arXiv subjects

Bojan Magajna

Publications and source records attributed to Bojan Magajna.

15 recordsLinked to original sources

arXiv2022

Which states can be reached from a given state by unital completely positive maps?

For a state $ω$ on a C$^*$-algebra $A$ we characterize all states $ρ$ in the weak* closure of the set of all states of the form $ω\circφ$, where $φ$ is a map on $A$ of the form $φ(x)=\sum_{i=1}^na_i^*xa_i,$ $\sum_{i=1}^na_i^*a_i=1$ ($a_i\in A$, $n=1,2,...$). These are precisely the states $ρ$ that satisfy $\|ρ|J\|\leq\|ω|J\|$ for each ideal $J$ of $A$. The corresponding question for normal states on a von Neumann algebra $R$ (with the weak* closure replaced by the norm closure) is also considered. All normal states of the form $ω\circψ$, where $ψ$ is a quantum channel on $R$ (that is, a map of the form $ψ(x)=\sum_ja_j^*xa_j$, where $a_j\in R$ are such that the sum $\sum_ja_j^*a_j$ converge to $1$ in the weak operator topology) are characterized. A variant of this topic for hermitian functionals instead of states is investigated. Maximally mixed states are shown to vanish on the strong radical of a C$^*$-algebra and for properly infinite von Neumann algebras the converse also holds.

math.OA↗

arXiv2022

Orthonormal pairs of operators

We consider pairs of operators $A,B\in B(H)$, where $H$ is a Hilbert space, such that there exist a linear isometry $f$ from the span of $\{A,B\}$ into $\mathbb{C}^2$ mapping $A,B$ into orthonormal vectors. We prove some necessary conditions for the existence of such an $f$ and determine all such pairs among commuting normal operators. Then we characterize all such pairs $A,B$ (in fact, we consider general sets instead of just pairs) under the additional requirement that $f$ is a complete isometry, when $H$ carries the column (or the row) operator space structure. We also metrically characterize elements in a C$^*$-algebra with orthogonal ranges.

math.FA↗

arXiv2015

Variance of operators and derivations

The variance of a bounded linear operator $a$ on a Hilbert space $H$ at a unit vector $h$ is defined by $D_h(a)=\|ah\|^2-| |^2$. We show that two operators $a$ and $b$ have the same variance at all vectors $h\in H$ if and only if there exist scalars $σ,λ$ with $|σ|=1$ such that $b=σa+\lambda1$ or $a$ is normal and $b=σa^*+\lambda1$. Further, if $a$ is normal, then the inequality $D_h(b)\leqκD_h(a)$ holds for some constant $κ$ and all unit vectors $h$ if and only if $b=f(a)$ for a Lipschitz function $f$ on the spectrum of $a$. Variants of these results for C$^*$-algebras are also proved. We also study the related, but more restrictive inequalities $\|bx-xb\|\leq \|ax-xa\|$ supposed to hold for all $x\in B(H)$ or for all $x\in B(H^n)$ and all positive integers $n$. We consider the connection between such inequalities and the range inclusion $d_b(B(H))\subseteq d_a(B(H))$, where $d_a$ and $d_b$ are the derivations on $B(H)$ induced by $a$ and $b$. If $a$ is subnormal, we study these conditions in particular in the case when $b$ is of the form $b=f(a)$ for a function $f$.

math.FA↗

arXiv2014

Bicommutants and Arens regularity

Let $C$ and $R$ be unital rings and $Z$ an injective cogenerator for right $C$-modules. For an $R,C$-bimodule $U$ let $U^*=Hom_C(U,Z)$, $S=End_R(U)$ and $Biend_R(U)=End_S(U)$, the biendomorphism ring of $U$. Under suitable requirements on $U$ we show that $B:=Biend_R(U)$ can be identified with a subring of $\tilde{B}:=Biend_R(U^*)$,study conditions for the reverse inclusion and density of $B$ in $\tilde{B}$. In the case $C$ is contained in the center of $R$ we describe $Biend_R(R^*)$ in terms of the Arens products on $R^{**}$ and study Arens regularity of $R$ in the context of duality of modules. We characterize Arens regular algebras over fields.

math.RA↗

arXiv2013

Bicommutants and ranges of derivations

Let $V$ be a vector space over a field $F$, $V^*$ its dual space and $L(V)$ the algebra of all linear operators on $V$. For an operator $a\in L(V)$ let $a*$ be its adjoint acting on $V*$, and for a subset $R$ of $L(V)$ let $R"$ be its bicommutant. If $R$ is the subalgebra of $L(V)$ generated by an operator $a$, we prove that the set $Z:={b*: b\in R}"$ is contained in ${b*: b\in R"}$; moreover $Z$ is described. This inclusion is equality if $V$ as a module over the polynomial algebra $R=F[t]$ via $t\mapsto a$ is nice enough (say torsion, or injective, or if it contains a copy of $R$ as a direct summand). Further, under the same assumption about $V$ for any $b\in L(V)$, $b\in(a)"$ if and only if the derivations $d_a$ and $d_b$ satisfy $d_b(F(V))\subseteq d_a(F(V))$, where $F(V)$ is the set of all finite rank operators on $V$. The inclusion $d_b(L(V))\subseteq d_a(L(V))$ also holds under these conditions.

math.RA↗

arXiv2011

Sums of products of positive operators and spectra of L\" uders operators

Each bounded operator T on an infinite dimensional Hilbert space H is a sum of three operators that are similar to positive operators; two such operators are sufficient if T is not a compact perturbation of a scalar. The spectra of Lüders operators (elementary operators on B(H) with positive coefficients) of lengths at least three are not necessarily contained in the set of all nonnegative real numbers. On the other hand, the spectra of such operators of lengths at most two contain only nonnegative real numbers, if the coefficients on one side commute.

math.FA↗

arXiv2011

Fixed points of normal completely positive maps on B(H)

Given a sequence of bounded operators $a_j$ on a Hilbert space $H$ with $\sum a_j^*a_j=1=\sum a_ja_j^*$, we study the map $Ψ$ defined on $B(H)$ by $Ψ(x)=\sum a_j^*xa_j$ and its restriction $Φ$ to the Hilbert-Schmidt class $C^2(H)$. In the case when the sum $\sum a_j^*a_j$ is norm-convergent we show in particular that the operator $Φ-1$ is not invertible if and only if the C$^*$-algebra $A$ generated by $(a_j)$ has an amenable trace. This is used to show that $Ψ$ may have fixed points in $B(H)$ which are not in the commutant $A'$ of $A$ even in the case when the weak* closure of $A$ is injective. However, if $A$ is abelian, then all fixed points of $Ψ$ are in $A'$ even if the operators $a_j$ are not positive.

math.OA↗

arXiv2009

Pointwise approximation by elementary complete contractions

A complete contraction on a C*-algebra A, which preserves all closed two sided ideals J, can be approximated pointwise by elementary complete contractions if and only if the induced map on the tensor product of B with A/J is contractive for every C*-algebra B, ideal J in A and C*-tensor norm on the tensor product. A lifting obstruction for such an approximation is also obtained.

math.OA↗

arXiv2009

Uniform approximation by elementary operators

On a separable C*-algebra A every (completely) bounded map, which preserves closed two sided ideals, can be approximated uniformly by elementary operators if and only if A is a finite direct sum of C*-algebras of continuous sections vanishing at infinity of locally trivial C*-bundles of finite type.

math.OA↗

arXiv2008

Weak* continuous states on Banach algebras

We prove that if a unital Banach algebra $A$ is the dual of a Banach space $\pd{A}$, then the set of weak* continuous states is weak* dense in the set of all states on $A$. Further, weak* continuous states linearly span $\pd{A}$.

math.FA↗

arXiv2008

Dual operator systems

We characterize weak* closed unital vector spaces of operators on a Hilbert space $H$. More precisely, we first show that an operator system, which is the dual of an operator space, can be represented completely isometrically and weak* homeomorphically as a weak* closed operator subsystem of $B(H)$. An analogous result is proved for unital operator spaces. Finally, we give some somewhat surprising examples of dual unital operator spaces.

math.OA↗

arXiv2005

Injective cogenerators among operator bimodules

Given C$^*$-algebras $A$ and $B$ acting cyclically on Hilbert spaces $\h$ and $\k$, respectively, we characterize completely isometric $A,B$-bimodule maps from $\bkh$ into operator $A,B$-bimodules. We determine cogenerators in some classes of operator bimodules. For an injective cogenerator $X$ in a suitable category of operator $A,B$-bimodules we show: if $A$, regarded as a C$^*$-subalgebra of $\al(X)$ (adjointable left multipliers on $X$), is equal to its relative double commutant in $\al(X)$, then $A$ must be a W$^*$-algebra.

math.OA↗

arXiv2004

On tensor products of operator modules

The injective tensor product of normal representable bimodules over von Neumann algebras is shown to be normal. The usual Banach module projective tensor product of central representable bimodules over an Abelian C$^*$-algebra is shown to be representable. A normal version of the projective tensor product is introduced for central normal bimodules.

math.OA↗

arXiv2004

Duality and Operator Algebras II: Operator Algebras as Banach Algebras

We answer, by counterexample, several open questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator spaces, as opposed to that of Banach spaces and Banach algebras. In particular, the `nonselfadjoint analogue' of a W*-algebra resides naturally in the category of dual operator spaces, as opposed to dual Banach spaces. We also show that an automatic w*-continuity result in the preceding paper of the authors is sharp.

math.OA↗

arXiv2004

Duality and operator algebras

We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras. In particular, we give several applications of operator space theory, based on the surprising fact that certain maps are always $w^*$-continuous on dual operator spaces. For example, this yields a new characterization of the $σ$-weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and it makes possible a generalization of the theory of $W^*$-modules to the framework of modules over such algebras. We also give a Banach module characterization of $σ$-weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.

math.OA↗