Model theory of operator algebras
... Hausdorff space X is an abelian (abstract) C∗ -algebra. The norm is the sup-norm. By a result of Gelfand and Naimark, these are all the unital abelian C∗ -algebras. ...
... Hausdorff space X is an abelian (abstract) C∗ -algebra. The norm is the sup-norm. By a result of Gelfand and Naimark, these are all the unital abelian C∗ -algebras. ...
SAM III General Topology
... written as π1 (X , x), is the group of automorphisms of x in the fundamental groupoid Π1 X of the space. Show that in a path-connected space, fundamental groups at any two distinct points are isomorphic to each other. Deduce this from a general observation about an abstract groupoid. ...
... written as π1 (X , x), is the group of automorphisms of x in the fundamental groupoid Π1 X of the space. Show that in a path-connected space, fundamental groups at any two distinct points are isomorphic to each other. Deduce this from a general observation about an abstract groupoid. ...
Subfactors and Modular Tensor Categories
... What looks like a group but is not a group? A naive but often fruitful idea in mathematics is to generalize mathematical objects by listing their essential properties as axioms, and then dropping one or more of them. By dropping the parallel postulate from Euclid’s geometry, one obtains interesting ...
... What looks like a group but is not a group? A naive but often fruitful idea in mathematics is to generalize mathematical objects by listing their essential properties as axioms, and then dropping one or more of them. By dropping the parallel postulate from Euclid’s geometry, one obtains interesting ...
A CATEGORY THEORETICAL APPROACH TO CLASSIFICATION
... prove a theorem gives an if and only if statement for X to have a universal covering. We should state that in this project we mainly followed the paper “Fundamental Group and Covering Spaces” by J. M. Møller. Finally, I would like to express my deep gratitude to my supervisor Prof. Ergün Yalçın fo ...
... prove a theorem gives an if and only if statement for X to have a universal covering. We should state that in this project we mainly followed the paper “Fundamental Group and Covering Spaces” by J. M. Møller. Finally, I would like to express my deep gratitude to my supervisor Prof. Ergün Yalçın fo ...
Solutions 1
... closed abelian subgroup (i.e. the family of closed abelian subgroups containing A, partially ordered with respect to inclusion, has a maximal element). (c) Prove that if there is a series 1 = G0 C G1 C · · · C Gn = G such that Gi /Gi+1 is abelian for each i, then there is such a series consisting of ...
... closed abelian subgroup (i.e. the family of closed abelian subgroups containing A, partially ordered with respect to inclusion, has a maximal element). (c) Prove that if there is a series 1 = G0 C G1 C · · · C Gn = G such that Gi /Gi+1 is abelian for each i, then there is such a series consisting of ...
Cohomology and K-theory of Compact Lie Groups
... left-invariant differential forms, which in turn have a natural correspondence with skewsymmetric multilinear forms on the Lie algebra of the Lie group. In this way the whole situation is reduced to computing the Lie algebra cohomology. One may further restrict to the bi-invariant differential forms ...
... left-invariant differential forms, which in turn have a natural correspondence with skewsymmetric multilinear forms on the Lie algebra of the Lie group. In this way the whole situation is reduced to computing the Lie algebra cohomology. One may further restrict to the bi-invariant differential forms ...
Superatomic Boolean algebras - Mathematical Sciences Publishers
... that a Boolean algebra generated by the union of finitely many superatomic subalgebras is superatomic, although the corresponding result does not hold for the concept "atomic". From this result, it follows that the direct product and free product of a set of superatomic Boolean algebras are superato ...
... that a Boolean algebra generated by the union of finitely many superatomic subalgebras is superatomic, although the corresponding result does not hold for the concept "atomic". From this result, it follows that the direct product and free product of a set of superatomic Boolean algebras are superato ...
HOMOLOGICAL PROPERTIES OF NON
... Definition 2.2. [4] A flow X consists of a compactly generated topological space PX, a discrete space X 0 , two continuous maps s and t called respectively the source map and the target map from PX to X 0 and a continuous and associative map ∗ : {(x, y) ∈ PX × PX; t(x) = s(y)} −→ PX such that s(x ∗ ...
... Definition 2.2. [4] A flow X consists of a compactly generated topological space PX, a discrete space X 0 , two continuous maps s and t called respectively the source map and the target map from PX to X 0 and a continuous and associative map ∗ : {(x, y) ∈ PX × PX; t(x) = s(y)} −→ PX such that s(x ∗ ...
On the Universal Enveloping Algebra: Including the Poincaré
... should also be noted that this bijective correspondence between modules gives us a faithful representation from g to U(g). So when we consider the universal enveloping algebra as a representation of g, there is no collapse of any important information pertaining to g. ...
... should also be noted that this bijective correspondence between modules gives us a faithful representation from g to U(g). So when we consider the universal enveloping algebra as a representation of g, there is no collapse of any important information pertaining to g. ...
Banach-Alaoglu theorems
... The best example to see the concept of the topology of the direct product is the space L∞ (X), which is the set of everywhere defined bounded functions. Note that there is no measure on X, there is no “almost everywhere”. With the usual supremum norm it is a normed space, hence metric, in particular ...
... The best example to see the concept of the topology of the direct product is the space L∞ (X), which is the set of everywhere defined bounded functions. Note that there is no measure on X, there is no “almost everywhere”. With the usual supremum norm it is a normed space, hence metric, in particular ...
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.