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C*-algebras --- Tensor products --- Tensor algebra --- Semigroups

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This is the first book to be dedicated entirely to Drinfeld's quasi-Hopf algebras. Ideal for graduate students and researchers in mathematics and mathematical physics, this treatment is largely self-contained, taking the reader from the basics, with complete proofs, to much more advanced topics, with almost complete proofs. Many of the proofs are based on general categorical results; the same approach can then be used in the study of other Hopf-type algebras, for example Turaev or Zunino Hopf algebras, Hom-Hopf algebras, Hopfish algebras, and in general any algebra for which the category of representations is monoidal. Newcomers to the subject will appreciate the detailed introduction to (braided) monoidal categories, (co)algebras and the other tools they will need in this area. More advanced readers will benefit from having recent research gathered in one place, with open questions to inspire their own research.

Hopf algebras. --- Tensor products. --- Tensor algebra.

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The Cuntz semigroup of a C^*-algebra is an important invariant in the structure and classification theory of C^*-algebras. It captures more information than K-theory but is often more delicate to handle. The authors systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C^*-algebra A, its (concrete) Cuntz semigroup mathrm{Cu}(A) is an object in the category mathrm{Cu} of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, the authors call the latter mathrm{Cu}-semigroups. The authors establish the existence of tensor products in the category mathrm{Cu} and study the basic properties of this construction. They show that mathrm{Cu} is a symmetric, monoidal category and relate mathrm{Cu}(Aotimes B) with mathrm{Cu}(A)otimes_{mathrm{Cu}}mathrm{Cu}(B) for certain classes of C^*-algebras. As a main tool for their approach the authors introduce the category mathrm{W} of pre-completed Cuntz semigroups. They show that mathrm{Cu} is a full, reflective subcategory of mathrm{W}. One can then easily deduce properties of mathrm{Cu} from respective properties of mathrm{W}, for example the existence of tensor products and inductive limits. The advantage is that constructions in mathrm{W} are much easier since the objects are purely algebraic.

C*-algebras. --- Tensor products. --- Tensor algebra. --- Semigroups.

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Renormalization group theory of tensor network states provides a powerful tool for studying quantum many-body problems and a new paradigm for understanding entangled structures of complex systems. In recent decades the theory has rapidly evolved into a universal framework and language employed by researchers in fields ranging from condensed matter theory to machine learning. This book presents a pedagogical and comprehensive introduction to this field for the first time. After an introductory survey on the major advances in tensor network algorithms and their applications, it introduces step-by-step the tensor network representations of quantum states and the tensor-network renormalization group methods developed over the past three decades. Basic statistical and condensed matter physics models are used to demonstrate how the tensor network renormalization works. An accessible primer for scientists and engineers, this book would also be ideal as a reference text for a graduate course in this area.

Renormalization group. --- Tensor products. --- Tensor algebra. --- Density matrices.

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Mathematical analysis --- 51 --- Mathematics --- Approximation theory. --- Banach spaces. --- Tensor products. --- 51 Mathematics