Previous talks this semester:
23.01.2014 at 11.15, room 105; Przemysław Ohrysko, On the limits of
Fourier coefficents along ultrafilters
Abstract. It will be shown that there are two distinct ultrafilters on the integers u and v such that
the u-limits and v-limits of Fourier coeffcients are the same. We know from different considerations that
the Gelfand space of the measure algebra with convolutions contains βN. The above shows that
nevertheless the most natural map from βN into the Gelfand space is not injective.
09.01, 16.01 at 11.15, room 105; Tomasz Kochanek, On the largest regular subalgebra of the measure algebra on a locally compact Abelian group
This talk is meant to be a continuation/extension of the recent talk by P. Ohrysko on the spectrum
of the Banach algebra M(T) of all bounded regular complex Borel measures on the unit circle.
First, we discuss
basic properties of the hull-kernel topology on Gelfand space, as well as the notion of regular algebra. From the
Wiener-Pitt phenomenon we deduce that for every non-discrete locally compact
Abelian group G the measure algebra M(G) is not regular,
and hence its spectrum cannot be totally disconnected
(tha fact we have already observed during the talk by P. Ohrysko). The same will be proved about the Banach algebras L_1(R+) and l_1(Z+).
In the second part of the talk we focus on the existence and properties of the largest regular subalgebra Reg(A) of
a given Banach algebra A, in particular, in the case A = M(G). Although no measure-theoretic characterisation of
Reg(M(G)) is known, we shall show that it is quite a huge subalgebra of M(G) containing,
for instance, all absolutely continuous and all discrete measures. We also identify the spectrum of Reg(M(G))
as a Hausdorff compactification of the dual group of G that does not coincide with the Bohr compactification,
unless G is discrete. Finally, we discuss the relation between Reg(A) and the Apostol algebra of A.
5.12, 12.12 at 11.15 am. room 105; Adam Kwela, on the paper " G. Horbaczewska, A. Skalski;
The Banach principle for ideal convergence in the classical and noncommutative context. J. Math. Anal. Appl. 342 (2008), no. 2, 1332–1341.
21.11, 28.11 - Piotr Koszmider, On certain Banach space
In my talk I willl present some random version of the split interval K and will describe some properities
of the Banach space C(K). It is the space of the paper
"P. Koszmider, On a problem of Rolewicz about Banach spaces that admit support sets; J. Funct. Anal. 257 (2009), no. 9, 2723–2741.
I hope the anti-Ramsey theoretic properties of the space can be exploited much more.
24.10, 31.10 - Przemysław Ohrysko, On the Arens - Royden theorem. This theorem implies that the space of
multiplicative funtionals on M(T) with the weak* topology is not homeomorphic to βN.
The participants should revise the following topics before the talks:
- Banach algebras, spectrum and spectral radius.
- Gelfand theory, Gelfand transform, semisimplicity, multiplicative linear functionals, functional calculus.
- Information about structure of the group of invertible elements as in book "Banach algebras" by W. Zelazko or in book "Functional Analysis" by W. Rudin.
- Additional definitions and theorems related to them:
- join spectrum of elements in commutative Banach algebras.
- multivariable functional calculus.
- Shilov idempotent theorem.
10.10.2013 and 17.10.2013. at 1pm, room 106
Tomasz Kochanek, Twisted sums with C(K)-spaces
Based on F. Cabello Sanchez, J.M.F. Castillo, N.J. Kalton, D.T. Yost, Twisted sums with C(K) spaces, TAMS 355 (2003), 4523-4541;
3.10.2013. at 1pm, room 106
Cristóbal Rodriguez (Paris 7);
The Open Coloring Axiom and embeddings into P(N)/fin.
Abstract; The Open Coloring Axiom (OCA), as introduced by Todorcevic in
1989, is a combinatorial principle that has been very successful in
answering questions related to the automorphism group of P(N)/fin, and
has found applications in other structures such as C*-algebras. In this
session we intend to introduce the axiom, showing some easy applications
and stating some important consequences, and then we will go on to show
a more elaborate use of the principle by sketching the proof of the
following theorem by Todorcevic: If I is a nonatomic analytic ideal on
N, then P(N)/I is not isomorphic to a subalgebra of P(N)/fin.
Talks in the second semester of 2012-13.
Talks in the first semester of 2012-13.
Talks in the second semester of 2011-12.
Talks in the first semester of 2011-12.