let $a$ be a $c^*$-algebra and $e$ be a left hilbert $a$-module. in this paper we define a product on $e$ that making it into a banach algebra and show that under the certain conditions $e$  is arens regular. we also study the relationship between derivations of $a$ and $e$.

After discussing some basic facts about generalized module maps, we use the representation theory of the algebra Ba(E) of adjointable operators on a Hilbert B–module E to show that the quotient of the group of generalized unitaries on E and its normal subgroup of unitaries on E is a subgroup of the group of automorphisms of the range ideal BE of E in B. We determine the kernel of the canonical ...

Let PN (R) be the space of all real polynomials in N variables with the usual inner product 〈 , 〉 on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form representing this inner product on the space of coefficient vectors of all polynomials in PN (R) of degree ≤ M . We exhibit two applications of this formula. First, given a...

Interacting quantum fields on spacetimes containing regions of closed time-like curves (CTCs) are subject to a non-unitary evolution X. Recently, a prescription has been proposed, which restores unitarity of the evolution by modifying the inner product on the final Hilbert space. We give a rigorous description of this proposal and note an operational problem which arises when one considers the ...

We characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left A-modules. We show that the category of right (left) comodules over A, relative to this coproduct, is isomorphic to the category of right (left) modules. This isomorphism enables a reformulation of the cotensor product of Eilenberg and Moore as a functor of modules rather than com...

Here, and throughout the paper, we useD to denote the α-order derivative, where α is a multi-index and we are using standard multi-index notation. Moreover, γα := |α|!/α! denotes the polynomial coefficient. [The naturality of this choice of the Sobolev inner product will be pointed out and discussed below.] We define the Sobolev space W (Ω) to be the closure of C(Ω̄) with respect to the above in...

This paper proposes a framework dedicated to the construction of what we call time elastic inner products allowing one to embed sets of non-uniformly sampled multivariate time series of varying lengths into vector space structures. This framework is based on a recursive definition that covers the case of multiple embedded time elastic dimensions. We prove that such inner products exist in our f...

Bayesian networks have become one of the major models used for statistical inference. We study the question whether the decisions computed by a Bayesian network can be represented within a low-dimensional inner product space. We focus on two-label classification tasks over the Boolean domain. As main results we establish upper and lower bounds on the dimension of the inner product space for Bay...

For two normed algebras $A$ and $B$ with the character space   $bigtriangleup(B)neq emptyset$  and a left $B-$module $X,$  a certain class of bounded linear maps from $A$ into $X$ is introduced. We set $CMH_B(A, X)$  as the set of all non-zero $B-$character module homomorphisms from $A$ into $X$. In the case where $bigtriangleup(B)=lbrace varphirbrace$ then $CMH_B(A, X)bigcup lbrace 0rbrace$ is...