Topological algebras with C<SUP>&#x002A;</SUP>-enveloping algebras

LetA be a complete topological &#x002A;algebra which is an inverse limit of Banach&#x002A;algebras. The (unique) enveloping algebra E(A) of A, providing a solution of the universal problem for continuous representations of A into bounded Hilbert space operators, is known to be an inverse limit of C&#x002A;-algebras. It is shown that S(A) is a C&#x002A;-algebra iff A admits greatest continuous C&#x002A;-seminorm iff the continuous states (respectively, continuous extreme states) constitute an equicontinuous set. A Q-algebra (i.e., one whose quasiregular elements form an open set) A has C&#x002A;-enveloping algebra. There exists (i) a Frechet algebra with C&#x002A;-enveloping algebra that is not a Q-algebra under any topology and (ii) a non-Q spectrally bounded algebra with C&#x002A;-enveloping algebra.A hermitian algebra with C&#x002A;-enveloping algebra turns out to be a Q-algebra. The property of having C&#x002A;-enveloping algebra is preserved by projective tensor products and completed quotients, but not by taking closed subalgebras. Several examples of topological algebras with C&#x002A;-enveloping algebras are discussed. These include several pointwise algebras of functions including well-known test function spaces of distribution theory, abstract Segal algebras and concrete convolution algebras of harmonic analysis, certain algebras of analytic functions (with Hadamard product) and Kothe sequence algebras of infinite type. The enveloping C&#x002A;-algebra of a hermitian topological algebra with an orthogonal basis is isomorphic to the C&#x002A;-algebra c0 of all null sequences

Complete positivity, tensor products and C<SUP>&#x002A;</SUP>-nuclearity for inverse limits of C<SUP>&#x002A;</SUP>-algebras

The paper aims at developing a theory of nuclear (in the topological algebraic sense) pro-C&#x002A;-algebras (which are inverse limits of C&#x002A;-algebras) by investigating completely positive maps and tensor products. By using the structure of matrix algebras over a pro-C&#x002A;-algebra, it is shown that a unital continuous linear map between pro-C&#x002A;-algebras A and B is completely positive iff by restriction, it defines a completely positive map between the C&#x002A;-algebras b(A) and b(B) consisting of all bounded elements of A and B. In the metrizable case, A and B are homeomorphically isomorphic iff they are matricially order isomorphic. The injective pro-C&#x002A;-topology &#945; and the projective pro-C&#x002A;-topology &#965; on A &#8855; B are shown to be minimal and maximal pro-C&#x002A;-topologies; and &#945; coincides with the topology of biequicontinous convergence iff either A or B is abelian. A nuclear pro-C&#x002A;-algebra A is one that satisfies, for any pro-C&#x002A;-algebra (or a C&#x002A;-algebra) B, any of the equivalent requirements; (i) &#945;= &#965; on A &#8707; B (ii) A is inverse limit of nuclear C&#x002A;-algebras (iii) there is only one admissible pro-C&#x002A;-topologyon A &#8855; B (iv) the bounded part b(A) of A is a nuclear C&#x002A;-algebra (v) any continuous complete state map A&#8594; B&#x002A; can be approximated in simple weak&#x002A; convergence by certain finite rank complete state maps. This is used to investigate permanence properties of nuclear pro-C&#x002A;-algebras pertaining to subalgebras, quotients and projective and inductive limits. A nuclearity criterion for multiplier algebras (in particular, the multiplier algebra of Pedersen ideal of a C&#x002A;-algebra) is developed and the connection of this C&#x002A;-algebraic nuclearity with Grothendieck's linear topological nuclearity is examined. A &#x0431;-C&#x002A;-algebra A is a nuclear space iff it is an inverse limit of finite dimensional C&#x002A;-algebras; and if abelian, then A is isomorphic to the algebra (pointwise operations) of all scalar sequences