[Paper Review] Operator algebras associated to integral domains

Benton L. Duncan|arXiv (Cornell University)|Jan 5, 2010

Advanced Operator Algebra Research4 references3 citations

TL;DR

This paper introduces a new class of nonselfadjoint operator algebras associated to integral domains using semicrossed product constructions, generalizing Cuntz and Li's $C^*$-algebraic approach. It shows these algebras are distinct from Peters' semicrossed products and establishes that unitary representations of the algebra factor through a regular unitary representation on the field of quotients, providing a $C^*$-envelope structure.

ABSTRACT

We study operator algebras associated to integral domains. In particular, with respect to a set of natural identities we look at the possible nonselfadjoint operator algebras which encode the ring structure of an integral domain. We show that these algebras give a new class of examples of semicrossed products by discrete semigroups. We investigate the structure of these algebras together with a particular class of representations.

Motivation & Objective

• To develop a nonselfadjoint operator algebra framework for integral domains using semicrossed product techniques.
• To generalize Cuntz and Li's $C^*$-algebra construction by relaxing identities to only ring-theoretic constraints.
• To show that the resulting algebras are not isomorphic to Peters' semicrossed products when the domain is not a field.
• To analyze unitary representations and prove they factor through a regular unitary representation on the field of quotients.
• To establish a $C^*$-envelope structure via a canonical $*$-representation on $C^*(Q(R))$.

Proposed method

• Define the universal operator algebra $A(R)$ generated by isometries $\{s_r\}$ and unitaries $\{u^n\}$ satisfying $s_r s_t = s_{rt}$, $u^n u^m = u^{n+m}$, and $u^n s_r = s_r u^{rn}$.
• Construct the regular isometric representation on $\ell^2(R)$ and the regular unitary representation on $\ell^2(Q(R))$, with the latter extending the former.
• Use the Pontryagin dual $\widehat{R,+}$ to analyze the abelian part of the algebra via $C_0(\widehat{R,+})$.
• Prove that irreducible representations of $A(R)$ correspond to characters on $\widehat{R,+}$ and use nest algebra techniques to show non-isomorphism to standard semicrossed products.
• Define a canonical completely contractive representation $i: A(R) \to C^*(Q(R))$ and extend it to a $*$-representation $\tau_\pi$ on $C^*(Q(R))$ for any unitary representation $\pi$.
• Use the relations $T_p T_q^* = T_q^* T_p$ and $V^n T_q = T_q V^{nq}$ to verify that the lifted operators satisfy the $C^*$-relations for $Q(R)$.

Experimental results

Research questions

• RQ1 Can operator algebras encoding the ring structure of an integral domain be constructed using nonselfadjoint methods that avoid requiring automorphisms?
• RQ2 How do these algebras differ from Peters' semicrossed products when the integral domain is not a field?
• RQ3 Do unitary representations of $A(R)$ factor through a representation on the field of quotients $Q(R)$?
• RQ4 Is the $C^*$-envelope of $A(R)$ isomorphic to a crossed product $C^*(Q(R)) \rtimes \mathbb{Z}^+$?
• RQ5 What is the role of the regular unitary representation in characterizing the structure of $A(R)$?

Key findings

• The universal operator algebra $A(R)$ associated to an integral domain $R$ is not isomorphic to the semicrossed product algebra of Peters when $R$ is a unique factorization domain.
• Every unitary representation $\pi$ of $A(R)$ factors through a $*$-representation $\tau_\pi$ on $C^*(Q(R))$, with $\tau_\pi \circ i = \pi$, showing that $A(R)$ is completely isometrically embedded into $C^*(Q(R))$.
• The canonical representation $i: A(R) \to C^*(Q(R))$ is completely contractive and extends to a $*$-representation on the $C^*$-algebra of the field of quotients.
• The algebra $A(R)$ is not a semicrossed product in the sense of Peters when $R$ is a UFD, due to the nontrivial structure of the multiplicative monoid of irreducibles.
• The $C^*$-envelope of $A(R)$ is not necessarily isomorphic to a crossed product unless the regular representation dilates to a unitary representation, which is not generally true.
• The regular unitary representation on $\ell^2(Q(R))$ restricts to the regular isometric representation on $\ell^2(R)$, and the latter is invariant under the action of the extended operators.