Positive Invertible Elements Research Articles

Ring Isomorphisms of $$\ast$$-Subalgebras of Murray–von Neumann Factors

The present paper is devoted to study of ring isomorphisms of $\ast$-subalgebras of Murray--von Neumann factors. Let $\cM,$ $\cN$ be von Neumann factors of type II$_1,$ and let $S(\cM),$ $S(\cN)$ be the $\ast$-algebras of all measurable operators affiliated with $\cM$ and $ \cN,$ respectively. Suppose that $\cA\subset S(\cM),$ $\cB\subset S(\cN)$ are their $\ast$-subalgebras such that $\cM\subset \cA,$ $\cN\subset \cB.$ We prove that for every ring isomorphism $\Phi: \cA \to \cB$ there exist a positive invertible element $a \in \cB$ with $a^{-1}\in \cB$ and a real $\ast$-isomorphism $\Psi: \cM \to \cN$ (which extends to a real $\ast$-isomorphism from $\cA$ onto $\cB$) such that $\Phi(x) = a\Psi(x)a^{-1}$ for all $x \in \cA.$ In particular, $\Phi$ is real-linear and continuous in the measure topology. In particular, noncommutative Arens algebras and noncommutative $\cL_{log}$-algebras associated with von Neumann factors of type II$_1$ satisfy the above conditions and the main Theorem implies the automatic continuity of their ring isomorphisms in the corresponding metrics. We also present an example of a $\ast$-subalgebra in $S(\cM),$ which shows that the condition $\cM\subset \cA$ is essential in the above mentioned result.

Isometries on Positive Definite Operators with Unit Fuglede-Kadison Determinant

In this paper we explore the structure of certain surjective generalized isometries (which are transformations that leave any given member of a large class of generalized distance measures invariant) of the set of positive invertible elements in a finite von Neumann factor with unit Fuglede-Kadison determinant. We conclude that any such map originates from either an algebra $^*$-isomorphism or an algebra $^*$-antiisomorphism of the underlying operator algebra.

The Poincaré half-space of a C$^*$-algebra

Let \mathcal{A} be a unital C ^* -algebra. Given a faithful representation \mathcal{A}\subset\mathcal B(\mathcal{L}) in a Hilbert space \mathcal{L} , the set G^+\subset\mathcal{A} of positive invertible elements can be thought of as the set of inner products in \mathcal{L} , related to \mathcal{A} , which are equivalent to the original inner product. The set G^+ has a rich geometry, it is a homogeneous space of the invertible group G of \mathcal{A} , with an invariant Finsler metric. In the present paper we study the tangent bundle TG^+ of G^+ , as a homogeneous Finsler space of a natural group of invertible matrices in M_2(\mathcal{A}) , identifying TG^+ with the Poincaré half-space \mathcal H of \mathcal{A} , \mathcal H=\{h\in\mathcal{A}: {\rm Im}(h)\ge 0, {\rm Im}(h) \hbox{ invertible}\}. We show that \mathcal H\simeq TG^+ has properties similar to those of a space of non-positive constant curvature.

Maps on Quantum States in -algebras Preserving von Neumann Entropy or Schatten -norm of Convex Combinations

AbstractVery recently, Karder and Petek completely described maps on density matrices (positive semidefinite matrices with unit trace) preserving certain entropy-like convex functionals of any convex combination. As a result, maps could be characterized that preserve von Neumann entropy or Schatten $p$-norm of any convex combination of quantum states (whose mathematical representatives are the density matrices). In this note we consider these latter two problems on the set of invertible density operators, in a much more general setting, on the set of positive invertible elements with unit trace in a $C^{\ast }$-algebra.

Maps between positive cones of operator algebras preserving a measure of the difference between arithmetic and geometric means

On the set of positive invertible elements in a finite von Neumann algebra carrying a faithful normalized trace $$\tau $$ the numerical quantity $$\begin{aligned} d_{\tau }(A,B)=\tau (A + B)/2 - \tau \left( A^{1/2}\left( A^{-1/2}BA^{-1/2}\right) ^{1/2}A^{1/2}\right) \end{aligned}$$can be viewed as a measure of the difference of the arithmetic and the geometric mean. In this paper, we study maps between the positive definite cones of operator algebras which respect the above distance measure. We obtain the interesting fact that any such map originates from a trace-preserving Jordan $${}^*$$-isomorphisms (either algebra $${}^*$$-isomorphism or algebra $${}^*$$-antiisomorphism in the more restrictive case of factors) between the underlying von Neumann algebras.

Norm-Additive Maps on the Positive Definite Cone of a $$\varvec{C^{*}}$$ C ∗ -Algebra

In this paper, we determine the structure of Schatten p-norm additive maps on the set of positive invertible elements of a $$C^{*}$$ -algebra carrying a faithful normalized trace. It turns out that any such transformation originates from a Jordan $$^*$$ -isomorphism of the underlying $$C^{*}$$ -algebra. In fact, our result can be viewed as a characterization of that sort of isomorphisms.

Piecewise ⁎-homomorphisms and Jordan maps on C⁎-algebras and factor von Neumann algebras

We investigate maps between C⁎-algebras that are well behaved with respect to mutually commuting elements. We contribute to the Mackey–Gleason problem by showing that any continuous bijection between self-adjoint parts of C⁎-algebras that preserves triple product (a,b)→aba, and is linear on commutative subspaces, is already linear. This allows us to describe such maps as direct differences of linear Jordan isomorphisms. We shall show that any weak⁎-continuous bijection between positive invertible elements of von Neumann factors (of dimension at least 9) that preserves products of commuting elements in both directions is of the form a→eψ(log⁡a)θ(ac), where θ is a linear Jordan ⁎-isomorphism, c nonzero real number and ψ is a hermitian continuous functional. In a similar way we describe the same type of bicontinuous maps between unitary groups of von Neumann factors. General form of the above mentioned maps on C⁎-algebras is also presented.

Barycentric maps for compactly supported measures

After outlining the way in which an intrinsic mean G={Gn} on a complete metric space gives rise to a contractive barycentric map on some class of Borel probability measures and some basic examples of this process, we show how the resulting barycentric map gives rise to a general theory of integration of measurable functions into the space. We apply this machinery to the cone of positive invertible elements of a C⁎-algebra equipped with the Thompson metric to derive barycentric maps and their basic properties arising from the power means. Finally we derive basic results for the Karcher barycenter including its approximation by the barycentric maps for power means and its satisfaction of the Karcher equation.

Maps on the positive definite cone of a C⁎-algebra preserving certain quasi-entropies

We describe the structure of those bijective maps on the cone of all positive invertible elements of a C⁎-algebra with a normalized faithful trace which preserve certain kinds of quasi-entropy. It is shown that essentially any such map is equal to a Jordan *-isomorphism of the underlying algebra multiplied by a central positive invertible element.

Examples and Applications of Generalized Gyrovector Spaces

In this paper we exhibit examples of generalized gyrovector spaces (GGVs) of the sets of positive invertible elements in unital C*-algebras. Then we employ a geometric inequality on the positive cones in these unital C*-algebras. By applying a Mazur–Ulam theorem for GGVs, we analyze Thompson-like isometries on GGVs of positive invertible elements.

On a characterization of commutativity for $$C^*$$ C ∗ -algebras via gyrogroup operations

We show that a unital \(C^*\)-algebra is commutative if and only if the gyrogroup of the set of positive invertible elements is in fact a group.

On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C<sup>*</sup>-Dynamical Systems

The analog of the Chern-Gauss-Bonnet theorem is studied for a $C^*$-dynamical system consisting of a $C^*$-algebra $A$ equipped with an ergodic action of a compact Lie group $G$. The structure of the Lie algebra $\mathfrak{g}$ of $G$ is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra $\mathcal{A} \subset A$ as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique $G$-invariant state on $A$, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on $\mathcal{A}$ and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case.

Similarity problems in noncommutative polydomains

In this paper we consider several problems of joint similarity to tuples of bounded linear operators in noncommutative polydomains and varieties associated with sets of noncommutative polynomials. We obtain analogues of classical results such as Rota's model theorem for operators with spectral radius less than one, Sz.-Nagy characterization of operators similar to isometries (or unitary operators), and the refinement obtained by Foiaş and by de Branges and Rovnyak for strongly stable contractions. We also provide analogues of these results in the context of joint similarity of commuting tuples of positive linear maps on the algebra of bounded linear operators on a separable Hilbert space. An important role in this paper is played by a class of noncommutative cones associated with positive linear maps, the Fourier type representation of their elements, and the constrained noncommutative Berezin transforms associated with these elements. It is shown that there is an intimate relation between the similarity problems and the existence of positive invertible elements in these noncommutative cones and the corresponding Berezin kernels.

On the standard K-loop structure of positive invertible elements in a [formula omitted]-algebra

We investigate the algebraic properties of the operation a∘b=aba on the set of all positive invertible elements of a C⁎-algebra A. We show that its commutativity, associativity and distributivity are each equivalent to the commutativity of A. We present abstract characterizations of the operation ∘ and a few related ones, too.

The Gauss–Bonnet theorem for noncommutative two tori with a general conformal structure

In this paper we give a proof of the Gauss–Bonnet theorem of Connes and Tretkoff for noncommutative two tori \mathbb{T}_{\theta}^2 equipped with an arbitrary translation invariant complex structure. More precisely, we show that for any complex number \tau in the upper half plane, representing the conformal class of a metric on \mathbb{T}_{\theta}^2 , and a Weyl factor given by a positive invertible element k \in C^{\infty}(\mathbb{T}_{\theta}^2) , the value at the origin, \zeta (0) , of the spectral zeta function of the Laplacian \triangle\mkern-.5mu ' attached to (\mathbb{T}_{\theta}^2, \tau, k) is independent of \tau and k .

A treatment of the Cauchy–Schwarz inequality in [formula omitted]-modules

We study the Cauchy–Schwarz and some related inequalities in a semi-inner product module over a C ⁎ -algebra A . The key idea is to consider a semi-inner product A -module as a semi-inner product A -module with respect to another semi-inner product. In this way, we improve some inequalities such as the Ostrowski inequality and an inequality related to the Gram matrix. The induced semi-inner products are also related to the notion of covariance and variance. Furthermore, we obtain a sequence of nested inequalities that emerges from the Cauchy–Schwarz inequality. As a consequence, we derive some interesting operator-theoretical corollaries. In particular, we show that the sequence arising from our construction, when applied to a positive invertible element of a C ⁎ -algebra, converges to its inverse.

Über Positive Resolventenwerte Positiver Operatoren

In this paper we study the positive resolvent values of positive operators respectively of positive elements in Banach lattice ordered algebras. In the matrix case these values are just the inverse M-matrices. One of the main results is the following: Let A be a Banach lattice ordered algebra. A positive invertible element x∈A is a resolvent value of a positive element y∈A if and only if the element x satisfies the negative principle: If a∈A, λ < 0 and xa≤λa then xa≤ 0.

Geometry of positive operators and uhlmann's approach to the geometric phase

In Uhlmann's description of the differential geometry of the space Ω of density operators, a relevant role is played by the parallel condition ω*ω = ω*ω, where ω is a lifting of a curve γ in Ω, i.e. ω( t)ω( t)* = γ( t) for all t. In this paper we get a principal bundle with a natural connection over the space G + of all positive invertible elements of a C*-algebra such that the parallel transport is ruled by Uhlmann's parallel equation.

Geometrical significance of the Löwner-Heinz inequality

It is proven that the Löwner-Heinz inequality ‖ A t B t ‖ ≤ ‖ A B ‖ t {\|A^{t}B^{t}\|\le \|AB\|^{t}} , valid for all positive invertible operators A , B {A, B} on the Hilbert space H {\mathcal H } and t ∈ [ 0 , 1 ] {t\in [0,1]} , has equivalent forms related to the Finsler structure of the space of positive invertible elements of L ( H ) {\mathcal L (\mathcal H )} or, more generally, of a unital C ∗ {C^{*}} -algebra. In particular, the Löwner-Heinz inequality is equivalent to some type of “nonpositive curvature" property of that space.

Exponential sets and their geometric motions

Let us call an “exponential set” in a C*-algebraA any set consisting of the exponentialse X of all the self-adjoint elementsX of a subspaceH ofA. For example, ifH = A the resulting exponential setG+ consists of all the positive invertible elements ofA, and all other exponential sets are contained in G+. An exponential setC ⊂ G+ inherits the geometric structure of the space G+ when the defining subspaceH has suitable properties. Here we investigate reasonable conditions onH that permit, for example, reduction of the canonical connection of G+ toC. As a consequence, in these cases the setC has a rich family of motions that are “rigid” for the geometry of G+. In particular we find thatC itself operates on C by the actionL g a = (g−1)*ag− of the groupG of all invertible elements ofA in G+, and that the subgroup generated byC is transitive. Similarly, in several cases the productscu withc e C andu unitary form a closed Lie subgroup ofG that acts onC, withC contained in it. This is the case forH, the space of elements of trace zero, when there is a trace.