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Register a new userOn $ast-$Reverse Derivable Maps
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Let $R$ be a ring with involution containing a nontrivial symmetric idempotent element $e$. Let $delta: Rrightarrow R$ be a mapping such that $delta(ab)=delta(b)a^{ast}+b^{ast}delta(a)$ for all $a,bin R$, we call $delta$ a $ast-$reverse derivable map on $R$. In this paper, our aim is to show that under some suitable restrictions imposed on $R$, every $ast-$reverse derivable map of $R$ is additive.
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We investigate the structures of Hopf $ast$-algebra on the Radford algebras over $mathbb {C}$. All the $*$-structures on $H$ are explicitly given. Moreover, these Hopf $*$-algebra structures are classified up to equivalence.
The generalized state space $ S_{mathcal{H}}(mathcal{mathcal{A}})$ of all unital completely positive (UCP) maps on a unital $C^*$-algebra $mathcal{A}$ taking values in the algebra $mathcal{B}(mathcal{H})$ of all bounded operators on a Hilbert space $mathcal{H}$, is a $C^ast$-convex set. In this paper, we establish a connection between $C^ast$-extreme points of $S_{mathcal{H}}(mathcal{A})$ and a factorization property of certain algebras associated to the UCP map. In particular, this factorization property of some nest algebras is used to give a complete characterization of those $C^ast$-extreme maps which are direct sums of pure UCP maps. This significantly extends a result of Farenick and Zhou [Proc. Amer. Math. Soc. 126 (1998)] from finite to infinite dimensional Hilbert spaces. Also it is shown that normal $C^ast$-extreme maps on type $I$ factors are direct sums of normal pure UCP maps if and only if an associated algebra is reflexive. Further, a Krein-Milman type theorem is established for $C^ast$-convexity of the set $ S_{mathcal{H}}(mathcal{A})$ equipped with bounded weak topology, whenever $mathcal{A}$ is a separable $C^ast$-algebra or it is a type $I$ factor. As an application, we provide a new proof of a classical factorization result on operator valued Hardy algebras.
Let $A$ be an algebra and let $f(x_1,...,x_d)$ be a multilinear polynomial in noncommuting indeterminates $x_i$. We consider the problem of describing linear maps $phi:Ato A$ that preserve zeros of $f$. Under certain technical restrictions we solve the problem for general polynomials $f$ in the case where $A=M_n(F)$. We also consider quite general algebras $A$, but only for specific polynomials $f$.
We study the problem of constructing a reverse nearest neighbor (RNN) heat map by finding the RNN set of every point in a two-dimensional space. Based on the RNN set of a point, we obtain a quantitative influence (i.e., heat) for the point. The heat map provides a global view on the influence distribution in the space, and hence supports exploratory analyses in many applications such as marketing and resource management. To construct such a heat map, we first reduce it to a problem called Region Coloring (RC), which divides the space into disjoint regions within which all the points have the same RNN set. We then propose a novel algorithm named CREST that efficiently solves the RC problem by labeling each region with the heat value of its containing points. In CREST, we propose innovative techniques to avoid processing expensive RNN queries and greatly reduce the number of region labeling operations. We perform detailed analyses on the complexity of CREST and lower bounds of the RC problem, and prove that CREST is asymptotically optimal in the worst case. Extensive experiments with both real and synthetic data sets demonstrate that CREST outperforms alternative algorithms by several orders of magnitude.
New upper bounds on the relative entropy are derived as a function of the total variation distance. One bound refines an inequality by Verd{u} for general probability measures. A second bound improves the tightness of an inequality by Csisz{a}r and Talata for arbitrary probability measures that are defined on a common finite set. The latter result is further extended, for probability measures on a finite set, leading to an upper bound on the R{e}nyi divergence of an arbitrary non-negative order (including $infty$) as a function of the total variation distance. Another lower bound by Verd{u} on the total variation distance, expressed in terms of the distribution of the relative information, is tightened and it is attained under some conditions. The effect of these improvements is exemplified.
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