Codd defined the relational algebra [3, 4] as the algebra with operations projection, join, restriction, union and difference. His projection operator can drop, permute and repeat columns of a relation. This permuting and repeating of columns does not really add expressive power to the relational algebra. Indeed, using the join operation, one can rewrite any relational algebra expression into a...

Let X be a Banach space and T be a bounded linear operator from X to itself (T ∈ B(X).) An operator D ∈ B(X) is a Drazin inverse of T if TD = DT , D = TD and T k = T D for some nonnegative integer k. In this paper we look at the Jörgens algebra, an algebra of operators on a dual system, and characterise when an operator in that algebra has a Drazin inverse that is also in the algebra. This resu...

We consider how a vertex operator algebra can be extended to an abelian intertwining algebra by a family of weak twisted modules which are simple currents associated with semisimple weight one primary vectors. In the case that the extension is again a vertex operator algebra, the rationality of the extended algebra is discussed. These results are applied to affine Kac-Moody algebras in order to...

We introduce notions of open-string vertex algebra, conformal open-string vertex algebra and variants of these notions. These are " open-string-theoretic, " " noncommutative " generalizations of the notions of vertex algebra and of conformal vertex algebra. Given an open-string vertex algebra, we show that there exists a vertex algebra, which we call the " meromorphic center, " inside the origi...

Let G be a complex, linear algebraic group acting on an algebraic space X . The purpose of this paper is to prove a Riemann-Roch theorem (Theorem 5.5) which gives a description of the completion of the equivariant Grothendieck group G0(G,X)⊗ C at any maximal ideal of the representation ring R(G) ⊗ C in terms of equivariant cycles. The main new technique for proving this theorem is our non-abeli...

We show that the matricial norms of a non-unital operator algebra determine those of the algebra obtained by adjoining a unit to it. As applications , we classify two-dimensional unital operator algebras and show that the algebra of bounded holomorphic functions on a strongly pseudoconvex domain has a contractive representation that is not completely contractive.

In this contribution, I explain the general principles of twisted modules for vertex operator algebras in its powerful formulation using formal series, and derive new general relations satisfied by twisted and non-twisted vertex operators. I prove new “equivalence” and “construction” theorems, identifying a very restricted set of sufficient conditions in order to have a twisted module for a ver...

In this note we announce some new methods and results in the theory of nonnormal Hilbert space operators and nonself ad joint operator algebras. A main difficulty in the subject has been the apparent absence of relations between, say, a nonself ad joint algebra of operators and its generated C*-algebra. For example, given full information about the norm-closed algebra P(T) generated by all poly...