LetA_e be the algebra obtained by adjoining identity to a non-unital Banach algebra (A, ∥ · ∥). Unlike the case for aC*-norm on a Banach *-algebra,A_e admits exactly one uniform norm (not necessarily complete) if so doesA. This is used to show that the spectral extension property carries over fromA to A_e. Norms onA_e that extend the given complete norm ∥ · ∥ onA are investigated. The operator seminorm ∥ · ∥_op onA_e defined by ∥ · ∥ is a norm (resp. a complete norm) iffA has trivial left annihilator (resp. ∥ · ∥_op restricted toA is equivalent to ∥ · ∥).

A completely positive operator valued linear map ϕ on a (not necessarily unital) Banach *-algebra with continuous involution admits minimal Stinespring dilation iff for some scalark > 0, ϕ(x)*ϕ(x) ≤ kϕ(x*x) for allx iff ϕ is hermitian and satisfies Kadison’s Schwarz inequality ϕ(h)²≤ kϕ(h²) for all hermitianh iff ϕ extends as a completely positive map on the unitizationA_eof A. A similar result holds for positive linear maps. These provide operator state analogues of the corresponding well-known results for representable positive functionals. Further, they are used to discuss (a) automatic Stinespring representability in Banach *-algebras, (b) operator valued analogue of Bochner-Weil-Raikov integral representation theorem, (c) operator valued analogue of the classical Bochner theorem in locally compact abelian groupG, and (d) extendability of completely positive maps from *-subalgebras. Evans’ result on Stinespring respresentability in the presence of bounded approximate identity (BAI) is deduced. A number of examples of Banach *-algebras without BAI are discussed to illustrate above results.

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC_c(G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC^∞-elementsC^∞(A), the analytic elementsC^ω(A) as well as the entire analytic elementsC^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI_α is constructed satisfyingA =C*-ind limI_α; and the locally convex inductive limit ind limI_α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK_a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.

Letf be a continuous function on the unit circle Γ, whose Fourier series is ω-absolutely convergent for some weight ω on the set of integersZ. If f is nowhere vanishing on Γ, then there exists a weightv onZ such that 1/f hadv-absolutely convergent Fourier series. This includes Wiener’s classical theorem. As a corollary, it follows that if φ is holomorphic on a neighbourhood of the range off, then there exists a weight Χ on Z such that φ ◯f has Χ-absolutely convergent Fourier series. This is a weighted analogue of Lévy’s generalization of Wiener’s theorem. In the theorems,v and Χ are non-constant if and only if ω is non-constant. In general, the results fail ifv or Χ is required to be the same weight ω.

For a compactly generated LCA group G, it is shown that the setH(G) of all generalized characters on G equipped with the compact-open topology is a LCA group andH(G) = Ĝ (the dual group ofG) if and only ifG is compact. Both results fail for arbitrary LCA groups. Further, ifG is second countable, then the Gel’fand space of the commutative convolution algebraC_c(G) equipped with the inductive limit topology is topologically homeomorphic toH(G).

Given a C∗-normed algebra A which is either a Banach ∗-algebra or a Frechet ∗-algebra, we study the algebras ∞A and A obtained by taking respectively the projective limit and the inductive limit of Banach ∗-algebras obtained by completing the universal graded differential algebra ∗A of abstract non-commutative differential forms over A. Various quantized integrals on ∞A induced by a K-cycle on A are considered. The GNS-representation of ∞A defined by a d-dimensional non-commutative volume integral on a d+-summable K-cycle on A is realized as the representation induced by the left action of A on ∗A. This supplements the representation A on the space of forms discussed by Connes (Ch. VI.1, Prop. 5, p. 550 of [C]).

Given a weighted discrete abelian semigroup $(S,\omega)$, the semigroup $M_\omega(S)$ of 𝜔-bounded multipliers as well as the Rees quotient $M_\omega(S)/S$ together with their respective weights $\overline{\omega}$ and $\overline{\omega}_q$ induced by 𝜔 are studied; for a large class of weights 𝜔, the quotient $\ell^1(M_\omega(S),\overline{\omega})/\ell^1(S,\omega)$ is realized as a Beurling algebra on the quotient semigroup $M_\omega(S)/S$; the Gel’fand spaces of these algebras are determined; and Banach algebra properties like semisimplicity, uniqueness of uniform norm and regularity of associated Beurling algebras on these semigroups are investigated. The involutive analogues of these are also considered. The results are exhibited in the context of several examples.