# Operator theory seminar

Title: Refining topologies on groupoids

Speaker: Ian Putnam, University of Victoria

Date and time:
14 Oct 2020,
3:30pm -
4:50pm

Location: ECS 125 and Zoom

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Abstract: The construction of C*-algebras from groupoids has been a fruitful source of examples of C*-algebras and also created an important link between C*-algebras and dynamical systems. If one takes a compact Hausdorff space which is separable, one can find a dense countable subset. Of course, the relative topology on this set is a disaster (not locally compact), but it has a finer topology, namely the discrete topology which is nicer. At the level of C*-algebras, the continuous functions on the original space lie in the multiplier algebra of the continuous functions vanishing at infinity of the subset. That is a fancy consequence of a simple fact: a continuous, bounded function on a topological space retains both of those features when the topology is replaced by a finer one. I will discuss a generalization of this fact to groupoids. I will explain the relevant terms, such as groupoid C*-algebra, multiplier algebra and give some examples of when such results are useful.

Please contact the organizer if you need the Zoom link.

Title: Noncommutative Geometry of lattices (part 2 of 2)

Speaker: Heath Emerson, University of Victoria

Date and time:
30 Sep 2020,
3:30pm -
4:50pm

Location: ECS 125 and via Zoom

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Abstract: A lattice G is a discrete subgroup of a
real vector space V with compact quotient. I am going
to show how to study the space of lattices using
the techniques of Noncommutative Geometry. The
idea is based on the properties of the harmonic
oscillator H, which gives rise to a suitable spectral
triple with a nice meromorphic extension property.
Applications of the Residue Index Formula give rise
to various topological invariants of a lattice which
are expressible in terms of cyclic cocycles. I will
also discuss a conjectured KK-duality for lattices,
which predicts in particular
a 0-dimensional duality between
the irrational rotation algebras A_h and A_{1/h},
based on Dirac-Schrodinger operators, rather than
on Connes' twisted Dolbeault operator.

Title: Noncommutative Geometry of lattices (part 1 of 2)

Speaker: Heath Emerson, University of Victoria

Date and time:
23 Sep 2020,
3:30pm -
4:50pm

Location: ECS 125

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Abstract: A lattice G is a discrete subgroup of a
real vector space V with compact quotient. I am going
to show how to study the space of lattices using
the techniques of Noncommutative Geometry. The
idea is based on the properties of the harmonic
oscillator H, which gives rise to a suitable spectral
triple with a nice meromorphic extension property.
Applications of the Residue Index Formula give rise
to various topological invariants of a lattice which
are expressible in terms of cyclic cocycles. I will
also discuss a conjectured KK-duality for lattices,
which predicts in particular
a 0-dimensional duality between
the irrational rotation algebras A_h and A_{1/h},
based on Dirac-Schrodinger operators, rather than
on Connes' twisted Dolbeault operator.

Title: Relative K-theory for C*-algebras

Speaker: Mitchell Haslehurst, University of Victoria

Date and time:
16 Sep 2020,
3:30pm -
4:50pm

Location: ECS 125

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Abstract: Using an approach to K-theory due to Karoubi, one may construct the relative K-theory between two C*-algebras which are related via a *-homomorphism. This construction generalizes that of the standard relative groups of a unital C*-algebra and an ideal therein, the noncommutative version of K^{*(X,Y)} of a compact pair (X,Y) in topological K-theory. As in the standard theory, a long exact sequence is obtained, and the relative groups satisfy Bott periodicity which causes the long exact sequence to collapse to a six-term exact sequence. I will present a portrait of the relative groups, describe the exact sequences that arise, and discuss examples.