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Kaplansky Density Theorem
In the theory of von Neumann algebras, the Kaplansky density theorem, due to Irving Kaplansky, is a fundamental approximation theorem. The importance and ubiquity of this technical tool led Gert Pedersen to comment in one of his books that, :''The density theorem is Kaplansky's great gift to mankind. It can be used every day, and twice on Sundays.'' Formal statement Let ''K''− denote the Strong operator topology, strong-operator closure of a set ''K'' in ''B(H)'', the set of bounded operators on the Hilbert space ''H'', and let (''K'')1 denote the intersection of ''K'' with the unit ball of ''B(H)''. :Kaplansky density theorem.Theorem 5.3.5; Richard Kadison, ''Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory'', American Mathematical Society. . If A is a self-adjoint algebra of operators in B(H), then each element a in the unit ball of the strong-operator closure of A is in the strong-operator closure of the unit ball of A. In other words, (A)_1^ = (A^)_1 ... [...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] |
Irving Kaplansky
Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andrews. http://www-history.mcs.st-andrews.ac.uk/Biographies/Kaplansky.html. Biography Kaplansky or "Kap" as his friends and colleagues called him was born in Toronto, Ontario, Canada, to Polish-Jewish immigrants; his father worked as a tailor, and his mother ran a grocery and, eventually, a chain of bakeries. He went to Harbord Collegiate Institute receiving the Prince of Wales Scholarship as a teenager. He attended the University of Toronto as an undergraduate and finished first in his class for three consecutive years. In his senior year, he competed in the first William Lowell Putnam Mathematical Competition, becoming one of the first five recipients of the Putnam Fellowship, which paid for graduate studies at Harvard University. Administe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gert Pedersen
Gert is a mainly masculine given name ( short form of Gerrit, Gerard, etc.) with some female bearers (short for Gertrude). Since 1993 no one in Sweden has been baptised as Gert according to the Swedish Bureau of Census, so the name is becoming increasingly rare. In 2010 around 12,000 in Sweden had the name as their first name according to the same source. Gert is most common in Sweden among males over 50 years of age. Around 400 females in Sweden have Gert as their first name according to the Swedish Bureau of Census. It may refer to: Men *Gert Aandewiel (born 1969), Dutch football player and manager *Gert Alberts (1836–1927), South African Voortrekker *Gert Andersen (born 1939), Danish handball player *Gert Bals (1936–2016), Dutch footballer *Gert Bastian (1923–1992), German military officer and politician *Gert Bender (born 1948), German motorcycle racer *Gerrit Gert van den Berg (cyclist) (1903-?), Dutch cyclist *Gerrit Gert van den Berg (politician) (born 1935), Dutc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Operator Topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ''x'' varies in ''H''. Equivalently, it is the coarsest topology such that, for each fixed ''x'' in ''H'', the evaluation map T\mapsto Tx (taking values in ''H'') is continuous in T. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets U(T_0,x,\epsilon) = \ (where ''T0'' is any bounded operator on ''H'', ''x'' is any vector and ε is any positive real number). In concrete terms, this means that T_i\to T in the strong operator topology if and only if \, T_ix-Tx\, \to 0 for each ''x'' in ''H''. The SOT is stronger than the weak operator topology and weaker than the norm topology. The SOT lacks some of the nicer properties that the weak operator topology has, but being ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Kadison
Richard Vincent Kadison (July 25, 1925 – August 22, 2018)Foreign Members list. . Accessed January 12, 2010 was an American known for his contributions to the study of s. Work Born in New York City in 1925, Kadison wa ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Operator Topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ''x'' varies in ''H''. Equivalently, it is the coarsest topology such that, for each fixed ''x'' in ''H'', the evaluation map T\mapsto Tx (taking values in ''H'') is continuous in T. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets U(T_0,x,\epsilon) = \ (where ''T0'' is any bounded operator on ''H'', ''x'' is any vector and ε is any positive real number). In concrete terms, this means that T_i\to T in the strong operator topology if and only if \, T_ix-Tx\, \to 0 for each ''x'' in ''H''. The SOT is stronger than the weak operator topology and weaker than the norm topology. The SOT lacks some of the nicer properties that the weak operator topology has, but being ... [...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] |
Self-adjoint Operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. If ''V'' is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of ''A'' is a Hermitian matrix, i.e., equal to its conjugate transpose ''A''. By the finite-dimensional spectral theorem, ''V'' has an orthonormal basis such that the matrix of ''A'' relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Functional Calculus
In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra. Theorem Theorem. Let ''x'' be a normal element of a C*-algebra ''A'' with an identity element e. Let ''C'' be the C*-algebra of the bounded continuous functions on the spectrum σ(''x'') of ''x''. Then there exists a unique mapping π : C → A, where ''π(f)'' is denoted ''f(x)'', such that π is a unit-preserving morphism of C*-algebras and π(1) = e and π(id) = ''x'', where id denotes the function ''z'' → ''z'' on σ(''x''). In particular, this theorem implies that bounded normal operators on a Hilbert space have a continuous functional calculus. Its proof is almost immediate from the Gelfand representation: it suffices to assume ''A'' is the C*-algebra of continuous functions on some compact space ''X'' and define : \pi(f) = f \circ x. Uniqueness follows ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobson Density Theorem
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring . The theorem can be applied to show that any primitive ring can be viewed as a "dense" subring of the ring of linear transformations of a vector space.Isaacs, Corollary 13.16, p. 187 This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson. This can be viewed as a kind of generalization of the Artin-Wedderburn theorem's conclusion about the structure of simple Artinian rings. Motivation and formal statement Let be a ring and let be a simple right -module. If is a non-zero element of , (where is the cyclic submodule of generated by ). Therefore, if are non-zero elements of , there is an element of that induces an endomorphism of transforming to . The natural question now is whether this can be ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kadison, Richard
Richard Vincent Kadison (July 25, 1925 – August 22, 2018)Foreign Members list. . Accessed January 12, 2010 was an American known for his contributions to the study of s. Work Born in New York City in 1925, Kadi ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |