• Volume 108, Issue 3

October 1998,   pages  217-303

• Rational curves on moduli spaces of vector bundles

We identify the spaces Homi(ℙ1,M) fori = 1, 2, whereM is the moduli space of vector bundles of rank 2 and determinant isomorphic to$$\mathcal{O}(x_0 )$$,x0X, on a compact Riemann surface of genusg ≥ 2.

• The Chow ring of a singular surface

We construct the Chow ringCH*(X) =CH0(X)⊕CH1(X)⊕CH2(X) of a reduced, quasi-projective surfaceX, together with Chern class mapsci :K0(X) → CHi(X), with the usual properties. As a consequence, we show that the cycle mapCH2(X)→ F0K0(X) is an isomorphism. Our treatment is greatly influenced by an unpublished 1983 preprint of Levine’s, but is much simpler, since we deal only with surfaces.

• AbsoluteN-summability of the series conjugate to a Fourier series

The object of the paper is to study the absolute N-summability of the series conjugate to a Fourier series, generalising a known result.

• Multidimensional modified fractional calculus operators involving a general class of polynomials

In the present work, we introduce and study essentially a class of multi-dimensional modified fractional calculus operators involving a general class of polynomials in the kernel. These operators are considered in the space of functionsMγ(R+n). Some mapping properties and fractional differential formulas are obtained. Also images of some elementary and special functions are established.

• Stinespring representability and Kadison’s Schwarz inequality in non-unital Banach star algebras and applications

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 &gt; 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 unitizationAeof 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.

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