Let X. Y be compact Hausdorff spaces, and E, F be Banach spaces. A linear map T¡GC (X¡AE)¡÷C (Y¡AF) is separating if ¡üTf(y)¡ü¡üTg(y)¡ü¡×0 whenever ¡üf(x)¡ü¡üg(x)¡ü¡×0, for every x belonging to X, y belonging to Y. Gau, Jeang and Wong prove that a biseparating linear bijection T is a weighted composition oprator Tf¡×hf¡³£p where h is a function from Y into the set of inveritable linear operators from E onto F and £p is a homeomorphism from Y onto X. In this thesis, we extend this result to the case that continuous functions are defined to a locally compact Hausdorff space, which is either £m-compact or first countable. Moreover, we give a short proof of a recent result of Mrcun. Finally, we give an alternative approach to an Araujo's result concerning biseparating maps of smooth functions appeared in Adv. Math.

There are various notions of positivity for matrices and linear matrix-valued maps that play important roles in quantum information theory. The cones of positive semidefinite matrices and completely positive linear maps, which represent quantum states and quantum channels respectively, are the most ubiquitous positive cones. There are also many natural cones that can been regarded as "more" or "less" positive than these standard examples. In particular, entanglement theory deals with the cones of separable operators and entanglement witnesses, which satisfy very strong and weak positivity properties respectively. Rather complementary to the various cones that arise in entanglement theory are norms. The trace norm (or operator norm, depending on context) for operators and the diamond norm (or completely bounded norm) for superoperators are the typical norms that are seen throughout quantum information theory. In this work our main goal is to develop a family of norms that play a role analogous to the cone of entanglement witnesses. We investigate the basic mathematical properties of these norms, including their relationships with other well-known norms, their isometry groups, and their dual norms. We also make the place of these norms in entanglement theory rigorous by showing that entanglement witnesses arise from minimal operator systems, and analogously our norms arise from minimal operator spaces. Finally, we connect the various cones and norms considered here to several seemingly unrelated problems from other areas. We characterize the problem of whether or not non-positive partial transpose bound entangled states exist in terms of one of our norms, and provide evidence in favour of their existence. We also characterize the minimum gate fidelity of a quantum channel, the maximum output purity and its completely bounded counterpart, and the geometric measure of entanglement in terms of these norms. / Natural Sciences and Engineering Research Council (Canada Graduate Scholarship), Brock Scholarship

This dissertation is devoted to groups generated by bounded automata and geometric objects related to these groups (limit spaces, Schreier graphs, etc.). It is shown that groups generated by bounded automata are contracting. We introduce the notion of a post-critical set of a finite automaton and prove that the limit space of a contracting self-similar group generated by a finite automaton is post-critically finite (finitely-ramified) if and only if the automaton is bounded. We show that the Schreier graphs on levels of automaton groups can be constructed by an iterative procedure of inflation of graphs. This was used to associate a piecewise linear map of the form fK(v) = minA∈KAv, where K is a finite set of nonnegative matrices, with every bounded automaton. We give an effective criterium for the existence of a strictly positive eigenvector of fK. The existence of nonnegative generalized eigenvectors of fK is proved and used to give an algorithmic way for finding the exponents λmax and λmin of the maximal and minimal growth of the components of f(n) K (v). We prove that the growth exponent of diameters of the Schreier graphs is equal to λmax and the orbital contracting coefficient of the group is equal to 1/λmin . We prove that the simple random walks on orbital Schreier graphs are recurrent. A number of examples are presented to illustrate the developed methods with special attention to iterated monodromy groups of quadratic polynomials. We present the first example of a group whose coefficients λmin and λmax have different values.

This dissertation is devoted to groups generated by bounded automata and geometric objects related to these groups (limit spaces, Schreier graphs, etc.). It is shown that groups generated by bounded automata are contracting. We introduce the notion of a post-critical set of a finite automaton and prove that the limit space of a contracting self-similar group generated by a finite automaton is post-critically finite (finitely-ramified) if and only if the automaton is bounded. We show that the Schreier graphs on levels of automaton groups can be constructed by an iterative procedure of inflation of graphs. This was used to associate a piecewise linear map of the form fK(v) = minA[set]KAv, where K is a finite set of nonnegative matrices, with every bounded automaton. We give an effective criterium for the existence of a strictly positive eigenvector of fK. The existence of nonnegative generalized eigenvectors of fK is proved and used to give an algorithmic way for finding the exponents λmax and λmin of the maximal and minimal growth of the components of fK(n)(v). We prove that the growth exponent of diameters of the Schreier graphs is equal to λmax and the orbital contracting coefficient of the group is equal to 1/λmin . We prove that the simple random walks on orbital Schreier graphs are recurrent. A number of examples are presented to illustrate the developed methods with special attention to iterated monodromy groups of quadratic polynomials. We present the first example of a group whose coefficients λmin and λmax have different values.