arXiv subjects

Tatiana Shulman

Publications and source records attributed to Tatiana Shulman.

31 records · Page 2Linked to original sources

arXiv2016

On superactivation of one-shot zero-error quantum capacity and the related property of quantum measurements

We begin with a detailed description of a low dimensional quantum channel ($d_A=4, d_E=3$) demonstrating the symmetric form of superactivation of one-shot zero-error quantum capacity. This means appearance of a noiseless (perfectly reversible) subchannel in the tensor square of a channel having no noiseless subchannels. Then we describe a quantum channel $Φ$ such that $\,\bar{Q}_0(Φ)=0$ and $\,\bar{Q}_0(Φ\otimesΦ)\geq\log n\,$ for any $\,n\leq+\infty$. We also show that the superactivation of one-shot zero-error quantum capacity of a channel can be reformulated in terms of quantum measurements theory as appearance of an indistinguishable subspace for tensor product of two observables having no indistinguishable subspaces.

quant-ph↗

arXiv2015

On subspaces of invariant vectors

Let $X_π$ be the subspace of fixed vectors for a uniformly bounded representation $π$ of a group $G$ on a Banach space $X$. We study the problem of the existence and uniqueness of a subspace $Y$ that complements $X_π$ in $X$. Similar questions for $G$-invariant complement to $X_π$ are considered. We prove that every non-amenable discrete group $G$ has a representation with non-complemented $X_π$ and find some conditions that provide an $G$-invariant complement. A special attention is given to representations on $C(K)$ that arise from an action of $G$ on a metric compact $K$.

math.GR↗

arXiv2014

On superactivation of zero-error capacities and reversibility of a quantum channel

We propose examples of low dimensional quantum channels demonstrating different forms of superactivation of one-shot zero-error capacities, in particular, the extreme superactivation (this complements the recent result of T.S.Cubitt and G.Smith). We also describe classes of quantum channels whose zero-error classical and quantum capacities cannot be superactivated. We consider implications of the superactivation of one-shot zero-error capacities to analysis of reversibility of a tensor-product channel with respect to families of pure states. Our approach based on the notions of complementary channel and of transitive subspace of operators makes it possible to study the superactivation effects for infinite-dimensional channels as well.

quant-ph↗

arXiv2012

Approximations of subhomogeneous algebras

Let $n$ be a natural number. Recall that a C*-algebra is said to be $n$-subhomogeneous if all its irreducible representations have dimension at most $n$. In this short note, we give various approximation properties characterising $n$-subhomogeneous C*-algebras.

math.OA↗

arXiv2011

Lifting algebraic contractions in C*-algebras

Let p be a polynomial in one variable. It is shown that the universal C*-algebra of the relation p(x)=0, \|x\| \le C is semiprojective, residually finite-dimensional and has trivial extension group.

math.OA↗

arXiv2011

Noncommutative Semialgebraic Sets in Nilpotent Variables

We solve the lifting problem in C^*-algebras for many sets of relations that include the relations x_j^{N_j} = 0 on each variable. The remaining relations must be of the form \| p(x_1,...,x_n) \| \leq C for C a positive constant and p a noncommutative *-polynomial that is in some sense homogeneous. For example, we prove liftability for the set of relations x^3=0, y^4=0, z^5=0, xx^*+yy^*+zz^* \leq 1. Thus we find more noncommutative semialgebraic sets that have the topology of noncommutative absolute retracts.

math.OA↗

arXiv2010

A Generalized Spectral Radius Formula and Olsen's Question

Let $A$ be a $C^*$-algebra and $I$ be a closed ideal in $A$. For $x\in A$, its image under the canonical surjection $A\to A/I$ is denoted by $\dot x$, and the spectral radius of $x$ is denoted by $r(x)$. We prove that $$\max\{r(x), \|\dot x\|\} = \inf \|(1+i)^{-1}x(1+i)\|$$ (where infimum is taken over all $i\in I$ such that $1+i$ is invertible), which generalizes spectral radius formula of Murphy and West \cite{MurphyWest} (Rota for $\mathcal{B(H)}$ \cite{Rota}). Moreover if $r(x)< \|\dot x\|$ then the infimum is attained. A similar result is proved for commuting family of elements of a $C^*$-algebra. Using this we give a partial answer to an open question of C. Olsen: if $p$ is a polynomial then for "almost every" operator $T\in B(H)$ there is a compact perturbation $T+K$ of $T$ such that $$\|p(T+K)\| = \|p(T)\|_e.$$ We show also that if operators $A,B$ commute, $A$ is similar to a contraction and $B$ is similar to a strict contraction then they are simultaneously similar to contractions.

math.OA↗

arXiv2010

Semiprojectivity of universal C*-algebras generated by algebraic elements

Let $p$ be a polynomial in one variable whose roots either all have multiplicity more than 1 or all have multiplicity exactly 1. It is shown that the universal $C^*$-algebra of a relation $p(x)=0$, $\|x\| \le 1$ is semiprojective. In the case of all roots multiple it is shown that the universal $C^*$-algebra is also residually finite-dimensional. Applications to polynomially compact operators are given.

math.FA↗

arXiv2010

Noncommutative Semialgebraic sets and Associated Lifting Problems

We solve a class of lifting problems involving approximate polynomial relations (soft polynomial relations). Various associated C*-algebras are therefore projective. The technical lemma we need is a new manifestation of Akemann and Pedersen's discovery of the norm adjusting power of quasi-central approximate units. A projective C*-algebra is the analog of an absolute retract. Thus we can say that various noncommutative semialgebraic sets turn out to be absolute retracts. In particular we show a noncommutative absolute retract results from the intersection of the approximate locus of a homogeneous polynomial with the noncommutative unit ball. By unit ball we are referring the C*-algebra of the universal row contraction. We show projectivity of alternative noncommutative unit balls. Sufficiently many C*-algebras are now known to be projective that we are able to show that the cone over any separable C*-algebra is the inductive limit of C*-algebras that are projective.

math.OA↗

arXiv2008

On algebras generated by inner derivations

We look for an effective description of the algebra D_{Lie}(X,B) of operators on a bimodule X over an algebra B, generated by inner derivations. It is shown that in some important examples D_{Lie}(X,B) consists of all elementary operators x\to \sum_i a_ixb_i satisfying the conditions $\sum_i a_ib_i = \sum_i b_ia_i = 0. The Banach algebraic versions of these results are also obtained and applied to the description of closed Lie ideals in some Banach algebras, and to the proof of a density theorem for Lie algebras of operators on Hilbert space.

math.OA↗

arXiv2008

The C*-algebras qA\otimes K and S^2A\otimes K are asymptotically equivalent

Let $A$ be a separable $C^*$-algebra. We prove that its stabilized second suspension $S^2A\otimes \mathcal K$ and the $C^*$-algebra $qA\otimes \mathcal K$ constructed by Cuntz in the framework of his picture of KK-theory are asymptotically equivalent. This means that there exist asymptotic morphisms from each to the other whose compositions are homotopic to the identity maps. This result yields an easy description of the natural transformation from KK-theory to E-theory. One more corollary is the following. T. Loring ([3]) proved that any asymptotic morphism from $\qC$ to any $C^*$-algebra $B$ is homotopic to a $\ast$-homomorphism. We prove that the same is true when $\C$ is replaced by any nuclear $C^*$-algebra $A$ and when $B$ is stable.

math.OA↗

arXiv2007

Lifting of nilpotent contractions

It is proved that that every nilpotent contraction in a quotient C*-algebra can be lifted to a nilpotent contraction. As a consequence we get that the universal C*-algebra generated by a nilpotent contraction is projective. This answers the question posed by T. Loring.

math.OA↗

arXiv2007

On universal C^*-algebras generated by n projections with scalar sum

We study the universal C^*-algebras generated by n projections $p_1, >..., p_n$ subject to the relation $p_1+... p_n = λ1$, $λ\in \mathbb R$. The questions of when these C^*-algebras are type I, nuclear or exact are considered. It is proved also that among these algebras there is continuum of mutually nonisomorphic ones.

math.OA↗