Two examples of Banach algebras

05/10/2018, 14:00 - 15:00 in MAP/01/002

Shkarin, Stanislav (Queen’s University Belfast)


Abstract:

The objective of the talk is to construct two peculiar Banach algebras. The first algebra B is a commutative unital semisimple Banach algebra whose underlying Banach space is isomorphic to a separable Hilbert space and whose spectrum is homeomorphic to the Hilbert cube. This is the first example of a reflexive Banach algebra with the spectrum containing the Hilbert cube.

The second Banach algebra A is obtained from B (the first one) as the direct limit of a sequence of quotients of B and has the most peculiar combination of properties:

  1. A is a separable infinite dimensional radical weakly amenable commutative Banach algebra whose underlying Banach space is a Hilbert space (first ever example with these properties);
  2. There is an element a of A such that both {can: c a complex number, n=1,2,…} and {a(1+a)n: n=1,2,…} are dense in A.

As a result the operator M on A of multiplication by a (Mx=ax) is supercyclic and the operator I+M is hypercyclcic, while the direct sum of M with itself is non-cyclic (as a multiplication operator on a commutative Banach algebra). This is not the first example of the sort, but is the first one with small spectrum (M is quasinilpotent), which provides an answer to a question of Bayart and Matheron. Furthermore the semigroup Tt=exp(tS) with S=ln (I+M) provides an example of a hypercyclic strongly continuous operator semigroup on a Hilbert space with the direct sum of Tt with itself being non-hypercyclic, answering another question of Bayart and Matheron.