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mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension
Separable_algebra
Algebraic structure of set algebra
{F}}).} A separable σ {\displaystyle \sigma } -algebra (or separable σ {\displaystyle \sigma } -field) is a σ {\displaystyle \sigma } -algebra F {\displaystyle
Σ-algebra
Algebraic field extension
contained in a separably-closed algebraic extension field. It is unique (up to isomorphism). The separable closure is the full algebraic closure if and
Algebraic_closure
Type of algebraic field extension
In field theory, a branch of algebra, an algebraic field extension E / F {\displaystyle E/F} is called a separable extension if for every α ∈ E {\displaystyle
Separable_extension
Algebraic structure
{\displaystyle K} is separable. Every finite extension of K {\displaystyle K} is separable. Every algebraic extension of K {\displaystyle K} is separable. Either K
Perfect_field
Ring that is also a vector space or a module
called the bidimension of A, measures the failure of separability. Let A be a finite-dimensional algebra over a field k. Then A is an Artinian ring. As A
Associative_algebra
Topological complex vector space
classification is possible, for separable simple nuclear C*-algebras. We begin with the abstract characterization of C*-algebras given in the 1943 paper by
C*-algebra
*-algebra of bounded operators on a Hilbert space
Neumann algebras are the direct integral of properly infinite factors. A von Neumann algebra that acts on a separable Hilbert space is called separable. Note
Von_Neumann_algebra
Construction in algebra
Hopf algebra. The axioms are partly chosen so that the category of H-modules is a rigid monoidal category. The unit H-module is the separable algebra HL
Hopf_algebra
Topological space with a dense countable subset
In mathematics, a topological space is called separable if it contains a countable dense subset; that is, there exists a sequence ( x n ) n = 1 ∞ {\displaystyle
Separable_space
commutative algebra, an étale algebra over a field is a special type of algebra, one that is isomorphic to a finite product of finite separable field extensions
Étale_algebra
Mathematical expression for linear operators
associative algebra over the field K {\displaystyle K} with Jacobson radical J {\displaystyle J} . Then A / J {\displaystyle A/J} is separable if and only
Jordan–Chevalley decomposition
Jordan–Chevalley_decomposition
Algebraic structure with "nice" duality properties
example of a separable algebra extension since e = ∑ i = 1 n a i ⊗ B b i {\textstyle e=\sum _{i=1}^{n}a_{i}\otimes _{B}b_{i}} is a separability element satisfying
Frobenius_algebra
Relation algebra Relational algebra Rota–Baxter algebra Schur algebra Semisimple algebra Separable algebra Shuffle algebra Sigma-algebra Simple algebra Structurable
List_of_algebras
Canonical commutation or anticommutation relations
. It is well known that the CCR algebra is a simple (unless the sympletic form is degenerate) non-separable algebra and is unique up to isomorphism.
CCR_and_CAR_algebras
Type of vector space in math
Hilbert space is separable provided it contains a dense countable subset. Along with Zorn's lemma, this means a Hilbert space is separable if and only if
Hilbert_space
Group of unitary matrices
field extension can be replaced by any degree 2 {\displaystyle 2} separable algebra, most notably a degree 2 {\displaystyle 2} extension of a finite field;
Unitary_group
Polynomial coprime with its derivative
mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct roots
Separable_polynomial
Topics referred to by the same term
Look up separable in Wiktionary, the free dictionary. Separability may refer to: Separable algebra, a generalization to associative algebras of the notion
Separability
considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces. Note
Abelian_von_Neumann_algebra
Mathematical concept
For a commutative C*-algebra, A ^ ≅ Prim ( A ) . {\displaystyle {\hat {A}}\cong \operatorname {Prim} (A).} Let H be a separable infinite-dimensional
Spectrum_of_a_C*-algebra
Class of mathematical sets
complement. Then we can define the Borel σ-algebra over X {\displaystyle X} to be the smallest σ-algebra containing all open sets of X {\displaystyle
Borel_set
Every polynomial has a real or complex root
(hence it is a Galois extension, as every algebraic extension of a field of characteristic 0 is separable). Let G be the Galois group of this extension
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Branch of functional analysis
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with
Operator_algebra
Technique for solving differential equations
differential equation for the unknown f ( x ) {\displaystyle f(x)} is separable if it can be written in the form d d x f ( x ) = g ( x ) h ( f ( x ) )
Separation_of_variables
Universal C*-algebra
certain relations. These algebras were introduced as the first concrete examples of a separable infinite simple C*-algebra, meaning that as a Hilbert
Cuntz_algebra
8-dimensional real non-associative division algebra. A similar construction works for any cubic alternative separable algebra over a field containing a primitive
Okubo_algebra
Algebraic structure with addition, multiplication, and division
exponential function exp : F → F×). For fields that are not algebraically closed (or not separably closed), the absolute Galois group Gal(F) is fundamentally
Field_(mathematics)
Mathematical representation in functional analysis
If A is a separable C*-algebra, the weak-* topology is metrizable on bounded subsets. Thus the spectrum of a separable commutative C*-algebra A can be
Gelfand_representation
Generalization of the concept of a direct sum in mathematics
von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field
Direct_integral
Field extension that is not algebraic
transcendence basis S such that L is a separable algebraic extension over K(S). A field extension L / K is said to be separably generated if it admits a separating
Transcendental_extension
Algebraic structure with addition and multiplication
Semisimplicity is closely related to separability. A unital associative algebra A over a field k is said to be separable if the base extension A ⊗ k F {\displaystyle
Ring_(mathematics)
Dimension of the column space of a matrix
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal
Rank_(linear_algebra)
In mathematics, vector space of linear forms
for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, there is a subspace
Dual_space
Finite dimensional algebra over a field whose central elements are that field
areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A that is simple, and for which the center
Central_simple_algebra
Algebraic structure modeling logical operations
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties
Boolean_algebra_(structure)
Algebraic field extension
mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the
Galois_extension
Normed vector space that is complete
of a separable Banach space need not be separable, but: Theorem—Let X {\displaystyle X} be a normed space. If X ′ {\displaystyle X'} is separable, then
Banach_space
analysis, the Calkin algebra, named after John Williams Calkin, is the quotient of B(H), the ring of bounded linear operators on a separable infinite-dimensional
Calkin_algebra
Theory in mathematics
of K-homology and K-theory as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician Gennadi
KK-theory
Subject area in mathematics
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic
Algebraic_K-theory
Neumann algebra is injective. It is amenable as a Banach algebra. (For separable algebras) It is isomorphic to a C*-subalgebra B of the Cuntz algebra 𝒪2
Nuclear_C*-algebra
Local ring in which Hensel's lemma holds
depends on the choice of a separable algebraic closure of the residue field of A, and automorphisms of this separable algebraic closure correspond to automorphisms
Henselian_ring
compactification. Multiplier algebras were introduced by Busby (1968). For example, if A is the C*-algebra of compact operators on a separable Hilbert space, M(A)
Multiplier_algebra
C*-algebra of all bounded operators on a Hilbert space H. A C*-algebra is exact if and only if every separable sub-C*-algebra is exact. A separable C*-algebra
Exact_C*-algebra
Quantum states that are not entangled
In quantum mechanics, separable states are multipartite quantum states that can be written as a convex combination of product states. Product states are
Separable_state
glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary
Glossary of commutative algebra
Glossary_of_commutative_algebra
C*-algebra
theorem for AF-algebras serves as a prototype for classification results for larger classes of separable simple amenable stably finite C*-algebras. Its proof
Approximately finite-dimensional C*-algebra
Approximately_finite-dimensional_C*-algebra
Branch of mathematics that studies algebraic structures
extension Algebraic extension Splitting field Algebraically closed field Algebraic element Algebraic closure Separable extension Separable polynomial
List of abstract algebra topics
List_of_abstract_algebra_topics
Specific algebraic group
enough r {\displaystyle r} . In general one has to use separable closures instead of algebraic closures. If F {\displaystyle F} is a field then the multiplicative
Algebraic_torus
Function in algebra
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size
Valuation_(algebra)
In mathematics, element that equals its square
idempotent a of R is called a full idempotent if RaR = R. A separability idempotent; see Separable algebra. Any non-trivial idempotent a is a zero divisor (because
Idempotent_(ring_theory)
condition over a separable algebra is a weak Hopf algebra, and conversely a weak Hopf algebra H is a Hopf algebroid over its separable subalgebra HL. The
Hopf_algebroid
Algebraic geometry
formally étale. Finite separable field extensions are formally étale. More generally, any (commutative) flat separable A-algebra B is formally étale. Formally
Formally_étale_morphism
Extension of a mathematical field with polynomial roots
In mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that
Algebraic_extension
Mathematical ring whose elements are matrices
the algebra B(H) of continuous operators; this identifies Mn(A) with a subalgebra of B(H⊕n). For simplicity, if we further suppose that H is separable and
Matrix_ring
Algebraic construct of interest in theoretical physics
compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do
Quantum_group
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
we have that The residue field k ( x ) {\displaystyle k(x)} is a separable algebraic extension of k ( y ) {\displaystyle k(y)} . f # ( m y ) O x , X =
Unramified_morphism
True when either but not both inputs are true
the XOR function requires a second layer because XOR is not a linearly separable function. Similarly, XOR can be used in generating entropy pools for hardware
Exclusive_or
Mathematical function
x_{m})=0.} In one variable, algebraic functions are closely related to algebraic curves and their function fields; in the separable case, they may also be
Algebraic_function
Mathematical theorem in the study of analysis
Weierstrass approximation theorem, one can show that the space C[a, b] is separable: the polynomial functions are dense, and each polynomial function can
Stone–Weierstrass_theorem
of A as an (Aop ⊗R A)-module. For example, an algebra has bidimension zero if and only if it is separable. boolean A boolean ring is a ring in which every
Glossary_of_ring_theory
Net in a normed algebra
approximate identity and a C*-algebra with a sequential approximate identity is called σ-unital. Every separable C*-algebra is σ-unital, though the converse
Approximate_identity
Field theory theorem
every finite separable field extension is simple, i.e. generated by a single element. This theorem implies in particular that all algebraic number fields
Primitive_element_theorem
German mathematician (1882–1935)
German mathematician who made many important contributions to abstract algebra. She also proved Noether's first and second theorems, which are fundamental
Emmy_Noether
Concept in ring theory
the opposite algebra of A {\displaystyle A} . The center of A {\displaystyle A} is R {\displaystyle R} , and A {\displaystyle A} is separable. A {\displaystyle
Azumaya_algebra
Field of knowledge
including number theory (the study of integers and their properties), algebra (the study of operations and the structures they form), geometry (the study
Mathematics
Classification in abstract algebra
) {\displaystyle Z(q)} is either a separable quadratic extension field of F {\displaystyle F} or the split algebra F ⊕ F {\displaystyle F\oplus F} . If
Classification of Clifford algebras
Classification_of_Clifford_algebras
Construction of a larger algebraic field by "adding elements" to a smaller field
property. An algebraic extension L / K {\displaystyle L/K} is called separable if the minimal polynomial of every element of L over K is separable, i.e., has
Field_extension
Subject in mathematics
sets. For non-separable spaces it can happen that the vector addition is no longer measurable to the product algebra of borel σ-algebras because in general
Measure theory in topological vector spaces
Measure_theory_in_topological_vector_spaces
A finitely generated k-algebra A is 0-smooth over k if and only if Spec A is a smooth scheme over k. A separable algebraic field extension L of k is
Smooth_algebra
Type of feedforward neural network
layers, notable for being able to distinguish data that is not linearly separable. Modern neural networks are trained using backpropagation and are colloquially
Multilayer_perceptron
On invertibility of polynomial maps (mathematics)
Stuart Sui-Sheng (August 1980), "A Jacobian criterion for separability", Journal of Algebra, 65 (2): 453–494, doi:10.1016/0021-8693(80)90233-1 Bass, Hyman;
Jacobian_conjecture
Unique von Neumann algebra
the Clifford algebra of an infinite separable Hilbert space. If p is any non-zero finite projection in a hyperfinite von Neumann algebra A of type II
Hyperfinite_type_II_factor
K0 group of A. One example of a UHF C*-algebra is the CAR algebra. It is defined as follows: let H be a separable complex Hilbert space H with orthonormal
Uniformly_hyperfinite_algebra
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
Graph_C*-algebra
Subspace preserved by a linear mapping
ideal in A. The invariant subspace problem concerns the case where V is a separable Hilbert space over the complex numbers, of dimension > 1, and T is a bounded
Invariant_subspace
Concept in topology
the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic
Polish_space
{\overline {k}}} denotes an algebraic closure of k. X × k k s {\displaystyle X\times _{k}k_{s}} is irreducible for a separable closure k s {\displaystyle
Geometrically (algebraic geometry)
Geometrically_(algebraic_geometry)
Field theory is the branch of algebra that studies fields
F is also in E. Separable extension An algebraic extension in which the minimal polynomial of every element of E over F is a separable polynomial, that
Glossary_of_field_theory
necessarily separable) Hilbert space. The problem has been solved in the affirmative for special cases (specifically for separable and Type-I C*-algebras). Akemann
Naimark's_problem
Mathematical problem in von Neumann algebra theory
conjecture in C*-algebra theory Tsirelson's problem in quantum information theory The predual of any (separable) von Neumann algebra is finitely representable
Connes_embedding_problem
dual space is a measurable function in the usual (strong) sense. For separable spaces, the notions of weak and strong measurability agree. If ( X , Σ
Weakly_measurable_function
Term in mathematics
and take T to be a maximal torus containing S defined over the separable algebraic closure K of k. Then G(K) has a Dynkin diagram with respect to some
Satake_diagram
Structure in algebraic geometry
In algebraic geometry, a motive (or sometimes motif, following French usage) is an abstract object introduced by Alexander Grothendieck in the 1960s as
Motive_(algebraic_geometry)
Equivalence relation on rings
ISBN 0-387-97845-3. Zbl 0765.16001. DeMeyer, F.; Ingraham, E. (1971). Separable algebras over commutative rings. Lecture Notes in Mathematics. Vol. 181.
Morita_equivalence
Length in a vector space
norms in additive combinatorics Kadec norm – All infinite-dimensional, separable Banach spaces are homeomorphicPages displaying short descriptions of redirect
Norm_(mathematics)
In algebraic geometry, a morphism f : X → S {\displaystyle f:X\to S} between schemes is said to be smooth if (i) it is locally of finite presentation
Smooth_morphism
Open convex self-dual cones
a Jordan C* algebra. The complexification of a simple Euclidean Jordan algebra is a simple complex Jordan algebra which is also separable, i.e. its trace
Symmetric_cone
and the Borel algebra on B) if and only if it is both weakly measurable and almost surely separably valued. In the case that B is separable, since any subset
Bochner_measurable_function
Mathematical group
In Galois theory, a branch of abstract algebra, the Galois group of a certain type of field extension is a symmetry group characterizing how it extends
Galois_group
Algebraic structure
whose minimal polynomial is separable. To use a piece of jargon, finite fields are perfect. Finite fields are quasi-algebraically closed: every degree d homogeneous
Finite_field
Study of Galois symmetry groups of differential fields
{\displaystyle e^{-x^{2}}} . For a differential field F, if G is a separable algebraic extension of F, the derivation of F uniquely extends to a derivation
Differential_Galois_theory
Roots of an algebraic element's minimal polynomial
list of each element is the separable degree [L:K(α)]sep. A theorem of Kronecker states that if α is a nonzero algebraic integer such that α and all of
Conjugate element (field theory)
Conjugate_element_(field_theory)
Function that returns cardinal numbers
algebraic field extensions, algebraic degree and separable degree are often employed (the algebraic degree equals the dimension of the extension as a
Cardinal_function
Tensor product space endowed with a special inner product
that inner product's isotropy subspaces. This definition is almost never separable, in part because, in physical applications, "most" of the space describes
Tensor product of Hilbert spaces
Tensor_product_of_Hilbert_spaces
Theorem in axiomatic quantum field theory
{\displaystyle \vert \Omega \rangle } is a cyclic vector for the field algebra A ( O ) {\displaystyle {\mathcal {A}}({\mathcal {O}})} corresponding to
Reeh–Schlieder_theorem
Expected value of a random variable given that certain conditions are known to occur
for creating a regular probability measure, which are separability and completeness. The σ-algebra H {\displaystyle {\mathcal {H}}} controls the "granularity"
Conditional_expectation
group is not special because there exist Azumaya algebras, which are trivial over a finite separable extension, but not over the base field. Special groups
Special group (algebraic group theory)
Special_group_(algebraic_group_theory)
SEPARABLE ALGEBRA
SEPARABLE ALGEBRA
Boy/Male
Indian, Marathi
Separate
Boy/Male
Sikh
Triumph for gods name, Triumph of the inseparable creator
Girl/Female
Muslim/Islamic
Inseparable friend
Girl/Female
Muslim
Inseparable friend
Biblical
a parable; governing
Girl/Female
Arabic, Muslim, Sindhi
Inseparable Friend
Girl/Female
Indian, Punjabi, Sikh
Triumph of the Inseparable Creator
Girl/Female
Arabic, Muslim
Example; Allegory; Parable
Girl/Female
Indian
Inseparable
Girl/Female
Arabic, Muslim
Inseparable Friend
Boy/Male
Arabic, Australian, Muslim
Considerate; Inseparable Friend
Girl/Female
Indian, Punjabi, Sikh
Love of the Inseparable Creator
Surname or Lastname
English
English : variant spelling of Rimer 1.German : variant of Riemer.German : habitational name for someone from Riem (now a suburb of Munich; formerly a separate town).
Girl/Female
Biblical
A parable, governing.
Boy/Male
Muslim/Islamic
Inseparable friend
Boy/Male
Sikh
Triumph for gods name, Triumph of the inseparable creator
Boy/Male
Muslim
Considerate, Inseparable friend
Surname or Lastname
English
English : occupational name for a maker of arms and armor, from Anglo-Norman French armer ‘arms-maker’ (Old French armier). Originally this was a separate name from Armour, but in due course the two became inextricably confused.
Girl/Female
Muslim
Example, Allegory, Parable
Girl/Female
Arabic
Separate
SEPARABLE ALGEBRA
SEPARABLE ALGEBRA
Boy/Male
Hindu
The one who wears Moon on head, Meaning Lord Shiva
Male
English
 Short form of English Leonard, LEN means "lion-strong." Compare with another form of Len.
Girl/Female
Arabic, Indian, Muslim
Beautiful
Girl/Female
Hindu, Indian
A Small Ruby
Male
Greek
Modern short form of Greek Evangelos, VANGELIS means "good angel" or "good messenger."
Boy/Male
Muslim
Gods warrior
Boy/Male
Hindu
Joyful, Kings of the hills, Kind hearted a sweet
Boy/Male
Tamil
Hrydayesh | ஹà¯à®°à¯à®¯à¯à®¤à®¯à¯‡à®·
King of heart, Lord of hearts
Boy/Male
Hindu
Giver of beauty, Lord Kuber
Boy/Male
Arabic, Muslim
Visitor
SEPARABLE ALGEBRA
SEPARABLE ALGEBRA
SEPARABLE ALGEBRA
SEPARABLE ALGEBRA
SEPARABLE ALGEBRA
a.
Capable of being overcome or conquered; surmountable.
a.
Invariably attached to some word, stem, or root; as, the inseparable particle un-.
a.
Capable of being prepared.
n.
See Sperable.
p. a.
Disunited from the body; disembodied; as, a separate spirit; the separate state of souls.
adv.
In an inseparable manner or condition; so as not to be separable.
a.
Not separable; incapable of being separated or disjoined.
a.
That may be secured.
a.
Capable of being spoken; fit to be spoken.
a.
Able to speak.
a.
Separable.
a.
Reparable.
a.
Inseparable.
a.
Capable of being, or proper to be , repaid; due; as, a loan repayable in ten days; services repayable in kind.
a.
Capable of being repaired, restored to a sound or good state, or made good; restorable; as, a reparable injury.
a.
Capable of being separated, disjoined, disunited, or divided; as, the separable parts of plants; qualities not separable from the substance in which they exist.
adv.
In a reparable manner.
a.
Capable of being severed.
n.
A kind of small nail used by shoemakers.
v. t.
To represent by parable.