Partial Isometries Research Articles

EP elements, partial isometries and solutions of some equations

In this paper, we firstly present new charactrizations of EP elements, partial isometries. Next, we investigate the general solutions of some equations. Finally, we discuss the relation between the consistency of certain equations and SEP elements.

Some studies on partial isometry in rings with involution

This paper mainly gives some sufficient and necessary conditions for an element in a ring with involution to be partial isometry and strongly EP element by using some invertible elements and solutions of certain equations.

On a locally compact monoid of cofinite partial isometries of ℕ with adjoined zero

Abstract Let 𝒞ℕ be a monoid which is generated by the partial shift α : n↦n +1 of the set of positive integers ℕ and its inverse partial shift β : n + 1 ↦n. In this paper we prove that if S is a submonoid of the monoid Iℕ∞ of all partial cofinite isometries of positive integers which contains Cscr;ℕ as a submonoid then every Hausdorff locally compact shift-continuous topology on S with adjoined zero is either compact or discrete. Also we show that the similar statement holds for a locally compact semitopological semigroup S with an adjoined compact ideal.

Some characterizations of strongly partial isometry elements in rings with involutions

In this paper, we study an element which is both group invertible and Moore Penrose invertible to be EP, partial isometry and strongly EP by discussing the existence of solutions in a definite set of some given constructive equations. Mainly, let a ? R# ? R+. Then we firstly show that an element a ? REP if and only if and Equation : axa+ + a+ax = 2x has at least one solution in ?a = {a, a#, a+, a+, (a#)+, (a+)+}. Next, a ? RSEP if and only if Equation: axa+ + a+ax = 2x has at least one solution in ?a. Finally, a ? RPI if and only if Equation: aya+x = xy has at least one solution in ?2a , where ?a = {a, a#, a+, (a#)+, (a+)+}.

Infinite sum relations on universal C*-algebras

We extend the usual theory of universal C*-algebras from generators and relations in order to allow some relations to be described using the strong operator topology. In particular, we can allow some infinite sum relations. We prove a universal property for the algebras we define and we show how the Cuntz algebra of infinite isometries as well as the Exel-Laca algebras can be described using infinite sum relations. Finally, we give some sufficient conditions for when a C*-algebra generated by projections and partial isometries is a universal C*-algebra using only norm relations, in case one still wants to avoid using relations with respect to the strong operator topology.

Partial isometries, duality, and determinantal point processes

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures [Formula: see text] on a space S with measure [Formula: see text], whose correlation functions are all given by determinants specified by an integral kernel K called the correlation kernel. We consider a pair of Hilbert spaces, [Formula: see text], which are assumed to be realized as [Formula: see text]-spaces, [Formula: see text], [Formula: see text], and introduce a bounded linear operator [Formula: see text] and its adjoint [Formula: see text]. We show that if [Formula: see text] is a partial isometry of locally Hilbert–Schmidt class, then we have a unique DPP [Formula: see text] associated with [Formula: see text]. In addition, if [Formula: see text] is also of locally Hilbert–Schmidt class, then we have a unique pair of DPPs, [Formula: see text], [Formula: see text]. We also give a practical framework which makes [Formula: see text] and [Formula: see text] satisfy the above conditions. Our framework to construct pairs of DPPs implies useful duality relations between DPPs making pairs. For a correlation kernel of a given DPP our formula can provide plural different expressions, which reveal different aspects of the DPP. In order to demonstrate these advantages of our framework as well as to show that the class of DPPs obtained by this method is large enough to study universal structures in a variety of DPPs, we report plenty of examples of DPPs in one-, two- and higher-dimensional spaces S, where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter ([Formula: see text]) series of infinite DPPs on [Formula: see text] and [Formula: see text] are discussed, which we call the Euclidean and the Heisenberg families of DPPs, respectively, following the terminologies of Zelditch.

Star Order and Partial Isometries in $$\boldsymbol{C}^{\mathbf{\ast}}$$-Algebras

Star Order and Partial Isometries in $$\boldsymbol{C}^{\mathbf{\ast}}$$-Algebras

K-Theory for Semigroup C*-Algebras and Partial Crossed Products

Using the Baum–Connes conjecture with coefficients, we develop a K-theory formula for reduced C*-algebras of strongly 0-E-unitary inverse semigroups, or equivalently, for a class of reduced partial crossed products. This generalizes and gives a new proof of previous K-theory results of Cuntz, Echterhoff and the author. Our K-theory formula applies to a rich class of C*-algebras which are generated by partial isometries. For instance, as new applications which could not be treated using previous results, we discuss semigroup C*-algebras of Artin monoids, Baumslag-Solitar monoids and one-relator monoids, as well as C*-algebras generated by right regular representations of semigroups of number-theoretic origin, and C*-algebras attached to tilings.

SOME NEW CHARACTERIZATIONS OF PARTIAL ISOMETRIES IN RINGS WITH INVOLUTION

In this paper, some characterizations of partial isometries, normal elements and strongly $EP$ elements are given by the construction of $EP$ elements. At the same time, the partial isometry elements are characterized by the existence of solutions of equations in rings in a given set, and also by the general form of solutions of given equations.

On a conjecture by Mbekhta about best approximation by polar factors

The polar factor of a bounded operator acting on a Hilbert space is the unique partial isometry arising in the polar decomposition. It is well known that the polar factor might not be a best approximant to its associated operator in the set of all partial isometries, when the distance is measured in the operator norm. We show that the polar factor of an arbitrary operator T T is a best approximant to T T in the set of all partial isometries X X such that dim ⁡ ( ker ⁡ ( X ) ∩ ker ⁡ ( T ) ⊥ ) ≤ dim ⁡ ( ker ⁡ ( X ) ⊥ ∩ ker ⁡ ( T ) ) \dim (\ker (X)\cap \ker (T)^\perp )\leq \dim (\ker (X)^\perp \cap \ker (T)) . We also provide a characterization of best approximations. This work is motivated by a recent conjecture by M. Mbekhta, which can be answered using our results.

Extreme Points and Factorizability for New Classes of Unital Quantum Channels

We introduce and study two new classes of unital quantum channels. The first class describes a 2-parameter family of channels given by completely positive (CP) maps $M_3({\bf C}) \mapsto M_3({\bf C})$ which are both unital and trace-preserving. Almost every member of this family is factorizable and extreme in the set of CP maps which are both unital and trace-preserving, but is not extreme in either the set of unital CP maps or the set of trace-preserving CP maps. We also study a large class of maps which generalize the Werner-Holevo channel for $d = 3$ in the sense that they are defined in terms of partial isometries of rank $d-1$. Moreover, we extend this to maps whose Kraus operators have the form $t |e_j \rangle \langle e_j | \oplus V $ with $V \in M_{d-1} ({\bf C}) $ unitary and $t \in (-1,1)$. We show that almost every map in this class is extreme in both the set of unital CP maps and the set of trace-preserving CP maps. We analyze in detail a particularly interesting subclass which is extreme unless $t = -1/(d-1)$. For $d = 3$, this includes a pair of channels which have a dual factorization in the sense that they can be obtained by taking the partial trace over different subspaces after using the same unitary conjugation in $M_3({\bf C}) \otimes M_3({\bf C})$.

Dye’s theorem for tripotents in von Neumann algebras and JBW$$^*$$-triples

We study morphisms of the generalized quantum logic of tripotents in JBW\(^*\)-triples and von Neumann algebras. Especially, we establish a generalization of celebrated Dye’s theorem on orthoisomorphisms between von Neumann lattices to this new context. We show the existence of a one-to-one correspondence between the following maps: (1) quantum logic morphisms between the posets of tripotents preserving reflection \(u\rightarrow - u\) (2) maps between triples that preserve tripotents and are real linear on sets of elements with bounded range tripotents. In a more general description we show that quantum logic morphisms on structure of tripotents are given by a family of Jordan *-homomorphisms on Peirce 2-subspaces. By examples we demonstrate optimality of the results. Besides we show that the set of partial isometries with its partial order and orthogonality relation is a complete Jordan invariant for von Neumann algebras.

Weak group inverses and partial isometries in proper *-rings

ABSTRACT A weak group element is introduced in a proper *-ring. Several equivalent conditions of weak group elements are investigated. We prove that an element is pseudo core invertible if it is both partial isometry and weak group invertible. Reverse order law and additive property of the weak group inverse are presented. Finally, under certain assumption on a, equivalent conditions of are presented by using the normality of the group invertible part of an element in its group-EP decomposition.

On partial isometries with circular numerical range

Abstract In their LAMA 2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on ℂ n cannot have a circular numerical range with a non-zero center, and proved this conjecture for n ≤ 4. We prove it for operators with rank A = n − 1 and any n. The proof is based on the unitary similarity of A to a compressed shift operator SB generated by a finite Blaschke product B. We then use the description of the numerical range of SB as intersection of Poncelet polygons, a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenter of the vertices of Poncelet polygons involving elliptic functions.

Partial isometry and strongly EP elements

EP elements are important research objects in the ring theory. This paper mainly gives sufficient and necessary conditions for an element in a ring to be an EP element, partial isometry, and strongly EP element by using solutions of certain equations.

Generalized inverses and solutions to related equations

In this paper, some new characterizations of partial isometries and strongly EP elements are investigated. Especially, we discuss the existence of the solutions of certain equations in a given set to characterize partial isometries, strongly EP elements and so on.

Interplay of Simple and Selfadjoint-ideal semigroups in B(H)

This paper investigates a question of Radjavi: Which multiplicative semigroups in B(H) have all their ideals selfadjoint (called herein selfadjoint-ideal (SI) semigroups)? We proved this property is a unitary invariant for B(H)-semigroups, which invariant we believe is new. We characterize those SI semigroups S singly generated by T, for T a normal operator and for T a rank one operator. When T is nonselfadjoint and normal or rank one: S is an SI semigroup if and only if it is simple, except in one special rank one partial isometry case when our characterization yields S that are SI but not simple. So SI and simplicity are not equivalent notions. When T is selfadjoint, it is straightforward to see that S is always an SI semigroup, but we prove by examples they may or may not be simple, but for this case we do not have a characterization. The study of SI semigroups involves solving certain operator equations in the semigroups. A central theme of this paper is to study when and when not SI is equivalent to simple.

Moore-Penrose inverses in rings and weighted partial isometries in [formula omitted]algebras

We explore the existence of MP-inverse in rings with involution. In addition, we give some equivalent conditions for an element of a C*−algebra to be a weighted-EP element and weighted partial isometry.

Partial isometries in an absolute order unit space

In this paper, we extend the notion of orthogonality to the general elements of an absolute matrix order unit space and relate it to the orthogonality among positive elements. We introduce the notion of a partial isometry in an absolute matrix order unit space. As an application, we describe the comparison of order projections. We also discuss finiteness of order projections.

Polyhomomorphisms of locally compact groups

Let and be locally compact groups with fixed two-sided invariant Haar measures. A polyhomomorphism is a closed subgroup with fixed Haar measure, whose marginals on and are dominated by the Haar measures on and . A polyhomomorphism can be regarded as a multi-valued map sending points to sets equipped with ‘uniform’ measures. For two polyhomomorphisms and there is a well-defined product . The set of polyhomomorphisms is a metrizable compact space with respect to the Chabauty-Bourbaki topology and the product is separately continuous. A polyhomomorphism determines a canonical operator , which is a partial isometry up to a scalar factor. For example, we consider locally compact linear spaces over finite fields and examine the closures of groups of linear operators in semigroups of polyhomomorphisms. Bibliography: 40 titles.