A duality for Moore group

Abstract

We suggest a new generalization of Pontryagin duality from the category of Abelian locally compact groups to a category which includes all Moore groups, i.e. groups whose irreducible representations are finite-dimensional. Objects in this category are pro-C*-algebras with a structure of Hopf algebras (in the strict algebraic sense) with respect to a certain topological tensor product.2010 Mathematics Subject Classification. 22D35, 22D25, 16T05.

Continuous and smooth envelopes of topological algebras

“…Later Yu. N. Kuznetsova obtained analogous results in [41], where she constructed a variant of duality theory in topology with homomorphisms into C * -algebras as observation tools. Again, up to later correction in [4] and in this paper, the results of [41] can be interpreted as a way for categorical construction of topology.…”

“…The similarity with the second column is that we describe here some already published facts, namely the results of the Yu. N. Kuznetsova paper [41]. But the difference is that we retell here the material of [41] in detail, with formulations and proofs.…”

C ∞(M) as a smooth envelope of its subalgebras

“…A detailed proof of this result is presented in [5]. The author plans to use this construction further in the generalization of duality theory in differential geometry by analogy with the mentioned duality theories in complex analysis [3] and in topology [10].…”

“…In the theory of involutive topological algebras, there is a similar construction which is called the pro-C * -envelope, which turns a topological algebra A into the projective limit Env C A of its C * -quotient algebras (see [7], [9], [13], [10]). The envelope Env C A can be treated as the continuous algebra 2 nearest to A on the outside.…”

“…Both envelopes are used in generalizations of Pontryagin duality from commutative locally compact groups to the classes of noncommutative groups, in particular quantum groups ( [3], [10]). Philosophically these theories are in some sense projections of functional analysis into the "noncommutative complex analysis" (considered as a geometric interpretation of the theory of Banach algebras, see [14]) and into the "noncommutative topology" (considered as a geometric interpretation of the theory of C * -algebras, see [6]).…”