We generalize the Serre-Swan theorem to non-commutative C∗algebras. For a Hilbert C∗-module X over a C∗-algebra A, we introduce a hermitian vector bundle EX associated to X . We show that there is a linear subspace ΓX of the space of all holomorphic sections of EX and a flat connection D on EX with the following properties: (i) ΓX is a Hilbert A-module with the action of A defined by D, (ii) th...

let a be a banach algebra and e be a banach a-bimodule then s = a  e, the l1-direct sum of a and e becomes a module extension banach algebra when equipped with the algebras product (a; x):(a′; x′) = (aa′; a:x′ + x:a′). in this paper, we investigate △-amenability for these banach algebras and we show that for discrete inverse semigroup s with the set of idempotents es, the module extension bana...

In general, a tensor is a multilinear transformation defined over an underlying finite dimensional vector space. In this brief introduction, tensor spaces of all integral orders will defined inductively. Initially the underlying vector space, V, will be assumed to be an inner product space in order to simplify the discussion. Subsequently, the presentation will be generalized to vector spaces w...

with a symmetric and positive definite operator A defined on a Hilbert space V with dimV = N and its preconditioned version PCG. We derive the CG from the projection in A-inner product point of view. We shall use (·, ·) for the standard inner product and (·, ·)A for the inner product introduced by the SPD operator A. When we say ‘orthogonal’ vectors, we refer to the default (·, ·) inner product...

In this paper, a new definition of numerical radius for adjointable operators in Hilbert -module space will be introduced. We also give a new proof of numerical radius inequalities for Hilbert space operators.