional, we show that, when $X, Y \in \tilde{\mathcal{A}}^{sa}$ are assumed to be $\mathcal{B}$-free, t…
Let G be a group, let U(G) denote the set of unbounded operators on L^2(G) which are affiliated to the group von Neumann algebra W(G) of G, and let D(G) denote the division closure of CG in U(G). Thus D(G) is the smallest subring of U(G) containing CG which is closed under taking inverses. If G is a free group then D(G) is a division ring, and in this case we shall give a criterion for an element of U(G) to be in D(G). This extends a result of Duchamp and Reutenauer, which was concerned with proving a conjecture of Connes.
The operator algebra is introduced based on the framework of logarithmic representation of infinitesimal generators. In conclusion a set of generally-unbounded infinitesimal generators is characterized as a module over the Banach algebra.
Let $M$ be a finite von Neumann algebra. In the first part, we give asymptotic results about $M$-stable sequences of weak*-continuous mappings which are related with operators belonging to $M$. In the second part, we extend, by a shorter way, similarity results given in [CaFa2] to unbounded semigroups of operators contained in a finite von Neumann algebra.
