Cardinalities and ranks of $\pi $-bases in topological spaces

It is shown that $rank (P^{*} A Q) = rank (P^{*} A) + rank (A Q) - rank (A),$ where $A$ is idempotent, $[P, Q]$ has full row rank and $P^{*} Q = 0$ . Some applications of the rank formula to generalized inverses of matrices are also presented.

Trace and determinant in Banach algebras

Bernard Aupetit, H. Mouton (1996)

Studia Mathematica

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We show that the trace and the determinant on a semisimple Banach algebra can be defined in a purely spectral and analytic way and then we obtain many consequences from these new definitions.

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