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In operator algebras, a hereditary C *-subalgebra of a C *-algebra ''A'' is a particular type of C *-subalgebra whose structure is closely related to that of ''A''. A C *-subalgebra ''B'' of ''A'' is a hereditary C *-subalgebra if for all ''a'' ∈ ''A'' and ''b'' ∈ ''B'' such that 0 ≤ ''a'' ≤ ''b'', we have ''a'' ∈ ''B''. If a C *-algebra ''A'' contains a projection ''p'', then the C *-subalgebra ''pAp'', called a corner, is hereditary. Slightly more generally, given a positive ''a'' ∈ ''A'', the closure of the set ''aAa'' is the smallest hereditary C *-subalgebra containing ''a'', denoted by Her(''a''). If ''A'' is unital and the positive element ''a'' is invertible, we see that Her(''a'') = ''A''. This suggests the following notion of strict positivity for the non-unital case: ''a'' ∈ ''A'' is said to be strictly positive if Her(''a'') = ''A''. For instance, in the C *-algebra ''K''(''H'') of compact operators acting on Hilbert space ''H'', ''c'' ∈ ''K''(''H'') is strictly positive if and only if the range of ''c'' is dense in ''H''. There is a bijective correspondence between closed left ideals and hereditary C *-subalgebras of ''A''. If ''L'' ⊂ ''A'' is a closed left ideal, let ''L *'' denote the image of ''L'' under the (·) * operation. The set ''L *'' is a right ideal and ''L *'' ∩ ''L'' is a C *-subalgebra. In fact, ''L *'' ∩ ''L'' is hereditary and the map ''L'' ''L *'' ∩ ''L'' is a bijection. It follows from the correspondence between closed left ideals and hereditary C *-subalgebras that a closed ideal, which is a C *-subalgebra, is hereditary . Another corollary is that a hereditary C *-subalgebra of a simple C *-algebra is also simple. A hereditary C *-subalgebra of an approximately finite-dimensional C *-algebra is also AF. This is not true for subalgebras that are not hereditary. For instance, every abelian C *-algebra can be embedded into an AF C *-algebra. Two C *-algebras are stably isomorphic if they contain stably isomorphic hereditary C *-subalgebras. Also hereditary C *-subalgebras are those C *-subalgebras in which the restriction of any irreducible representation is also irreducible. == References == "Operator algebras: theory of C *-algebras and von Neumann algebras", B. Blackadar, Def II.3.4.1 p. 75 "A Course in Operator Theory", John B. Conway, def. 5.2 p. 21 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hereditary C*-subalgebra」の詳細全文を読む スポンサード リンク
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