Volume 101, Issue 3
December 1991, pages 149-230
pp 149-167 December 1991
Complete positivity, tensor products and C^{*}-nuclearity for inverse limits of C^{*}-algebras
Subhash J Bhatt Dinesh J Karia
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.
pp 169-177 December 1991
A multiplier theorem for the sublaplacian on the Heisenberg group
A multiplier theorem for the sublaplacian on the Heisenberg group is proved using Littlewood-Paley-Stein theory ofg-functions.
pp 179-181 December 1991
A note on the multidimensional Weyl fractional operator
The purpose of the present paper is to establish a connection theorem involving the multidimensional Weyl fractional operator and the classical multidimensional Laplace transform. This provides an extension of a result due to Raina and Koul [6].
pp 183-194 December 1991
An analysis of the scattering of horizontally polarized shear wave by a semi-infinite crack running with uniform velocity along the interface of two dissimilar semi-infinite elastic media has been carried out. The mixed boundary value problem has been solved completely by the Wiener-Hopf technique. The effect of different values of the material parameter, the angle of incidence of incident wave and the crack propagation velocity on the stress intensity factor have been illustrated graphically.
pp 195-198 December 1991
Continuous dependence for integrodifferential equations with infinite delay
Continuous dependence for integrodifferential equation with infinite delay$$\begin{gathered} \dot x = h(t,x) + \int_{ \sim \infty }^t {q(t,s,x(s))ds} + F(t,x(t),Sx(t))t \geqslant 0 \hfill \\ x(t) = \Phi (t) \hfill \\ \end{gathered} $$ where$$Sx(t) = \int_{ \sim \infty }^t {k(t,s,x(s))} ds$$ is studied under the assumption of existence of unique solution.
pp 199-213 December 1991
Curves on threefolds with trivial canonical bundle
C H Clemens has shown that homologically trivial codimension two cycles on a general hypersurface of degree five and dimension three form a subgroup of infinite rank inside the intermediate jacobian. We generalize this to other complete intersection threefolds with trivial canonical bundle.
pp 215-218 December 1991
Self-dual connections, hyperbolic metrics and harmonic mappings on Riemann surfaces
The Sampson-Wolf model of Teichmüller space (using harmonic mappings) is shown to be exactly the same as the more recent Hitchin model (utilizing self-dual connections). Indeed, it is noted how the self-duality equations become the harmonicity equations. An interpretation of the modular group action in this model is mentioned.
pp 219-226 December 1991
Subordination properties of certain integrals
Let B_{1}(μ,β) denote the class of functions f(z)= z + a_{2}z^{2}+ h+ a_{n}z^{m}+… that are analytic in the unit disc Δ and satisfy the condition Ref′(z)(f(z)/z)^{⧎-1} > β, zεΔ, for some ⧎>0 and β< 1. Denote by S*(0)for B_{1}(0,0). For μ andc such thatc > -μ, letF =I_{gm,c}(f) be defined by$$F(z) = \left[ {\frac{{\mu + c}}{{Z^c }}\int_0^z {f^\mu (t)} t^{c - 1} dt} \right]^{1/\mu } ,z \in \Delta .$$ The author considers the following two types of problems: (i) To find conditions on ⧎,c and ρ > 0 so thatfεB_{1}(μ -ρ) implies I_{μ},c(f<εS*(0); (ii) To determine the range of μ and δ > 0 so that fεB_{1} (μ -δ) impliesI_{μο}(f)εS*(0); We also prove that if / satisfies Re(f′(z) +zf′’(z)) > 0 in Δ then the nth partial sumf_{n} off satisfiesf_{n}(z)/z≺ -1 -(2/z)log(l -z)in Δ. Here, ≺ denotes the subordination of analytic functions with univalent analytic functions. As applications we also give few examples.
pp 227-230 December 1991
The seminormality property of circular complexes
In this paper we prove that the ring R[X, Y]/(X. Y, Y.X) is seminormal, whereR is a Cohen-Macaulay normal domain andX, Y are matrices of indeterminates.
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