Cuntz Algebra
In mathematics, the Cuntz algebra \mathcal_n , named after Joachim Cuntz, is the universal C*-algebra generated by n isometries of an infinite-dimensional Hilbert space \mathcal satisfying certain relations. These algebras were introduced as the first concrete examples of a separable infinite simple C*-algebra, meaning as a Hilbert space, \mathcal_n is isometric to the sequence space :l^2(\mathbb) and it has no nontrivial closed ideals. These algebras are fundamental to the study of simple infinite C*-algebras since any such algebra contains, for any given ''n'', a subalgebra that has \mathcal_n as quotient. Definitions Let ''n'' ≥ 2 and \mathcal be a separable Hilbert space. Consider the C*-algebra \mathcal generated by a set ::\_^n of isometries (i.e. S_i^*S_i = 1) acting on \mathcal satisfying ::\sum_^n S_i S_i^* = 1. This universal C*-algebra is called the Cuntz algebra, denoted by \mathcal_n . A simple C*-algebra is said to be purely infinite if every hereditary ...
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ...
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Approximately Finite-dimensional C*-algebra
In mathematics, an approximately finite-dimensional (AF) C*-algebra is a C*-algebra that is the inductive limit of a sequence of finite-dimensional C*-algebras. Approximate finite-dimensionality was first defined and described combinatorially by Ola Bratteli. Later, George A. Elliott gave a complete classification of AF algebras using the ''K''0 functor whose range consists of ordered abelian groups with sufficiently nice order structure. The classification theorem for AF-algebras serves as a prototype for classification results for larger classes of separable simple amenable stably finite C*-algebras. Its proof divides into two parts. The invariant here is ''K''0 with its natural order structure; this is a functor. First, one proves ''existence'': a homomorphism between invariants must lift to a *-homomorphism of algebras. Second, one shows ''uniqueness'': the lift must be unique up to approximate unitary equivalence. Classification then follows from what is known as ''the inter ...
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Signal Processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, digital storage efficiency, correcting distorted signals, subjective video quality and to also detect or pinpoint components of interest in a measured signal. History According to Alan V. Oppenheim Alan Victor Oppenheim''Alan Victor Oppenheim'' was elected in 1987

K-graph C*-algebra
In mathematics, for k \in \mathbb, a k-graph (also known as a higher-rank graph or graph of rank k) is a countable category \Lambda together with a functor d : \Lambda \to \mathbb^k, called the degree map, which satisfy the following factorization property: if \lambda \in \Lambda and m,n \in \mathbb^k are such that d(\lambda) = m + n , then there exist ''unique'' \mu,\nu \in \Lambda such that d( \mu ) = m , d( \nu ) = n, and \lambda = \mu\nu . An immediate consequence of the factorization property is that morphisms in a k-graph can be factored in multiple ways: there are also unique \mu',\nu' \in \Lambda such that d( \mu' ) = m , d( \nu' ) = n, and \mu \nu = \lambda = \nu' \mu' . A 1-graph is just the path category of a directed graph. In this case the degree map takes a path to its length. By extension, k-graphs can be considered higher-dimensional analogs of directed graphs. Another way to think about a k-graph is as a k-colored directed graph together with addit ...
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Graph C*-algebra
In mathematics, a graph C*-algebra is a universal C*-algebra constructed from a directed graph. Graph C*-algebras are direct generalizations of the Cuntz algebras and Cuntz-Krieger algebras, but the class of graph C*-algebras has been shown to also include several other widely studied classes of C*-algebras. As a result, graph C*-algebras provide a common framework for investigating many well-known classes of C*-algebras that were previously studied independently. Among other benefits, this provides a context in which one can formulate theorems that apply simultaneously to all of these subclasses and contain specific results for each subclass as special cases. Although graph C*-algebras include numerous examples, they provide a class of C*-algebras that are surprisingly amenable to study and much more manageable than general C*-algebras. The graph not only determines the associated C*-algebra by specifying relations for generators, it also provides a useful tool for describing an ...
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Preadditive Category
In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom-set Hom(''A'',''B'') in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas: f\circ (g + h) = (f\circ g) + (f\circ h) and (f + g)\circ h = (f\circ h) + (g\circ h), where + is the group operation. Some authors have used the term ''additive category'' for preadditive categories, but here we follow the current trend of reserving this term for certain special preadditive categories (see below). Examples The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. Note that commutativity is crucial here; it ensures that the sum of two gro ...
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Biproduct
In category theory and its applications to mathematics, a biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects. The biproduct is a generalization of finite direct sums of modules. Definition Let C be a category with zero morphisms. Given a finite (possibly empty) collection of objects ''A''1, ..., ''A''''n'' in C, their ''biproduct'' is an object A_1 \oplus \dots \oplus A_n in C together with morphisms *p_k \!: A_1 \oplus \dots \oplus A_n \to A_k in C (the '' projection morphisms'') *i_k \!: A_k \to A_1 \oplus \dots \oplus A_n (the ''embedding morphisms'') satisfying *p_k \circ i_k = 1_, the identity morphism of A_k, and *p_l \circ i_k = 0, the zero morphism A_k \to A_l, for k \neq l, and such that *\left( A_1 \oplus \dots \oplus A_n, p_k \right) is a product for the A_k, and *\left( A_1 \oplus \dots \op ...
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Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group is a group homomorphism . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set ''S'' to itself. In any category, the composition of any two endomorphisms of is again an endomorphism of . It follows that the set of all endomorphisms of forms a monoid, the full transformation monoid, and denoted (or to emphasize the category ). Automorphisms An invertible endomorphism of is called an automorphism. The set of all automorphisms is a subset of with a group structure, called the automorphism group of and denoted . In the following diagram, the arrows denote implication: Endomorphism rings Any two endomorphisms of an abelian group, , can be ad ...
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Crossed Product
In mathematics, and more specifically in the theory of von Neumann algebras, a crossed product is a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a group. It is related to the semidirect product construction for groups. (Roughly speaking, ''crossed product'' is the expected structure for a group ring of a semidirect product group. Therefore crossed products have a ring theory aspect also. This article concentrates on an important case, where they appear in functional analysis.) Motivation Recall that if we have two finite groups G and ''N'' with an action of ''G'' on ''N'' we can form the semidirect product N \rtimes G. This contains ''N'' as a normal subgroup, and the action of ''G'' on ''N'' is given by conjugation in the semidirect product. We can replace ''N'' by its complex group algebra ''C'' 'N'' and again form a product C \rtimes G in a similar way; this algebra is a sum of subspaces ''gC'' 'N''as ''g'' runs through t ...
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Direct Limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms ( group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects. The direct limit of the objects A_i, where i ranges over some directed set I, is denoted by \varinjlim A_i . (This is a slight abuse of notation as it suppresses the system of homomorphisms that is crucial for the structure of the limit.) Direct limits are a special case of the concept of colimit in category theory. Direct limits are dual to inverse limits, which are also a special case of limits in category theory. Formal definition We will first give the definition for algebraic structures like groups and modules, and then the general defini ...
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Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, an ...
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