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Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets
The problem of construction of Barabanov norms for analysis of properties of
the joint (generalized) spectral radius of matrix sets has been discussed in a
number of publications. The method of Barabanov norms was the key instrument in
disproving the Lagarias-Wang Finiteness Conjecture. The related constructions
were essentially based on the study of the geometrical properties of the unit
balls of some specific Barabanov norms. In this context the situation when one
fails to find among current publications any detailed analysis of the
geometrical properties of the unit balls of Barabanov norms looks a bit
paradoxical. Partially this is explained by the fact that Barabanov norms are
defined nonconstructively, by an implicit procedure. So, even in simplest cases
it is very difficult to visualize the shape of their unit balls. The present
work may be treated as the first step to make up this deficiency. In the paper
two iteration procedure are considered that allow to build numerically
Barabanov norms for the irreducible matrix sets and simultaneously to compute
the joint spectral radius of these sets.Comment: 17 pages, 36 bibliography references, 3 figures; shortened version,
new LaTeX style, fixed typos, accepted in DCDS-
Operator system structures on the unital direct sum of C*-algebras
This work is motivated by Radulescu's result on the comparison of C*-tensor
norms on C*(F_n) x C*(F_n). For unital C*-algebras A and B, there are natural
inclusions of A and B into their unital free product, their maximal tensor
product and their minimal tensor product. These inclusions define three
operator system structures on the internal sum A+B, the first of which we
identify as the coproduct of A and B in the category of operator systems.
Partly using ideas from quantum entanglement theory, we prove various
interrelations between these three operator systems. As an application, the
present results yield a significant improvement over Radulescu's bound on
C*(F_n) x C*(F_n). At the same time, this tight comparison is so general that
it cannot be regarded as evidence for a positive answer to the QWEP conjecture.Comment: 17 pages, to appear in Rocky Mountain J. Mat
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