Abstract. The ε -pseudospectrum Λ ε (a) of an element a of an arbitrary Banach algebra A is studied. Its relationships with the spectrum and numerical range of a are given. Characterizations of scalar, Hermitian and Hermitian idempotent elements by means of their pseudospectra are given. The stability of the pseudospectrum is discussed. It is shown that the pseudospectrum has no isolated points, and has a finite number of components, each containing an element of the spectrum of a . Suppose for some ε > 0 and a,b ∈ A, Λ ε (ax) = Λ ε (bx) ∀x ∈ A . It is shown that a = b if:(ii) a is Hermitian idempotent.(iii) a is the product of a Hermitian idempotent and an invertible element.(iv) A is semisimple and a is the product of an idempotent and an invertible element.(vii) A is a commutative semisimple Banach algebra.Mathematics subject classification (2010): 47A10, 46H05, 47A12.

We introduce a new distributional invariance principle, called 'partial spreadability', which emerges from the representation theory of the Thompson monoid F + in the framework of W*algebraic probability spaces. We show that partial spreadability implies Markovianity of noncommutative stationary processes (in the sense of B. Kümmerer and as they are considered by U. Haagerup and M. Musat within the context of factorisable Markov maps). Conversely we show that a large class of noncommutative stationary Markov processes provides representations of the Thompson monoid F + . In the particular case of a classical probability space, our approach anticipates the availability of a de Finetti theorem for recurrent stationary Markov chains with values in a standard Borel space, generalizing an early result of P.

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We show that representations of the Thompson group F in the automorphisms of a noncommutative probability space yield a large class of bilateral stationary noncommutative Markov processes. As a partial converse, bilateral stationary Markov processes in tensor dilation form yield representations of F. As an application, and building on a result of Kümmerer, we canonically associate a representation of F to a bilateral stationary Markov process in classical probability.