Type I -algebras of real rank zero

We show that a separable -algebra of type I has real rank zero if and only if where is a modified dimension. We also show that a separable -algebra of type I has real rank zero if and only if it is an AF-algebra.

     

A Hilbert *-module method for Morita equivalence of twisted crossed products

We present a new proof for Morita equivalence of twisted crossed products by coactions within the abstract context of crossed products of Hilbert -modules. In this context we are free from representing all -algebras and Hilbert -modules on Hilbert spaces.

     

Bilocal derivations of standard operator algebras

In this paper, we shall show the following two results: (1) Let be a standard operator algebra with , if is a linear mapping on which satisfies that maps into for all , then is of the form for some in . (2) Let be a Hilbert space, if is a norm-continuous linear mapping on which satisfies that maps into for all self-adjoint projection in , then is of the form for some in .

     

Fixed points of the mapping class group in the moduli spaces

Let be a compact oriented surface with or without boundary components. In this note we prove that if then there exist infinitely many integers such that there is a point in the moduli space of irreducible flat connections on which is fixed by any orientation preserving diffeomorphism of . Secondly we prove that for each orientation preserving diffeomorphism of and each there is some such that has a fixed point in the moduli space of irreducible flat connections on . Thirdly we prove that for all there exists an integer such that the 'th power of any diffeomorphism fixes a certain point in the moduli space of irreducible flat connections on .

     

A Strict Version of the Non-commutative Urysohn Lemma

Given a pair , of -commuting, hereditary -subalgebras of a unital -algebra , such that is -unital and , there is an element in , with , such that is strictly positive in and is strictly positive in in . Moreover, is strictly positive in in .

     

Fixed point property and normal structure for Banach spaces associated to locally compact groups

In this paper we investigate when various Banach spaces associated to a locally compact group have the fixed point property for nonexpansive mappings or normal structure. We give sufficient conditions and some necessary conditions about for the Fourier and Fourier-Stieltjes algebras to have the fixed point property. We also show that if a -algebra has the fixed point property then for any normal element of , the spectrum is countable and that the group -algebra has weak normal structure if and only if is finite.

     

Invariance of spectrum for representations of -algebras on Banach spaces

Let be a Banach space, a unital -algebra, and an injective, unital homomorphism. Suppose that there exists a function such that, for all , and all ,

(a) ,

(b) ,

(c) .
Then for all , the spectrum of in equals the spectrum of as a bounded linear operator on . If satisfies an additional requirement and is a -algebra, then the Taylor spectrum of a commuting -tuple of elements of equals the Taylor spectrum of the -tuple in the algebra of bounded operators on . Special cases of these results are (i) if is a closed subspace of a unital -algebra which contains as a unital -subalgebra such that , and only if , then for each , the spectrum of in is the same as the spectrum of left multiplication by on ; (ii) if is a unital -algebra and is an essential closed left ideal in , then an element of is invertible if and only if left multiplication by on is bijective; and (iii) if is a -algebra, is a Hilbert -module, and is an adjointable module map on , then the spectrum of in the -algebra of adjointable operators on is the same as the spectrum of as a bounded operator on . If the algebra of adjointable operators on is a -algebra, then the Taylor spectrum of a commuting -tuple of adjointable operators on is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on .

     

The structure of hypersurfaces with some curvature conditions

Let be a hypersurface in , and let denote the mean curvature and the scalar curvature of respectively. We show that if is compact and , then is diffeomorphic to . Also we prove that if is complete, is constant and , then is or or .

     

Continuous Fell bundles associated to measurable twisted actions

Given a twisted action of a second-countable, locally compact group on a separable -algebra , we prove the existence of a topology on making it a Fell bundle, whose cross sectional -algebra is isomorphic to the Busby-Smith-Packer-Raeburn crossed product.

     

Commutativity criteria for Banach -algebras

Let be a Banach -algebra with an identity. Necessary and sufficient conditions are given for to be commutative modulo its -radical and for to be commutative if has a faithful -representation as operators on a Hilbert space.

     

An invariance of domain result for multi-valued maximal monotone operators whose domains do not necessarily contain any open sets

Let be a real, reflexive, locally uniformly convex Banach space with locally uniformly convex. Let be a maximal monotone operator and open and bounded. Assume that is pathwise connected and such that and Then If, moreover, is of type () on then may be replaced above by The significance of this result lies in the fact that it holds for multi-valued mappings which do not have to satisfy It has also been used in this paper in order to establish a general ``invariance of domain' result for maximal monotone operators, and may be applied to a greater variety of problems involving partial differential equations. No degree theory has been used. In addition to the above, necessary and sufficient conditions are given for the existence of a zero (in an open and bounded set ) of a completely continuous perturbation of a maximal monotone operator such that is locally monotone on

     

On a class of subalgebras of and the intersection of their free maximal ideals

Let be a Tychonoff space and a subalgebra of containing . Suppose that is the set of all functions in with compact support. Kohls has shown that is precisely the intersection of all the free ideals in or in . In this paper we have proved the validity of this result for the algebra . Gillman and Jerison have proved that for a realcompact space , is the intersection of all the free maximal ideals in . In this paper we have proved that this result does not hold for the algebra , in general. However we have furnished a characterisation of the elements that belong to all the free maximal ideals in . The paper terminates by showing that for any realcompact space , there exists in some sense a minimal algebra for which becomes -compact. This answers a question raised by Redlin and Watson in 1987. But it is still unsettled whether such a minimal algebra exists with respect to set inclusion.

     

Less Saturated Ideals

We prove the following:

(1)

If is weakly inaccessible then is not -saturated.

(2)

If is weakly inaccessible and is regular then is not -saturated.

(3)

If is singular then is not -saturated.

Combining this with previous results of Shelah, one obtains the following:

(A)

If then is not -saturated.

(B)

If then is not -saturated.

     

On functions arising as potentials on spaces of homogeneous type

On a space of homogeneous type we consider functions in , , which are potentials of order of functions. We show that these functions belong to the class of smooth functions of Calderón-Scott. This result has applications to tangential convergence.

     

Beals-Cordes-type characterizations of pseudodifferential operators

We show that, if is the representation of on given by (2.11), and is a bounded operator on , then belongs to if and only if

is a function on with values in the Banach space .

     

Factorisation in nest algebras

We give a necessary and sufficient condition on an operator for the existence of an operator in the nest algebra of a continuous nest satisfying (resp. . We also characterise the operators in which have the following property: For every continuous nest there exists an operator in satisfying (resp. .

     

Gröbner duality and multiple points in linearly general position

It is proved for each , , that a primary -dimensional scheme in of degree in linearly general position lies in a rational normal scroll of dimension .

     

Finite linear groups and theorems of Minkowski and Schur

Let be a finite group with a faithful rational valued character of degree . A theorem of I. Schur gives a bound for the order of in terms of , generalizing an earlier result of H. Minkowski who showed that the same bound holds if . This note contains strengthened versions of these results which in particular show that a -subgroup of of maximum possible order contains a reflection.

     

A note on GK dimension of skew polynomial extensions

Let be a finitely generated commutative domain over an algebraically closed field , an algebra endomorphism of , and a -derivation of . Then if and only if is locally algebraic in the sense that every finite dimensional subspace of is contained in a finite dimensional -stable subspace.

Similarly, if is a finitely generated field over , a -endomorphism of , and a -derivation of , then if and only if is an automorphism of finite order.

     

The expected value of the number of real zeros of a random sum of Legendre polynomials

It is known that the expected number of zeros in the interval of the sum , in which is the normalized Legendre polynomial of degree and the coefficients are independent normally distributed random variables with mean 0 and variance 1, is asymptotic to for large . We improve this result and show that this expected number is for any positive .