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.

For a complex Banach algebra A with unit 1, we give several characterizations of the scalars, that is, multiples of the identity. To a large extent, this work is a continuation and generalization of the work done on characterizations of the radical in Banach algebras. In particular it is shown that if a ∈ A has the property that the number of elements in the spectrum of ax is less than or equal to the number of elements in the spectrum of x for all x in an arbitrary neighbourhood of 1, then a is a scalar. Moreover, as a consequence of some of the results, new spectral characterizations of commutative Banach algebras are obtained. In particular, A is commutative if and only if it has the property that the number of elements in the spectrum remains invariant under all permutations of three elements in some neighbourhood of the identity.

For C0semigroups of continuous linear operators acting in a Banach space criteria are available which are equivalent to uniform mean ergodicity of the semigroup, meaning the existence of the limit (in the operator norm) of the Cesàro or Abel averages of the semigroup. Best known, perhaps, are criteria due to Lin, in terms of the range of the innitesimal generator A being a closed subspace or, whether 0 belongs to the resolvent set of A or is a simple pole of the resolvent map λ → (λ − A) −1. It is shown in the setting of locally convex spaces (even in Fréchet spaces), that neither of these criteria remain equivalent to uniform ergodicity of the semigroup (i.e., the averages should now converge for the topology of uniform convergence on bounded sets). Our aim is to exhibit new results dealing with uniform mean ergodicity of C0semigroups in more general spaces. A characterization of when a complete, barrelled space with a basis is Montel, in terms of uniform mean ergodicity of certain C0semigroups acting in the space, is also presented.

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.

As a follow-up to a paper of D. Petz and J. Zemánek [4], 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.