Subhash J. Bhatt

Complete positivity, tensor products and C*-nuclearity for inverse limits of C*-algebras

The paper aims at developing a theory of nuclear (in the topological algebraic sense) pro-C*-algebras (which are inverse limits of C*-algebras) by investigating completely positive maps and tensor products. By using the structure of matrix algebras over a pro-C*-algebra, it is shown that a unital continuous linear map between pro-C*-algebrasA andB is completely positive iff by restriction, it defines a completely positive map between the C*-algebrasb(A) andb(B) consisting of all bounded elements ofA andB. In the metrizable case,A andB are homeomorphically isomorphic iff they are matricially order isomorphic. The injective pro-C*-topology α and the projective pro-C*-topology v on A⊗B are shown to be minimal and maximal pro-C*-topologies; and α coincides with the topology of biequicontinous convergence iff eitherA orB is abelian. A nuclear pro-C*-algebraA is one that satisfies, for any pro-C*-algebra (or a C*-algebra)B, any of the equivalent requirements; (i) α =v onA ⊗B (ii)A is inverse limit of nuclear C*-algebras (iii) there is only one admissible pro-C*-topologyon A⊗B (iv) the bounded partb(A) ofA is a nuclear C⊗-algebra (v) any continuous complete state map A→B* can be approximated in simple weak* convergence by certain finite rank complete state maps. This is used to investigate permanence properties of nuclear pro-C*-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*-algebra) is developed and the connection of this C*-algebraic nuclearity with Grothendieck’s linear topological nuclearity is examined. A σ-C*-algebraA is a nuclear space iff it is an inverse limit of finite dimensional C*-algebras; and if abelian, thenA is isomorphic to the algebra (pointwise operations) of all scalar sequences.

On unbounded subnormal operators

A minimal normal extension of unbounded subnormal operators is established and characterized and spectral inclusion theorem is proved. An inverse Cayley transform is constructed to obtain a closed unbounded subnormal operator from a bounded one. Two classes of unbounded subnormals viz analytic Toeplitz operators and Bergman operators are exhibited.

Admissibility of Weights on Non-normed *-Algebras

The notion of weights on (topological) *-algebras is defined and studied. The primary purpose is to define the notions of admissibility and approximate admissibility of weights, and to investigate when a weight is admissible or approximately admissible. The results obtained are applied to vector weights and tracial weight on unbounded operator algebras, as well as to weights on smooth subalgebras of a C * -algebra.

Köthe spaces and topological algebra with bases

Nuclear Köthe sequence spaceλ(P) its crossdualλ(P)x and their non-nuclear variants are examined as topological algebras. Modelling on them, a general theory of nuclear topological algebras with orthogonal basis is developed. As a by-product, abstract characterizations of sequence algebras ℓ∞ andc0 are obtained. In a topological algebra set-up, an abstract Grothendieck-Pietsch nuclearity criterion is developed.

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous*-automorphisms of a Frechet*-algebraA, it is shown that the enveloping ↡-C*-algebraE(S(ℝ, A∞, α)) of the smooth Schwartz crossed productS(ℝ,A∞, α) of the Frechet algebra A∞ of C∞-elements ofA is isomorphic to the Σ-C*-crossed productC*(ℝ,E(A), α) of the enveloping Σ-C*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK*(S(ℝ, A∞, α)) =K*(C*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC*-algebra defined by densely defined differential seminorms is given.

SECOND-ORDER NONCOMMUTATIVE DIFFERENTIAL AND LIPSCHITZ STRUCTURES DEFINED BY A CLOSED SYMMETRIC OPERATOR

The Banach

Topological algebras withC*-enveloping algebras

^{\ast }

Sufficiency and strong commutants in quantum probability theory

-operator algebras, exhibiting the second-order noncommutative differential structure and the noncommutative Lipschitz structure, that are determined by the unbounded derivation and induced by a closed symmetric operator in a Hilbert space, are explored.

UNBOUNDED C -SEMINORMS AND UNBOUNDED C -SPECTRAL ALGEBRAS

LetA be a complete topological *algebra which is an inverse limit of Banach *algebras. The (unique) enveloping algebraE(A) ofA, providing a solution of the universal problem for continuous representations ofA into bounded Hilbert space operators, is known to be an inverse limit ofC*-algebras. It is shown thatS(A) is aC*-algebra iffA admits greatest continuousC*-seminorm iff the continuous states (respectively, continuous extreme states) constitute an equicontinuous set. AQ-algebra (i.e., one whose quasiregular elements form an open set)A hasC*-enveloping algebra. There exists (i) a Frechet algebra with C*-enveloping algebra that is not aQ-algebra under any topology and (ii) a non-Q spectrally bounded algebra withC*-enveloping algebra.A hermitian algebra withC*-enveloping algebra turns out to be aQ-algebra. The property of havingC*-enveloping algebra is preserved by projective tensor products and completed quotients, but not by taking closed subalgebras. Several examples of topological algebras withC*-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 Köthe sequence algebras of infinite type. The envelopingC*-algebra of a hermitian topological algebra with an orthogonal basis is isomorphic to theC*-algebrac0 of all null sequences.

An intrinsic characterization of pro-C*-algebras and its applications

A probability algebra (A, *, ω) consisting of a*algebraA with a faithful state ω provides a framework for an unbounded noncommutative probability theory. A characterization of symmetric probability algebra is obtained in terms of an unbounded strong commutant of the left regular representation ofA. Existence of coarse-graining is established for states that are absolutely continuous or continuous in the induced topology. Sufficiency of a*subalgebra relative to a family of states is discussed in terms of noncommutative Radon-Nikodym derivatives (a form of Halmos-Savage theorem), and is applied to couple of examples (including the canonical algebra of one degree of freedom for Heisenberg commutation relation) to obtain unbounded analogues of sufficiency results known in probability theory over a von Neumann algebra.