## 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:

*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);- There is an element
*a*of*A*such that both {*ca*:^{n}*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 *T*_{t}=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 *T*_{t} with itself being non-hypercyclic, answering another question of Bayart and Matheron.