Non Commutative Topology and Local Structure of Operator Algebras

Starting with a $W^{*}$-algebra $M$ we use the inverse system obtained by cutting down $M$ by its (central) projections to define an inverse limit of $W^{*}$-algebras, and show that how this pro-$W^{*}$-algebra encodes the local structure of $M$. For the $C^{*}$-algebras we do the same thing using their atomic enveloping $W^{*}$-algebra . We investigate the relation of these ideas to the Akemann-Giles-Kummer non commutative topology. Finally we use these ideas to look at the local structure of Kac algebras.Comment: 23 page

Module Amenability for Semigroup Algebras

We extend the concept of amenability of a Banach algebra $A$ to the case that there is an extra $\mathfrak A$-module structure on $A$, and show that when $S$ is an inverse semigroup with subsemigroup $E$ of idempotents, then $A=\ell^1(S)$ as a Banach module over $\mathfrak A=\ell^1(E)$ is module amenable iff $S$ is amenable. When $S$ is a discrete group, $\ell^1(E)=\mathbb C$ and this is just the celebrated Johnson's theorem.Comment: 12 page

Tannaka-Krein duality for compact groupoids III, duality theory

In a series of papers, we have shown that from the \repn theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we introduce the Tannaka groupoid of a compact groupoid and show how to recover the original groupoid from its Tannaka groupoid.Comment: 9 page

On generalized Stone's Theorem

It is known that the generator of a strictly continuous one parameter unitary group in the multiplier algebra of a $C^{*}$-algebra is affiliated to that $C^{*}$-algebra . We show that under natural non degeneracy conditions, this self adjoint unbounded operator lies indeed in the (unbounded) multiplier algebra of the Pedersen's ideal of the $C^{*}$-algebra.Comment: 7 pages, no figure

Uniform Closure of Dual Banach Algebras

We give a characterization of the ''uniform closure'' of the dual of a $C^{*}$-algebra. Some applications in harmonic analysis are given.Comment: 6 page

Tannaka-Krein duality for compact groupoids II, Fourier transform

In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study the Fourier and Fourier-Plancherel transforms and prove the Plancherel theorem for compact groupoids. We also study the central functions in the algebra of square integrable functions on the isotropy groups.Comment: 14 page

Locally Compact Pro-C∗C^{*}-Algebras

Let $X$ be a locally compact non compact Hausdorff topological space. Consider the algebras $C(X)$, $C_b(X)$, $C_0(X)$, and $C_{00}(X)$ of respectively arbitrary, bounded, vanishing at infinity, and compactly supported continuous functions on $X$. From these, the second and third are $C^{*}$-algebras, the forth is a normed algebra, where as the first is only a topological algebra. The interesting fact about these algebras is that if one of them is given, the rest can be obtained using functional analysis tools. For instance, given the $C^{*}$-algebra $C_0(X)$, one can get the other three algebras by $C_{00}(X)=K(C_0(X))$, $C_b(X)=M(C_0(X))$, $C(X)=\Gamma(K(C_0(X)))$, that is by forming the Pedersen's ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen's ideal, respectively.In this article we consider the possibility of these transitions for general $C^{*}$-algebra . The difficult part is to start with a pro_$C^{*}$-algebra $A$ and to construct a $C^{*}$-algebra $A_0$ such that $A=\Gamma(K(A_0))$. The pro-$C^{*}$-algebras for which this is possible are called {\it locally compact} and we have characterized them using a concept similar to approximate identities.Comment: 21 pages, no figure

Control-target inversion property on Abelian groups

We show that the quantum Fourier transform on finite fields used to solve query problems is a special case of the usual quantum Fourier transform on finite abelian groups. We show that the control/target inversion property holds in general. We apply this to get a sharp query complexity separation between classical and quantum algorithms for a hidden homomorphism problem on finite Abelian groups.Comment: 9 pages, no figures, MSC: 81P6

Extensions and Dilations of module maps

We study completely positive module maps on $C^{*}$-algebras which are $C^*$-module over another $C^*$-algebra with compatible actions. We extend several well known dilation and extension results to this setup, including the Stinespring dilation theorem and Wittstock, Arveson, and Voiculescu extension theorems.Comment: 13 page

Quantum error-correction codes on Abelian groups

We prove a general form of bit flip formula for the quantum Fourier transform on finite abelian groups and use it to encode some general CSS codes on these groups.Comment: 13 pages, no figures, MSC: 81P6