On a new approach to Fredholm theory in unital C∗-algebras

Abstract

Motivated by Fredholm theory on the standard Hilbert module over a unital C∗ -algebra introduced by Mishchenko and Fomenko, we provide a new approach to axiomatic Fredholm theory in unital C∗ -algebras established by Kečkić and Lazović in [16]. Our approach is equivalent to the approach introduced by Kečkić and Lazović, however, we provide new proofs which are motivated by the proofs given by Mishchenko and Fomenko in [18]. Next,we extend Fredholm theory in von Neumann algebras established by Breuer in [4] and [5] to spectral Fredholm theory. We consider 2 by 2 upper triangular operator matrices with coefficients in a von Neumann algebra and give the relationship between the generalized essential spectra in the sense of Breuer of such matrices and of their diagonal entries, thus generalizing in this setting the result by Ðorđević in [6]. Finally, we prove that if a generalized Fredholm operator in the sense of Breuer has 0 as an isolated point of its spectrum, then the corresponding spectral projection is finite.