MSC: 15A60 46H05 46H10 46H15 47B10 Keywords: Rank Socle TraceAs a follow-up to a paper of D. Petz and J. Zemánek [1], a number of equivalent conditions which characterize the trace among linear functionals on matrix algebras, finite rank operators and the socle elements of semisimple Banach algebras in general are given. Moreover, the converse problem is also addressed; that is, given the equivalence of certain conditions which characterize the trace, what can be said about the structure of the socle? In particular, we characterize those socles isomorphic to matrix algebras in this manner, as well as those socles which are minimal two-sided ideals.

No abstract

No abstract

We consider a multiplicative variation on the classical Kowalski-S lodkowski Theorem which identifies the characters among the collection of all functionals on a Banach algebra A. In particular we show that, if A is a C * -algebra, and if φ : A → C is a continuous function satisfying φ(1) = 1 and φ(x)φ(y) ∈ σ(xy) for all x, y ∈ A (where σ denotes the spectrum), then φ generates a corresponding character ψ φ on A which coincides with φ on the principal component of the invertible group of A. We also show that, if A is any Banach algebra whose elements have totally disconnected spectra, then, under the aforementioned conditions, φ is always a character.2010 Mathematics Subject Classification. 15A60, 46H05, 46H10, 46H15, 47B10.

As a follow-up to a paper of Aupetit and Mouton (1996), we consider the spectral definitions of rank, trace and determinant applied to elements in a general Banach algebra. We prove a generalization of Sylvester's Determinant Theorem to Banach algebras and thereafter a generalization of the Frobenius inequality.