σ-unital C * algebra

The -unital C * -algebras are examined in the mathematical sub-area of functional analysis, they are C * -algebras with a certain countability condition . ${\displaystyle \sigma }$

definition

A C * -algebra is called -unital if it has a countable approximation of one . ${\displaystyle \sigma }$

Examples

• Separable C * algebras are -unital. ${\displaystyle \sigma }$
• C * algebras with a unit are -unital, so this term is only useful for C * algebras without a unit. ${\displaystyle \sigma }$
• The commutative C * algebra of the C ₀ functions on a locally compact Hausdorff space is -unital if and only if is σ-compact . If it is uncountable and discrete , then it is not -unital. ${\displaystyle C_{0}(X)}$ ${\displaystyle X}$ ${\displaystyle \sigma }$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle C_{0}(X)}$ ${\displaystyle \sigma }$

properties

The following statements about a C * -algebra are equivalent: ${\displaystyle A}$

• ${\displaystyle A}$is -unital. ${\displaystyle \sigma }$
• ${\displaystyle A}$has a strictly positive element, that is, there is a positive element , so that for all states on . ${\displaystyle a\in A}$ ${\displaystyle f(a)>0}$ ${\displaystyle f}$ ${\displaystyle A}$
• There is a positive element , so it is a dense subset . ${\displaystyle a\in A}$ ${\displaystyle aA\subset A}$
• There is a positive element so that the unity element in the enveloping Von Neumann algebra is the smallest projection with . ${\displaystyle a\in A}$ ${\displaystyle p}$ ${\displaystyle pa=a}$

Individual evidence

1. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-12-549450-5 , sentence 3.10.6
2. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 12.3.1
3. ↑ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-12-549450-5 , sentence 3.10.5