Bounded and unitary elements in pro-C^*-algebras

A pro-C^*-algebra is a (projective) limit of C^*-algebras in the category of topological *-algebras. From the perspective of non-commutative geometry, pro-C^*-algebras can be seen as non-commutative k-spaces. An element of a pro-C^*-algebra is bounded if there is a uniform bound for the norm of its images under any continuous *-homomorphism into a C^*-algebra. The *-subalgebra consisting of the bounded elements turns out to be a C^*-algebra. In this paper, we investigate pro-C^*-algebras from a categorical point of view. We study the functor (-)_b that assigns to a pro-C^*-algebra the C^*-algebra of its bounded elements, which is the dual of the Stone-\v{C}ech-compactification. We show that (-)_b is a coreflector, and it preserves exact sequences. A generalization of the Gelfand-duality for commutative unital pro-C^*-algebras is also presented.Comment: v2 (accepted

On strongly closed subalgebras of B(X)

Let X be a real or complex Banach space. The strong topology on the algebra B(X) of all bounded linear operators on X is the topology of pointwise convergence of nets of operators. It is given by a basis of neighbourhoods of the origin consisting of sets of the form (1) U(ε;x_{1},...,x_{n}) = {T ∈ B(X): ∥ Tx_{i}∥ <ε, i=1,...,n},$where$x_{1},...,x_{n}$ are linearly independent elements of X and ε is a positive real number. Closure in the strong topology will be called strong closure for short. It is well known that the strong closure of a subalgebra of B(X) is again a subalgebra. In this paper we study strongly closed subalgebras of B(X), in particular, maximal strongly closed subalgebras. Our results are given in Section 1, while in Section 2 we give the motivation for this study and pose several open questions

Concerning entire functions in B0B_{0}-algebras

We construct a non-m-convex non-commutative $B_0$-algebra on which all entire functions operate. Our example is also a Q-algebra and a radical algebra. It follows that some results true in the commutative case fail in general

Continuous characters and joint topological spectrum

It is well known that there is a one-to-one correspondence between the characters of a finitely generated commutative Banach algebra and the joint spectrum of its generators. In this paper we show that this fact is also true for an arbitrary semitopological algebra and its continuous characters, provided we replace the concept of a joint spectrum by concept of a topological joint spectrum. In particular, we show that a finitely generated semitopological algebra has a continuous character if and only if the topological joint spectrum of its generators is non-void