A properly infinite C*-algebra has no non-zero traces. We construct properly infinite Banach *-algebras with non-zero, bounded traces, and show that there are even such algebras which are fairly "close" to the class of C*-algebras, in the sense that they can be hermitian or *-semisimple.
We consider Fréchet algebras which are subalgebras of the algebra 𝔉 = ℂ [[X]] of formal power series in one variable and of 𝔉ₙ = ℂ [[X₁,..., Xₙ]] of formal power series in n variables, where n ∈ ℕ. In each case, these algebras are taken with the topology of coordinatewise convergence. We begin with some basic definitions about Fréchet algebras, (F)-algebras, and other topological algebras, and recall some of their properties; we discuss Michael's problem from 1952 on the continuity of characters...
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