A Finite Dimensional Introduction to Operator Algebra

  title={A Finite Dimensional Introduction to Operator Algebra},
  author={Paul S. Muhly},
This article surveys some recent advances in operator algebra that were inspired by considerations from ring theory, particularly the representation theory of finite dimensional algebras. 
Canonical Models for Representations of Hardy Algebras
Abstract.We develop a model theory for completely non coisometric representations of the Hardy algebra of a W*-correspondence defined over a von Neumann algebra. It follows very closely the model
A Halmos Doctrine and Shifts on Hilbert Space
This a survey of some recent work on noncommutative function theory related to tensor algebras that derives in part from Paul Halmos’s paper, Shifts on Hilbert space.
Nest Representations of Directed Graph Algebras
This paper is a comprehensive study of the nest representations for the free semigroupoid algebra LG of a countable directed graph G as well as its norm‐closed counterpart, the tensor algebra T+(G).
Universal operator algebras of directed graphs
We define and investigate properties of universal operator algebras of directed graphs. Results include free products decomposition and continuity of the construction with respect to direct limits.
On quotients of tensor algebras and their C*-envelopes
  • P. Muhly, B. Solel
  • Mathematics
    Proceedings of the Edinburgh Mathematical Society
  • 2000
Abstract We identify the C*-envelopes of certain quotients of tensor algebras over C*-correspondences.
Notes on C*-algebras of graphs
C*-algebras of graphs include up to strong Morita equivalence all af algebras, Cuntz-Krieger algebras and the Toeplitz algebra. Our survey discusses recent developments and related constructions.
Partly Free Algebras From Directed Graphs
We say that a nonselfadjoint operator algebra is partly free if it contains a free semigroup algebra. Motivation for such algebras occurs in the setting of what we call free semigroupoid algebras.
Progress in noncommutative function theory
In this expository paper, we describe the study of certain non-self-adjoint operator algebras, the Hardy algebras, and their representation theory. We view these algebras as algebras of (operator
Progress in noncommutative function theory
In this expository paper, we describe the study of certain non-self-adjoint operator algebras, the Hardy algebras, and their representation theory. We view these algebras as algebras of (operator
Cohomology of Hopf C*‐Algebras and Hopf Von Neumann Algebras
We define two canonical cohomology theories for Hopf C*‐algebras and for Hopf von Neumann algebras (with coefficients in their comodules). We then study the situations when these cohomologies vanish.


Limit algebras : an introduction to subalgebras of C[*]-algebras
Written by one of the key researchers in this field, this volume develops the theory of non-self adjoint limit algebras from scratch.
On the C*-envelope of approximately finite-dimensional operator algebras.
The C*-envelope of the limit algebra (or limit space) of a contractive regular system of digraph algebras (or digraph spaces) is shown to be an approximately finite C*-algebra and the direct system
Subspaces of C*-algebras
1. Modules 2. Multipliers and morphisms 3. Projections and unitaries 4. Tensor products 5. The KSGNS construction 6. Stabilisation or absorption 7. Full modules, Morita equivalence 8. Slice maps and
An Algebraic Characterization of Boundary Representations
We show that boundary representations of an operator algebra may be characterized as those (irreducible) completely contractive representations that determine Hilbert modules that are simultaneously
A Characterization of Operator Algebras
Representations of triangular subalgebras of groupoid C *-algebras
We investigate the invariant subspace structure of subalgebras of groupoid C *-algebras that are determined by automorphism groups implemented by cocycles on the groupoids. The invariant subspace
To a special embedding of circle algebras having the same spectrum, we associate an r-discrete, locally compact groupoid, similar to the Cuntz-Krieger groupoid. Its C algebra, denotedO , is a
Hilbert Modules over Function Algebras
A different point of view is outlined which may assist in guiding developments in the area of non-selfadjoint operator theory and which has largely eluded us.
Tree Algebras, Semidiscreteness, and Dilation Theory
We introduce a class of finite-dimensional algebras built from a partial order generated as a transitive relation from a finite tree. These algebras, known as tree algebras, have the property that