Partial $\ast$-algebras of distributions

The problem of multiplying elements of the conjugate dual of certain kind of commutative generalized Hilbert algebras, which are dense in the set of C ∞ -vectors of a self-adjoint operator, is considered in the framework of the so-called duality method. The multiplication is defined by identifying each distribution with a multiplication operator acting on the natural rigged Hilbert space. Certain spaces, that are an

MR3112896 Saichev, Alexander I.; Woyczyński, Wojbor A. Distributions in the physical and engineering sciences. Vol. 2. Linear and nonlinear dynamics in continuous media. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York, 2013. xxiv+409 pp. ISBN: 978-0-8176-3942-6; 978-0-8176-4652-3 (Reviewer: Francesco Tschinke)

MR2806473 (2012f:47002) Hirasawa, Go(J-IBARE) A metric for unbounded linear operators in a Hilbert space. (English summary) Integral Equations Operator Theory 70 (2011), no. 3, 363–378.

MR3198857 Han, Deguang; Larson, David R.; Liu, Bei; Liu, Rui Dilations for systems of imprimitivity acting on Banach spaces. J. Funct. Anal. 266 (2014), no. 12, 6914–6937. (Reviewer: Francesco Tschinke)

Colombeau Algebras and convolutions generated by self-adjoint operators

The role of convolution of functions in the construction of Colombeau algebras of generalized functions is analyzed, with particular referring to the commutative relation with the derivation operator. The possibility to consider the A-convolution, with A an unbounded self-adjoint operator in Hilbert space, is discussed. K

Unbounded C*-seminorms and biweights on partial *-algebras

Unbounded C*-seminorms generated by families of biweights on a partial *-algebra are considered and the admissibility of biweights is characterized in terms of unbounded C*-seminorms they generate. Furthermore, it is shown that, under suitable assumptions, when the family of biweights consists of all those ones which are relatively bounded with respect to a given C*-seminorm q, it can be obtained an expression for q analogous to that one which holds true for the norm of a C*-algebra.

Bounded elements in certain topological partial *-algebras

We continue our study of topological partial *algebras, focusing our attention to the interplay between the various partial multiplications. The special case of partial *-algebras of operators is examined first, in particular the link between the strong and the weak multiplications, on one hand, and invariant positive sesquilinear (ips) forms, on the other. Then the analysis is extended to abstract topological partial *algebras, emphasizing the crucial role played by appropriate bounded elements, called $\M$-bounded. Finally, some remarks are made concerning representations in terms of the so-called partial GC*-algebras of operators.

MR2903153 Roch, Steffen; Santos, Pedro A. Two points, one limit: homogenization techniques for two-point local algebras. J. Math. Anal. Appl. 391 (2012), no. 2, 552–566. (Reviewer: Francesco Tschinke)

Distributions Frames and bases

In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate, in particular, conditions for them to constitute a "continuous basis" for the smallest space $\mathcal D$ of a rigged Hilbert space. This analysis requires suitable extensions of familiar notions as those of frame, Riesz basis and orthonormal basis. A motivation for this study comes from the Gel'fand-Maurin theorem which states, under certain conditions, the existence of a family of generalized eigenvectors of an essentially self-adjoint operator on a domain $\mathcal D$ which acts like an orthonormal basis of the Hilbert space $\mathcal H$. The correspond…

Continuous *-homomorphisms of Banach Partial *-algebras

We continue the study of Banach partial *-algebras, in particular the question of the interplay between *-homomorphisms and biweights. Two special types of objects are introduced, namely, relatively bounded biweights and Banach partial *-algebras satisfying a certain Condition (S), which behave in a more regular way. We also present a systematic construction of Banach partial *-algebras of this type and exhibit several examples.

Partial Multiplication of Operators in Rigged Hilbert Spaces

The problem of the multiplication of operators acting in rigged Hilbert spaces is considered. This is done, as usual, by constructing certain intermediate spaces through which the product can be factorized. In the special case where the starting space is the set of C∞-vectors of a self-adjoint operator A, a general procedure for constructing a special family of interspaces is given. Their definition closely reminds that of the Bessel potential spaces, to which they reduce when the starting space is the Schwartz space \(\mathcal{S}(\mathbb{R}^n ).\) Some applications are considered.

MR2986428 Lebedev, Leonid P.(CL-UNC); Vorovich, Iosif I.; Cloud, Michael J. Functional analysis in mechanics. Second edition. Springer Monographs in Mathematics. Springer, New York, 2013. x+308 pp. ISBN: 978-1-4614-5867-8; 978-1-4614-5868-5

MR2677289 Takakura, Mayumi Noncommutative integration in partial O∗-algebras. Fukuoka Univ. Sci. Rep. 40 (2010), no. 1, 1–20. (Reviewer: Francesco Tschinke)

MR2859703 Liu, Zhe On some mathematical aspects of the Heisenberg relation. Sci. China Math. 54 (2011), no. 11, 2427–2452. (Reviewer: Francesco Tschinke)

A note on *-derivations of partial *-algebras

A definition of *-derivation of partial *-algebra through a sufficient family of ips-forms is proposed.

Biweights and *-homomorphisms of partial *-algebras

Consider two partial *-algebras, 1 and 2, and an *-homomorphism Φ from 1 into 2. Given a biweight ϕ on 2, we discuss conditions under which the natural composition ϕ∘Φ of ϕ and Φ is a biweight on 1. In particular, we examine whether the restriction of a biweight to a partial *-subalgebra is again a biweight.

C*-seminorms and representation on partial *-algebras

In this paper we investigate the *-representations of a partial *-algebra A. In particular, it is proved that, if A is semiassociative and if the set of right multipliers is dense with respect to a seminorm p on A, there exists a bounded and regular *-represenation on A.

Convolutions generated by self-adjoint unbounded operators

Faithfully representable topological *-algebras: some spectral properties

A faithfully representable topological *-algebra (fr*-algebra) A0 is characterized by the fact that it possesses sufficiently many *-representations. Some spectral properties are examined, by constructing a convenient quasi *-algebra A over A0, starting from the order bounded elements of A0.

MR3091813 Botelho, Fernanda; Jamison, James; Molnár, Lajos Surjective isometries on Grassmann spaces. J. Funct. Anal. 265 (2013), no. 10, 2226–2238. (Reviewer: Francesco Tschinke)

Multiplication of Operators in Rigged Hilbert Spaces: motivations and main Results

*-Algebre parziali di distribuzioni

Si illustra in sintesi il metodo per definire nello spazio delle distribuzioni temperate S8(R) una struttura *-algebra parziale non banale

Some results about operators in nested Hilbert spaces

With the use of interpolation methods we obtain some results about the domain of an operator acting on the nested Hilbert space {ℋf}f∈∑ generated by a self-adjoint operatorA and some estimates of the norms of its representatives. Some consequences in the particular case of the scale of Hilbert spaces are discussed.

A short note on O*-algebras and quantum dynamics

We review some recent results concerning algebraic dynamics and O*-algebras. We also give a perturbative condition which can be used, in connection with previous results, to define a time evolution via a limiting procedure.

MR3037568 Argerami, Martín; Massey, Pedro Schur-Horn theorems in II∞-factors. Pacific J. Math. 261 (2013), no. 2, 283–310. (Reviewer: Francesco Tschinke)

Some Notes About Distribution Frame Multipliers

Inspired by a recent work about distribution frames, the definition of multiplier operator is extended in the rigged Hilbert spaces setting and a study of its main properties is carried on. In particular, conditions for the density of domain and boundedness are given. The case of Riesz distribution bases is examined in order to develop a symbolic calculus.

C*-seminorms generated by families of biweights on partial *-algebras

If A[t] is a topological partial *-algebra with unit, topologized by the family of seminorms {p_a}, the notion of bounded element is defined, and some conditions to obtain an unbounded C*-seminorm q(x)=sup p_a(x) on A[t] with domain the subalgebra of bounded elements of A[t] are given.

Spectral Properties of Partial *-Algebras

We continue our study of topological partial *algebras focusing our attention to some basic spectral properties. The special case of partial *-algebras of operators is examined first, in order to find sufficient hints for the study of the abstract case. The outcome consists in the selection of a class of topological partial *-algebras (partial GC*-algebras) that behave well from the spectral point of view and that allow, under certain conditions, a faithful realization as a partial O*-algebra.

A note on partial*–algebras and spaces of distributions

Given a rigged Hilbert space (D,H,D'), the spaces D_{loc are considered. It is shown that, if D is a Hilbert *-algebra, D_{loc} carry out a natural structure of partial *-algebra. Furthermore, on D_{loc} it is defined a topology, so that D_{loc} is an interspace. Examples from distributions theory are considered.

Riesz-Fischer Maps, Semi-frames and Frames in Rigged Hilbert Spaces

In this note we present a review, some considerations and new results about maps with values in a distribution space and domain in a σ-finite measure space X. Namely, this is a survey about Bessel maps, frames and bases (in particular Riesz and Gel’fand bases) in a distribution space. In this setting, the Riesz-Fischer maps and semi-frames are defined and new results about them are obtained. Some examples in tempered distributions space are examined.

MR3257881 Reviewed Hadwin, Don Approximate double commutants in von Neumann algebras and C∗-algebras. Oper. Matrices 8 (2014), no. 3, 623–633. (Reviewer: Francesco Tschinke) 46L10 (46L05

In this paper, the author proves an asymptotic version of the double commutant theorem, in a particular set-up of commutative C∗ -algebras. More precisely, he considers the relative approximate double commutant of a C ∗-algebra with unit, and, using a theorem of characterization for a commutative C∗-subalgebra with unit (inspired by a well-known result due to Kadison for a von Neumann sub-algebra of type I), and from a theorem based on a Machado result, he proves that if A is a commutative C∗-subalgebra of a C∗-algebra B centrally prime with unit, then A is equal to its relative approximate double commutant. In the case where B is a von Neumann algebra, a distance formula is found.

MR3299506 Reviewed Rădulescu, Florin(I-ROME2) On unbounded, non-trivial Hochschild cohomology in finite von Neumann algebras and higher order Berezin's quantization. (English summary) Rev. Roumaine Math. Pures Appl. 59 (2014), no. 2, 265–292.

If (At)t>1 is a family of finite von Neumann algebras with a Chapman-Kolmogorov set of linear maps (symbol system) (Φs,t), and if αt:A→A are isomorphisms in a finite family of von Neumann algebras, the corresponding Hochschild cocycles are related to an obstruction to the deformation of the set of linear maps (Φs,t) in the corresponding Chapman-Kolmogorov system (Φs,t)˜ of completely positive maps. In this set-up, the author introduces an invariant (c,Z) for a finite von Neumann algebra M, consisting of a 2-Hochschild cohomology cocycle c and a coboundary unbounded operator Z for c. With some assumptions on c and Z=α+X+iY (α>0, Y is antisymmetric), the existence of an unbounded derivation δ…