Trace and determinant in Jordan-Banach algebras

Abstract

Using an appropriate definition of the multiplicity of a spectral value, we introduce a new definition of the trace and determinant of elements with finite spectrum in Jordan-Banach algebras. We first extend a result obtained by J. Zemánek in the associative case, on the connectedness of projections which are close to each other spectrally (Theorem 2.3). Secondly we show that the rank of the Riesz projection associated to a finite-rank element a and a finite subset of its spectrum is equal to the sum of the multiplicities of these spectral values (Theorem 2.6). Then we turn to the study of properties such as linearity and continuity of the trace and multiplicativity of the determinant.

On Automatic Surjectivity of Spectrum Preserving Additive Transformations

“…In fact B.Aupetit shows in [6] that this multiplicity is equal to the rank of the Riesz projection associated to T and α. Let α ∈ σ(T ) and Γ be a small curve isolating α from the rest of the spectrum of T .…”

“…The following lemma is a simple version of result of B.Aupetit in the case of B(X) (see [6] Corollary 3.6).…”

“…It suffices to see that AnnSocB(X) = {0}( [6] and [7]). The following lemma is due to B.Aupetit [7] Corollary 2.4.…”

“…We denote by F (X) the ideal of finite rank operators it coincides with SocB(X), the socle of B(X). We know by [6] that the socle of B(X) is linearly engendered by idempotent operators of rank one. AnnSocB(X) denotes the set of operators T ∈ B(X) satisfying T R = 0 for every R ∈ SocB(X).…”

The index for Fredholm elements in a Banach algebra via a trace

“…The existence of a continuous trace on an operator ideal of operators on a Banach space has long been known to provide a useful tool for developing the Fredholm theory of operators (see for instance the monograph of A. Pietsch [20] and the paper by the present authors [13]). The problem of defining traces on ideals in a Banach algebra attracted the attention of many authors (see the papers by Puhl [22] and Aupetit and Mouton [2]). The main thrust of these papers was to show that a trace and a Fredholm determinant exist on the socle of a semisimple Banach algebra ( [2] and [22]) and on the question of extending the trace or the determinant to larger ideals ( [22] and [3]).…”