Serdica Mathematical Journal

Volume 41, Number 1, 2015

This issue contains contributions to the
1st Summer School in Operator Theory
held on the island of Chios, at the University of the Aegean, 25–30 July 2011.

C O N T E N T S

A B S T R A C T S

THE C*-ALGEBRA OF A LOCALLY COMPACT GROUP
Ying-Fen Lin y.lin@qub.ac.uk

2010 Mathematics Subject Classification: 22D25, 22E25.
Key words: Group C*-algebras, unitary representations, locally compact groups.

In this note, we briefly introduce the C*-algebra of a locally compact group and present some important structural results.

INTERACTIONS BETWEEN HARMONIC ANALYSIS AND OPERATOR THEORY
I. G. Todorov i.todorov@qub.ac.uk

2010 Mathematics Subject Classification: Primary 47L05, 47L35; Secondary 43A45.
Key words: locally compact group, spectral synthesis, Schur multiplier, relexivity.

The artice is a survey of several topics that have led to fruitful interactions between Operator Theory and Harmonic Analysis, including operator and spectral synthesis, Schur and Herz-Schur multipliers, and reflexivity. Some open questions and directions are included in a separate section.

SHIFT OPERATORS
M. S. Anoussis mano@aegean.gr

2010 Mathematics Subject Classification: Primary 46L05; Secondary 47A15, 47B35.
Key words: shift operator, Toeplitz algebra, Wold decomposition, invariant subspace.

Shift operators play an important role in different areas of Mathematics such as Operator Theory, Dynamical Systems and Compex Analysis. In these lectures we discuss basic properties of these operators. We present Beurling's Theorem which describes the invariant subspaces of the shift. The structure of the C*-algebra generated by the shift is described. We also indicate how the shift operators appear in the analysis of isometries on a Hilbert space: Wold decomposition and Coburn's theorem.

OPERATOR ALGEBRAS: AN INTRODUCTION
Aristides Katavolos akatavol@math.uoa.gr

2010 Mathematics Subject Classification: Primary 46L05; Secondary 47L55.
Key words: C*-algebras, Gelfand Theory; masa bimodules.

The first part of these notes contains a sketch of the elementary parts of C*-algebra theory, culminating in the two Gelfand--Naimark theorems. The final section is a presentation of the basic facts of the theory of weak-* closed (possibly non-selfadjoint) unital algebras containing maximal abelian selfadjoint algebras (masas), or more generally bimodules over masas.

EP ELEMENTS IN RINGS AND IN SEMIGROUPS WITH INVOLUTION AND IN C*-ALGEBRAS
Sotirios Karanasios skaran@math.ntua.gr

2010 Mathematics Subject Classification: Primary 46L05, 46J05, 46H05, 46H30, 47A05, 47A53, 47A60, 15A09, 15A33, 16A28, 16A32, 16B99,16W10; Secondary 46E25,46K05, 47A12, 47A68.
Key words: EP operator, EP element, involution, strong involution, semigroup with involution, ring with involution, regular ring, antiregular ring, *-regular ring, faithful ring, C*-algebra, group inverse, Drazin inverse, generalized inverse, Moore-Penrose inverse, regular element, antiregular element, polar element, quasipolar element, global star cancelation law, annihilator.

This work includes a survey of most of the results concerning EP elements in semigroups and rings with involution and in C*-algebras

SOME NOTES ON MORITA EQUIVALENCE OF OPERATOR ALGEBRAS
G. K. Eleftherakis gelefth@math.upatras.gr

2010 Mathematics Subject Classification: Primary 47L40; Secondary 47L45, 47L55.
Key words: Morita equivalence, operator algebras.

In this paper we present some key moments in the history of Morita equivalence for operator algebras.

THE LOGIC OF QUANTUM MECHANICS
V. Felouzis felouzis@aegean.gr

2010 Mathematics Subject Classification: Primary 81P10, 4703; Secondary 0101.
Key words: logic, observable, linear operator.

These notes, written for the Summer school in Operator Theory (Chios 2010) provide a brief and elementary introduction to the Logic of Quantum mechanics and its connections with the theory of operators in a Hilbert space.