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.
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions, possibly in some generalized sense. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.
Algebraic quantum field theory (AQFT) is an application to local quantum physics of C*-algebra theory. Also referred to as the Haag–Kastler axiomatic framework for quantum field theory, because it was introduced by Rudolf Haag and Daniel Kastler (1964). The axioms are stated in terms of an algebra given for every open set in Minkowski space, and mappings between those.
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group, the finite abelian groups, and the additive group of the integers, the real numbers, and every finite-dimensional vector space over the reals or a p-adic field.
In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "mean".
In mathematics, a group is said to have the infinite conjugacy class property, or to be an ICC group, if the conjugacy class of every group element but the identity is infinite.
In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II1 subfactor. They also provide an appropriate algebraic framework for many knot invariants (in particular the Jones polynomial), and have been used in describing the properties of Khovanov homology with respect to tangle composition. Any subfactor planar algebra provides a family of unitary representations of Thompson groups. Any finite group (and quantum generalization) can be encoded as a planar algebra.
In mathematics and theoretical physics, a locally compact quantum group is a relatively new C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf-algebra approaches. Earlier attempts at a unifying definition of quantum groups using, for example, multiplicative unitaries have enjoyed some success but have also encountered several technical problems.
In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there are three such Jordan algebras up to isomorphism. One of them, which was first mentioned by Pascual Jordan, John von Neumann, and Eugene Wigner (1934) and studied by Albert (1934), is the set of 3×3 self-adjoint matrices over the octonions, equipped with the binary operation
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 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.
Iakov (Yan) Soibelman born 15 April 1956 is a Russian American mathematician, professor at Kansas State University, member of the Kyiv Mathematical Society (Ukraine), founder of Manhattan Mathematical Olympiad.
Adrian Ioana is a Romanian mathematician. He is currently a professor at the University of California, San Diego.
Sorin Teodor Popa is a Romanian American mathematician working on operator algebras. He is a professor at the University of California, Los Angeles.
Mihnea Popa is a Romanian-American mathematician at Harvard University, specializing in algebraic geometry. He is known for his work on complex birational geometry, Hodge theory, abelian varieties, and vector bundles.
Roberto Longo is an Italian mathematician, specializing in operator algebras and quantum field theory.
Edward George Effros was an American mathematician, specializing in operator algebras and representation theory. His research included "C*-algebras theory and operator algebras, descriptive set theory, Banach space theory, and quantum information."
Tsachik Gelander is an Israeli mathematician working in the fields of Lie groups, topological groups, symmetric spaces, lattices and discrete subgroups. He is a professor in Northwestern University.
Michel Enock-Levi is a French mathematician and a research director at CNRS, credited with the early development of Pontryagin-style dualities for non-commutative topological groups.
Leonid Iosifovich Vainerman is a Ukrainian and French mathematician, professor emeritus at University of Caen Normandy. Vainerman's research results are in functional analysis, ordinary differential equations, operator theory, topological groups, Lie groups, and abstract harmonic analysis. In the 1970s, he co-developed Pontryagin-style dualities for non-commutative topological groups, a set of results that served as a precursor for the modern theory of quantum groups.
