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Real rank (C*-algebras) : ウィキペディア英語版
Real rank (C*-algebras)
In mathematics, the real rank of a C
*-algebra
is a noncommutative analogue of Lebesgue covering dimension.
== Definition ==
The definition was first introduced by Lawrence G. Brown and Gert K. Pedersen. The real rank of a unital C
*-algebra ''A'' is the smallest non-negative integer ''n'', denoted RR(''A''), such that for every (''n'' + 1)-tuple (''x''0, ''x''1, ... ,''x''''n'') of self-adjoint elements of ''A'' and every ''ε'' > 0, there exists an (''n'' + 1)-tuple (''y''0, ''y''1, ... ,''y''''n'')
of self-adjoint elements of ''A'' such that \sum_^n y_i^2 is invertible and
\lVert \sum_^n (x_i - y_i)^2 \rVert < \varepsilon.
Nearly by definition, if ''X'' is a compact Hausdorff space, then RR(''C''(''X'')) = dim(''X''), where dim refers to the Lebesgue covering dimension of the space ''X''.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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