Relative double commutants in coronas of separable C*-algebras

We prove a double commutant theorem for separable subalgebras of a wide class of corona C*-algebras, largely resolving a problem posed by Pedersen. Double commutant theorems originated with von Neumann, whose seminal result evolved into an entire field now called von Neumann algebra theory. Voiculescu later proved a C*-algebraic double commutant theorem for subalgebras of the Calkin algebra. We prove a similar result for subalgebras of a much more general class of so-called corona C*-algebras

Locally quasi-nilpotent elementary operators

Let $A$ be a unital dense algebra of linear mappings on a complex vector space $X$. Let $\phi=\sum_{i=1}^n M_{a_i,b_i}$ be a locally quasi-nilpotent elementary operator of length $n$ on $A$. We show that, if $\{a_1,\ldots,a_n\}$ is locally linearly independent, then the local dimension of V(\phi)=\spa\{b_ia_j: 1 \leq i,j \leq n\} is at most $\frac{n(n-1)}{2}$. If \lDim V(\phi)=\frac{n(n-1)}{2} , then there exists a representation of $\phi$ as $\phi=\sum_{i=1}^n M_{u_i,v_i}$ with $v_iu_j=0$ for $i\geq j$. Moreover, we give a complete characterization of locally quasi-nilpotent elementary operators of length 3.Comment: 15

Spectral isometries on non-simple C*-algebras

We prove that unital surjective spectral isometries on certain non-simple unital C*-algebras are Jordan isomorphisms. Along the way, we establish several general facts in the setting of semisimple Banach algebras.Comment: 7 pages; paper available since July 201

A comparison between the methods of apportionment using power indices: the case of the U.S. presidential election

In this paper, we compare the five more famous methods of apportionment, the methods of Adams, Dean, Hill, Webster and Jefferson. The criteria used for this comparison is the minimization of a distance between a power vector and a population vector. The power is measured with the well-known Banzhaf power index. The populations are the ones of the different States of the U.S. We then compare the apportionment methods in terms of their ability to bring closer the power of the States to their relative population: this ensures that every citizen in the country gets the same power. The U.S. presidential election by Electors is studied through 22 censuses since 1790. Our analysis is largely based on the book written by Balinski and Young (2001). The empirical findings are linked with theoretical results.Banzhaf index, methods of apportionment, distances, balance population-power.

Configurations study for the Banzhaf and the Shapley-Shubik indices of power

How can we count and list all the Banzhaf or Shapley-Shubik index of power configurations for a given number of players? There is no formula in the literature that may give the cardinal of such a set, and moreover, even if this formula had existed, there is no formula which gives the configuration vectors. Even if we do not present such a formula, we present a methodology which enables to determine the set of configurations and its cardinality.

On the Chacteristic Numbers of Voting Games

This paper deals with the non-emptiness of the stability set for any proper voting game.We present an upper bound on the number of alternatives which guarantees the non emptiness of this solution concept. We show that this bound is greater than or equal to the one given by Le Breton and Salles [6] for quota games.voting game, core, stability set

Italian Senate apportionment: is the 2007 proposal fair?

Since the political collapse of the 90’s, and in particular since the bicameral commission experience of 1997, Italian governments have always tried to face the need for wide constitutional reform. Reductions in the number of deputies and senators have been planned on several occasions. The purpose of this paper is to analyze whether or not the proposed reforms to the apportionment of seats in the Italian senate is fair. We use the theory of power indices to compare different scenarios. We show that the intended reform produces an outcome that is worse than both the ideal situation and the actual situation.power index, Banzhaf, Italian Senate

On the performance of the Shapley Shubik and Banzhaf power indices for the allocations of mandates

A classical problem in the power index literature is to design a voting mechanism such as the distribution of power of the different players is equal (or closer) to a pre established target. This tradition is especially popular when considering two tiers voting mechanisms: each player votes in his own jurisdiction to designate a delegate for the upper tier; and the question is to assign a certain number of mandates for each delegate according the population of the jurisdiction he or she represents. Unfortunately, there exist several measures of power, which in turn imply different distributions of the mandates for the same pre established target. The purposes of this paper are twofold: first, we calculate the probability that the two most important power indices, the Banzhaf index and the Shapley-Shubik index, lead to the same voting rule when the target is the same. Secondly, we determine which index on average comes closer to the pre established target.Banzhaf, Shapley-Shubik, power indices