theory of graph algebras

We present an explicit version of Berger, Coburn and Lebow's classification result for pure pairs of commuting isometries in the sense of an explicit recipe for constructing pairs of commuting isometric multipliers with precise coefficients. We describe a complete set of (joint) unitary invariants and compare the Berger, Coburn and Lebow's representations with other natural analytic representations of pure pairs of commuting isometries. Finally, we study the defect operators of pairs of commuting isometries.

Pairs of commuting isometries - I

Nuclearity of semigroup C⁎-algebras

The structure of doubly non-commuting isometries

We prove a functoriality result for the full C*-algebras of right-LCM monoids with respect to monoid inclusions that are closed under factorization and preserve orthogonality, and use this to show that if a right-LCM monoid is amenable in the sense of Nica, then so are its submonoids. As applications, we complete the classification of Artin monoids with respect to Nica amenability by showing that only the right-angled ones are amenable in the sense of Nica and we show that the Nica amenability of a graph product of right-LCM semigroups is inherited by the factors.

Amenability and functoriality of right-LCM semigroup C*-algebras

The Baumslag-Solitar group is an example of an HNN extension. Spielberg showed that it has a natural positive cone, and that it is then a quasi-lattice ordered group in the sense of Nica. We give conditions for an HNN extension of a quasi-lattice ordered group $(G,P)$ to be quasi-lattice ordered. In that case, if $(G,P)$ is amenable as a quasi-lattice ordered group, then so is the HNN extension.

HNN extensions of quasi-lattice ordered groups and their operator algebras

Structure theorems for star-commuting power partial isometries

We give a new construction of a C*-algebra from a cancellative semigroup $P$ via partial isometric representations, generalising the construction from the second named author's thesis. We then study our construction in detail for the special case when $P$ is an LCM semigroup. In this case we realize our algebras as inverse semigroup algebras and groupoid algebras, and apply our construction to free semigroups and Zappa-Sz\'ep products associated to self-similar groups.

The C*-algebra of a cancellative semigroup

We give a new construction of a C*-algebra from a cancellative semigroup $P$ via partial isometric representations, generalising the construction from the second named author's thesis. We then study our construction in detail for the special case when $P$ is an LCM semigroup. In this case we realize our algebras as inverse semigroup algebras and groupoid algebras, and apply our construction to free semigroups and Zappa-Szep products associated to self-similar groups.

C*-algebras from partial isometric representations of LCM semigroups

We give a new formulation and proof of a theorem of Halmos and Wallen on the structure of power partial isometries on Hilbert space. We then use this theorem to give a structure theorem for a finite set of partial isometries which star-commute: each operator commutes with the others and with their adjoints.

Ilija Tolich

Papers

HNN extensions of quasi-lattice ordered groups and their operator algebras

Structure theorems for star-commuting power partial isometries

The C*-algebra of a cancellative semigroup

C*-algebras from partial isometric representations of LCM semigroups

Structure theorems for star-commuting power partial isometries