mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable...
30 KB (5,339 words) - 13:11, 26 March 2024
with the lower limit topology. A separable σ-algebra is a σ-algebra F {\displaystyle {\mathcal {F}}} that is a separable space when considered as a metric...
14 KB (2,071 words) - 12:06, 2 March 2024
{\displaystyle A} a separable algebra over K {\displaystyle K} . It is useful to describe separability in terms of the element p := σ ( 1 ) = ∑ a i ⊗ b...
11 KB (1,777 words) - 06:51, 26 April 2024
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...
30 KB (4,256 words) - 10:18, 18 August 2024
null set N and μ is a σ-finite measure) then L2(X, μ) is separable. The relationship between commutative von Neumann algebras and measure spaces is analogous...
10 KB (1,547 words) - 15:36, 16 February 2024
Borel set (redirect from Borel σ-algebra)
Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra. The Borel algebra on X is the smallest σ-algebra containing all open sets...
13 KB (1,793 words) - 02:48, 4 June 2024
Approximate identity (redirect from Σ-unital 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...
3 KB (474 words) - 21:29, 30 January 2023
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...
42 KB (5,905 words) - 00:34, 7 May 2024
Gelfand representation (redirect from C*-algebra representation)
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...
12 KB (1,778 words) - 15:40, 12 February 2024
Jordan–Chevalley decomposition (redirect from Jordan decomposition in a Lie algebra)
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...
41 KB (5,900 words) - 22:24, 9 July 2024