Pinchings and positive linear maps

Given a family of completely positive maps, indexed by a group, from aC*-algebra into itself, we are concerned with its dilation to a group of *-automorphisms on a larger algebra. A Schwarz-type inequality forn-positive *-linear mappings from an involutive algebra into the bounded linear operators on a hilbert space is obtained. Strongly continuous one-parameter semigroups and groups onC*-algebras, which have certain positivity properties, are studied.

In this paper, we push forward the conjecture on a noncommutative version of the maximal ergodic theorem for positive contractions due to Ackoglu. That is, we establish the one‐sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative ‐spaces for a fixed , which particularly applies to positive isometries, the convex hull of positive Lamperti contractions, and also the power‐bounded doubly Lamperti operators; moreover, it is known that this subclass recovers all positive contractions on the classical Lebesgue spaces . As an unexpected consequence, we deduce a completely bounded version of Ackoglu's theorem by making use of Ackoglu's original dilation. We also observe that the concrete examples of positive contractions without Akcoglu's dilation, which were constructed by Junge–Le Merdy [J. Funct. Anal. 249 (2007), no. 1, 220–252.], still satisfy the maximal ergodic inequalities. Together with the class of power‐bounded positive invertible operators considered by Hong–Liao–Wang [Duke Math. J. (2020). In press], we thereby verify the noncommutative Ackoglu ergodic theorem for all the positive operators that have naturally appeared in the literature up to the moment of writing. Based on the noncommutative Calderón transference principle established in [Duke Math. J. (2020). In press], the general pattern of the demonstration follows from the classical one due to Kan, but several new ideas are absolutely necessary in the noncommutative setting. Let us just mention three of them: the argument for the maximal ergodic theorem for positive isometries is highly nontrivial compared to the classical case; the novel simultaneous dilation property is indispensable to get the ergodic theorem for the convex hull of positive Lamperti contractions; numerous adjustments are truly needed in this new setting to conclude a desired structural theorem for positive doubly Lamperti operators since the orthogonal relations of operator algebras are completely different from those in classical measure theory.

An effective method of investigation of positive maps on the set of positive definite operators

Let r be a normal semifinite trace on a von Neumann algebra, and let f be a continuous convex function on the interval [0, oo) with f (0) = 0. For a positive element a of the algebra and a positive contraction a on the algebra, the following inequality is obtained: r(f (c((a))) Zf((Ari,r,i)), where f is a convex function on the real line, A is a selfadjoint operator, and {rqi} is an orthonormal set (see [11], for example). Slightly more general, if {Pi} is a pairwise orthogonal family of projections and we set a(A) = >nPiAPi for a selfadjoint operator A, then tr a (f (A)) > tr f (a (A)) (see [2, 6, and 9]). Going further in this direction Brown and Kosaki [3] have recently proved that r(f (v*av)) < r(v* f (a)v), where r is a faithful normal semifinite trace on a von Neumann algebra, v is a contraction, and a is a positive element of the algebra. (In fact, a may be unbounded and affiliated with the algebra.) In the present paper we obtain a similar inequality for an arbitrary positive contraction a on the algebra and show that (*) r(f (a (a))) < r(a(f (a))). Here f is supposed to be a continuous convex function with f(O) = 0. (Note that (*) holds without r if f is operator convex; cf. Theorem IV.1 in [1].) The inequality (*) supports the experience that inequalities involving operators are more simply treatable provided both sides are inside of a trace (see [3 and 8] for further evidences). The idea of our proof is to approximate the operators a and a (a) by diagonal ones by means of their spectral resolution and to apply the classical Jensen's inequality in Received by the editors November 19, 1984 and, in revised form, June 24, 1985 and December 17, 1985. 1980 Mathematics Subject Cai Secondary 26A51, 47B15.

Let $\mathcal{M}$ be a semifinite von Neumann algebra on a Hilbert space $\mathcal{H}$ equipped with a faithful normal semifinite trace $\tau$, $S(\mathcal{M},\tau)$ be the ${}^*$-algebra of all $\tau$-measurable operators. Let $S_0(\mathcal{M},\tau)$ be the ${}^*$-algebra of all $\tau$-compact operators and $T(\mathcal{M},\tau)=S_0(\mathcal{M},\tau)+\mathbb{C}I$ be the ${}^*$-algebra of all operators $X=A+\lambda I$ with $A\in S_0(\mathcal{M},\tau)$ and $\lambda \in \mathbb{C}$. It is proved that every operator of $T(\mathcal{M},\tau)$ that is left-invertible in $T(\mathcal{M},\tau)$ is in fact invertible in $T(\mathcal{M},\tau)$. It is a generalization of Sterling Berberian theorem (1982) on the subalgebra of thin operators in $\mathcal{B} (\mathcal{H})$. For the singular value function $\mu(t; Q)$ of $Q=Q^2\in S(\mathcal{M},\tau)$, the inclusion $\mu(t; Q)\in \{0\}\bigcup [1, +\infty)$ holds for all $t&amp;gt;0$. It gives the positive answer to the question posed by Daniyar Mushtari in 2010.

Every C*-algebra \(\mathfrak{A}\) has a faithful *-representation π in a Hilbert space \(\mathcal{H}\). Consequently it is natural to pose the following question: under which conditions, the completion of a C*-algebra in a weaker than the given one topology, can be realized as a quasi *-algebra of operators? The present paper presents the possibility of extending the well known Gelfand — Naimark representation of C*-algebras to certain Banach C*-modules.

Shannon like inequalities for f-connections of positive linear maps and positive operators

This paper presents a novel semantics for a quantum programming language by\noperator algebras, which are known to give a formulation for quantum theory\nthat is alternative to the one by Hilbert spaces. We show that the opposite\ncategory of the category of W*-algebras and normal completely positive\nsubunital maps is an elementary quantum flow chart category in the sense of\nSelinger. As a consequence, it gives a denotational semantics for Selinger's\nfirst-order functional quantum programming language QPL. The use of operator\nalgebras allows us to accommodate infinite structures and to handle classical\nand quantum computations in a unified way.\n

This paper presents a novel semantics for a quantum programming language by operator algebras, which are known to give a formulation for quantum theory that is alternative to the one by Hilbert spaces. We show that the opposite of the category of W*-algebras and normal completely positive subunital maps is an elementary quantum flow chart category in the sense of Selinger. As a consequence, it gives a denotational semantics for Selinger’s first-order functional quantum programming language. The use of operator algebras allows us to accommodate infinite structures and to handle classical and quantum computations in a unified way.

Numerical ranges of products of two positive contractions

We propose a definition of purity for positive linear maps between Euclidean Jordan Algebras (EJA) that generalizes the notion of purity for quantum systems. We show that this definition of purity is closed under composition and taking adjoints and thus that the pure maps form a dagger category (which sets it apart from other possible definitions.) In fact, from the results presented in this paper, it follows that the category of EJAs with positive contractive linear maps is a dagger-effectus, a type of structure originally defined to study von Neumann algebras in an abstract categorical setting. In combination with previous work this characterizes EJAs as the most general systems allowed in a generalized probabilistic theory that is simultaneously a dagger-effectus. Using the dagger structure we get a notion of dagger-positive maps of the form f = g*g. We give a complete characterization of the pure dagger-positive maps and show that these correspond precisely to the Jordan algebraic version of the sequential product that maps (a,b) to sqrt(a) b sqrt(a). The notion of dagger-positivity therefore characterizes the sequential product.

A note on inequalities for positive linear maps

Let F be a closed sublattice of a Banach lattice G. We show that any positive linear mapping from F into L 1 ( μ ) {L^1}(\mu ) or C ( X ) C(X) for a Stonian space X has a positive norm preserving extension to G. A dual result for positive norm preserving liftings is also established. These results are applied to obtain extension and lifting theorems for order summable and majorizing linear mappings. We also obtain some partial results concerning positive extensions and liftings of compact linear mappings.

Operator algebras come in many flavours. For the purpose of this article, however, the term is only used for one of two kinds of self-adjoint algebras of operators on Hilbert space, viz., C*- algebras (which are norm-closed) or von Neumann algebras (which are closed in the topology of pointwise strong convergence or equivalently, in the weak-* topology it inherits as a result of being a Banach dual space). To be fair, there are a number of people in India (eg., Gadadhar Misra, Tirthankar Bhattacharyya, Jaydeb Sarkar, Santanu Dey, etc.,) who work on non-selfadjoint algebras, mostly from the point of view of connections with complex function theory; but in the interest of restricting the size of this paper, I confine myself here to selfadjoint algebras. I apologise for ways in which my own personal taste and limitations colour this depiction of operator algebras. Another instance of this arbitrary personal taste is a decision to concentrate on the work of younger people. Thus, the work of the more senior people who have worked in operator algebras is only seen via their collaborations with younger people: e.g., KRP via Srinivasan and Rajarama Bhat, Kalyan Sinha via Debashish, Partha, Arup, Raja, etc,. and me via Vijay, Srinivasan and Panchugopal. Not long ago, interest in operator algebras in India was restricted to the three centres of the Indian Statistical Institute. Now, I am happy to note that it has spread to IMSc, some IITs, IISERs, NISER, JNU,.... My role in this article has been merely that of compiling inputs from many active Indian operator algebraists that came to my mind. I wrote soliciting a response from a certain number of them, then put together the responses received. (I apologise to those people who were omitted in this process). My colleague, Partha, with the help of his collaborator Arup, agreed to take care of the C*-related inputs, while I take care of the von Neumann-related ones with the help of my collaborator Vijay. What follows are some areas of ongoing research done in von Neumann algebras in India and some names of people doing such work: (a) subfactors and planar algebras, (Vijay Kodiyalam of IMSc, Chennai); (b) quantum dynamical systems and complete positivity (Rajarama Bhat of ISI, Bengaluru); (c) E0 semigroups (R. Srinivasan of CMI, Chennai, and Panchugopal Bikram of NISER, Bhubhaneswar), and (d) Masas in II1 von Neumann algebras and free Araki-Woods factors (Kunal Mukherjee of IIT, Chennai, and Panchugopal Bikram of NISER, Bhubhaneswar).

Let A and B be unital JB-algebras. We study the extremal structure of the convex set S (A,B) of all identity preserving positive linear maps from A to B. We show that every unital Jordan homomorphism from A to B is an extreme point of S (A,B). An extreme point of S (A,B) need not be a homomorphism and we show that, given A, every extreme point of S (A,B) is a homomorphism for any B if, and only if, dim A ≤ 2. We also determine when S (A,B) is a simplex.