We give a spectral theorem for unital representations of Hermitian commutative unital *-algebras by possibly unbounded operators in a pre-Hilbert space. A better result is known for the case in which the *-algebra is countably generated.
We consider commutative C* -algebras of Toeplitz operators in the weighted Bergman space on the unit ball in $\mathbb{C}^{\mathbf{n}}$. For the algebras of elliptic type we find a new representation, namely as the algebra of operators which are functions of certain collections of commuting unbounded self-adjoint operators in the Bergman space.
We define Schatten classes of adjointable operators on Hilbert modules over abelian $C^*$-algebras. Many key features carry over from the Hilbert space case. In particular, the Schatten classes form two-sided ideals of compact operators and are equipped with a Banach norm and a $C^*$-valued trace with the expected properties. For trivial Hilbert bundles, we show that our Schatten-class operators can be identified bijectively with Schatten-norm-continuous maps from the base space into the Schatten classes on the Hilbert space fiber, with the fiberwise trace. As applications, we introduce the $C…
Let $\ell$ be a commutative ring with involution $*$ containing an element $λ$ such that $λ+λ^*=1$ and let $\operatorname{Alg}^*_\ell$ be the category of $\ell$-algebras equipped with a semilinear involution and involution preserving homomorphisms. We construct a triangulated category $kk^h$ and a functor $j^h:\operatorname{Alg}^*_\ell\to kk^h$ that is homotopy invariant, matricially and hermitian stable and excisive and is universal initial with these properties. We prove that a version of Karoubi's fundamental theorem holds in $kk^h$. By the universal property of the latter, this implies tha…
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…
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 explore factorizations of noncommutative Riemannian spin geometries over commutative base manifolds in unbounded KK-theory. After setting up the general formalism of unbounded KK-theory and improving upon the construction of internal products, we arrive at a natural bundle-theoretic formulation of gauge theories arising from spectral triples. We find that the unitary group of a given noncommutative spectral triple arises as the group of endomorphisms of a certain Hilbert bundle; the inner fluctuations split in terms of connections on, and endomorphisms of, this Hilbert bundle. Moreover, we …
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…
From Kümmerer's investigations on stationary Markov processes has emerged an operator algebraic definition of white noises which captures many examples from classical as well as from non-commutative probability. Within non-commutative $L^p$-spaces associated to a white noise, the role of (non-)commutative Lévy processes is played by additive cocycles for the white noise shift, and moreover, the notion for exponentials of classical Lévy processes is generalized by unitary cocycles. As a main result we report a bijective correspondence between additive and unitary cocycles for white noise shifts…
The theory of Toeplitz quantization presented in our previous paper is extended and further developed to include diverse and interesting non-commutative realizations of the classical Euclidean plane. This is done using Hilbert spaces of entire functions, where polynomials in one complex variable form a dense subspace. The complex coordinate naturally acts as an unbounded multiplication operator generating, together with its adjoint, a highly non-commutative *-algebra of operators. The Toeplitz operators are then geometrically constructed as special elements from this algebra; they are associat…
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…
In this paper we propose a new approach to the central limit theorem (CLT), based on functions of bounded Féchet variation for the continuously differentiable linear statistics of random matrix ensembles which relies on: a weaker form of a large deviation principle for the operator norm; a Poincaré-type inequality for the linear statistics; and the existence of a second-order limit distribution. This approach frames into a single setting many known random matrix ensembles and, as a consequence, classical central limit theorems for linear statistics are recovered and new ones are established, e…
The classical Gelfand--Naimark theorems provide important insight into the structure of general and of commutative C*-algebras. It is shown that these can be generalized to certain ordered *-algebras. More precisely, for $σ$-bounded closed ordered *-algebras a faithful representation as operators is constructed. Similarly, for commutative such algebras, a faithful representation as complex-valued functions is constructed if an additional necessary regularity condition is fulfilled. These results generalize the Gelfand--Naimark representation theorems to classes of *-algebras larger than C*-alg…
We give a counterexample to a trilinear version of the operator space Grothendieck theorem. In particular, we show that for trilinear forms on $\ell_\infty$, the ratio of the symmetrized completely bounded norm and the jointly completely bounded norm is in general unbounded, answering a question of Pisier. The proof is based on a non-commutative version of the generalized von Neumann inequality from additive combinatorics.
Let $G$ be a group and $\ell$ a commutative unital $\ast$-ring with an element $λ\in \ell$ such that $λ+ λ^\ast = 1$. We introduce variants of hermitian bivariant $K$-theory for $\ast$-algebras equipped with a $G$-action or a $G$-grading. For any graph $E$ with finitely many vertices and any weight function $ω\colon E^1 \to G$, a distinguished triangle for $L(E)=L_\ell(E)$ in the hermitian $G$-graded bivariant $K$-theory category $kk^h_{G_{\mathrm{gr}}}$ is obtained, describing $L(E)$ as a cone of a matrix with coefficients in $\mathbb{Z}[G]$ associated to the incidence matrix of $E$ and the w…
We initiate a study of non-commutative Choquet boundary for spaces of unbounded operators. We define the notion of local boundary representations for local operator systems in locally C$^*$-algebras and prove that local boundary representations provide an intrinsic invariant for a particular class of local operator systems. An appropriate analog of purity of local completely positive maps on local operator systems is used to characterize local boundary representations for local operator systems in Frechet locally C$^*$-algebras.
Having in view the study of a version of Gel'fand-Neumark duality adapted to the context of Alain Connes' spectral triples, in this very preliminary review, we first present a description of the relevant categories of geometrical spaces, namely compact Hausdorff smooth finite-dimensional orientable Riemannian manifolds (or more generally Hermitian bundles of Clifford modules over them); we give some tentative definitions of the relevant categories of algebraic structures, namely "propagators" and "spectral correspondences" of commutative Riemannian spectral triples; and we provide a constructi…
We produce a variety of odd bounded Fredholm modules and odd spectral triples on Cuntz-Krieger algebras by means of realizing these algebras as "the algebra of functions on a non-commutative space" coming from a sub shift of finite type. We show that any odd $K$-homology class can be represented by such an odd bounded Fredholm module or odd spectral triple. The odd bounded Fredholm modules that are constructed are finitely summable. The spectral triples are $θ$-summable although their bounded transform, when constructed using the sign-function, will already on the level of analytic $K$-cycles …
