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Connes Embedding Problem
Connes' embedding problem, formulated by Alain Connes in the 1970s, is a major problem in von Neumann algebra theory. During that time, the problem was reformulated in several different areas of mathematics. Dan-Virgil Voiculescu, Dan Voiculescu developing his free entropy theory found that Connes’ embedding problem is related to the existence of microstates. Some results of von Neumann algebra theory can be obtained assuming positive solution to the problem. The problem is connected to some basic questions in quantum theory, which led to the realization that it also has important implications in computer science. The problem admits a number of equivalent formulations. Notably, it is equivalent to the following long standing problems: * Kirchberg's QWEP conjecture in C*-algebra theory * Tsirelson's bound#Tsirelson's problem, Tsirelson's problem in quantum information theory * The predual of any (separable) von Neumann algebra is finitely representable in the trace class. In Jan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vanderbilt University. He was awarded the Fields Medal in 1982. Career Source: Academic career timeline: (1966–1970) – Bachelor's degree from the École Normale Supérieure (now part of Paris Sciences et Lettres University). (1973) – doctorate from Pierre and Marie Curie University, Paris, France (1970–1974) – appointment at the French National Centre for Scientific Research, Paris (1975) – Queen's University at Kingston, Ontario, Canada (1976–1980) – the University of Paris VI (1979 – present) – the Institute of Advanced Scientific Studies, Bures-sur-Yvette, France (1981–1984) – the French National Centre for Scientific Research, Paris (1984–2017) – the , Paris (2003–2011) – Vanderbilt University, Na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows: *The ring L^\infty(\mathbb R) of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space L^2(\mathbb R) of square-integrable functions. *The algebra \mathcal B(\mathcal H) of all bounded operators on a Hilbert space \mathcal H is a von Neumann algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dan-Virgil Voiculescu
Dan-Virgil Voiculescu (born 14 June 1949) is a Romanian professor of mathematics at the University of California, Berkeley. He has worked in single operator theory, operator K-theory and von Neumann algebras. More recently, he developed free probability theory. Education and career Voiculescu studied at the University of Bucharest, receiving his PhD in 1977 under the direction of Ciprian Foias. He was an assistant at the University of Bucharest (1972–1973), a researcher at the Institute of Mathematics of the Romanian Academy (1973–1975), and a researcher at INCREST (1975–1986). He came to Berkeley in 1986 for the International Congress of Mathematicians, and stayed on as visiting professor. Voiculescu was appointed professor at Berkeley in 1987. Awards and honors He received the 2004 NAS Award in Mathematics from the National Academy of Sciences (NAS) for “the theory of free probability, in particular, using random matrices and a new concept of entropy to sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous linear operators on a complex Hilbert space with two additional properties: * ''A'' is a topologically closed set in the norm topology of operators. * ''A'' is closed under the operation of taking adjoints of operators. Another important class of non-Hilbert C*-algebras includes the algebra C_0(X) of complex-valued continuous functions on ''X'' that vanish at infinity, where ''X'' is a locally compact Hausdorff space. C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tsirelson's Bound
A Tsirelson bound is an upper limit to quantum mechanical correlations between distant events. Given that quantum mechanics violates Bell inequalities (i.e., it cannot be described by a local hidden-variable theory), a natural question to ask is how large can the violation be. The answer is precisely the Tsirelson bound for the particular Bell inequality in question. In general, this bound is lower than the bound that would be obtained if more general theories, only constrained by "no-signalling" (i.e., that they do not permit communication faster than light), were considered, and much research has been dedicated to the question of why this is the case. The Tsirelson bounds are named after Boris S. Tsirelson (or Cirel'son, in a different transliteration), the author of the article in which the first one was derived. Bound for the CHSH inequality The first Tsirelson bound was derived as an upper bound on the correlations measured in the CHSH inequality. It states that if we have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Complexity Theory
Quantum complexity theory is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational model based on quantum mechanics. It studies the hardness of computational problems in relation to these complexity classes, as well as the relationship between quantum complexity classes and classical (i.e., non-quantum) complexity classes. Two important quantum complexity classes are BQP and QMA. Background A complexity class is a collection of computational problems that can be solved by a computational model under certain resource constraints. For instance, the complexity class P (complexity), P is defined as the set of problems solvable by a Turing machine in polynomial time. Similarly, quantum complexity classes may be defined using quantum models of computation, such as the quantum computer, quantum circuit model or the equivalent quantum Turing machine. One of the main aims of quantum complexity theory is to find ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications Of The ACM
''Communications of the ACM'' is the monthly journal of the Association for Computing Machinery (ACM). It was established in 1958, with Saul Rosen as its first managing editor. It is sent to all ACM members. Articles are intended for readers with backgrounds in all areas of computer science and information systems. The focus is on the practical implications of advances in information technology and associated management issues; ACM also publishes a variety of more theoretical journals. The magazine straddles the boundary of a science magazine, trade magazine, and a scientific journal. While the content is subject to peer review, the articles published are often summaries of research that may also be published elsewhere. Material published must be accessible and relevant to a broad readership. From 1960 onward, ''CACM'' also published algorithms, expressed in ALGOL. The collection of algorithms later became known as the Collected Algorithms of the ACM. See also * ''Journal of the A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on P. If X is an arbitrary set, its power set \wp(X), ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on \wp(X) are usually called X.If X happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on \wp(X) or an (ultra)filter just on X is meant; both kinds of (ultra)filters are quite different. Some authors use "(ultra)filter" ''of'' a partial ordered set" vs. "''on'' an arbitrary set"; i.e. they write "(ultra)filter on X" to abbreviate "(ultra)filter of \wp(X)". An ultrafilter on a set X may be considered as a finitely additive measure on X. In this view, every subset of X is either considered "almost everything" (has measure 1) or "almos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperfinite Type II Factor
In mathematics, there are up to isomorphism exactly two separably acting hyperfinite type II factors; one infinite and one finite. Murray and von Neumann proved that up to isomorphism there is a unique von Neumann algebra that is a factor of type II1 and also hyperfinite; it is called the hyperfinite type II1 factor. There are an uncountable number of other factors of type II1. Connes proved that the infinite one is also unique. Constructions *The von Neumann group algebra of a discrete group with the infinite conjugacy class property is a factor of type II1, and if the group is amenable and countable the factor is hyperfinite. There are many groups with these properties, as any locally finite group is amenable. For example, the von Neumann group algebra of the infinite symmetric group of all permutations of a countable infinite set that fix all but a finite number of elements gives the hyperfinite type II1 factor. *The hyperfinite type II1 factor also arises from the group-m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uffe Haagerup
Uffe Valentin Haagerup (19 December 1949 – 5 July 2015) was a mathematician from Denmark. Biography Uffe Haagerup was born in Kolding, but grew up on the island of Funen, in the small town of Fåborg. The field of mathematics had his interest from early on, encouraged and inspired by his older brother. In fourth grade Uffe was doing trigonometric and logarithmic calculations. He graduated as a student from Svendborg Gymnasium in 1968, whereupon he relocated to Copenhagen and immediately began his studies of mathematics and physics at the University of Copenhagen, again inspired by his older brother who also studied the same subjects at the same university. Early university studies in Einstein's general theory of relativity and quantum mechanics, sparked a lasting interest in the mathematical field of operator algebra, in particular Von Neumann algebra and Tomita–Takesaki theory. In 1974 he received his Candidate's degree ( cand. scient.) from the University of Copenhagen a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microstate (statistical Mechanics)
In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations. In contrast, the macrostate of a system refers to its macroscopic properties, such as its temperature, pressure, volume and density. Treatments on statistical mechanics define a macrostate as follows: a particular set of values of energy, the number of particles, and the volume of an isolated thermodynamic system is said to specify a particular macrostate of it. In this description, microstates appear as different possible ways the system can achieve a particular macrostate. A macrostate is characterized by a probability distribution of possible states across a certain statistical ensemble of all microstates. This distribution describes the probability of finding the system in a certain microstate. In the thermodynamic limit, the microstates visited by a macroscopic system during ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
