Mathematics 442C
Suggested solutions to exercise sheet 3
1. (a) Since T is a topology, it is closed under nite intersections and arbitrary
unions. Since S is the collection of unions of nite intersections of sets
in S and S T , we have S T .
The empty set
442C Banach algebras 200910
1
1.1
Introduction to Banach algebras
Denitions and examples
Let us adopt the convention that all vector spaces and Banach spaces are
over the eld of complex numbers.
1.1.1 Denition. A Banach algebra is a vector space A such th
4
4.1
C*-algebras
Denitions and examples
4.1.1 Denition. Let A be a Banach algebra. An involution on A is a map
A A, a a such that for all a, b A and C we have:
(i). (a) = a and (a + b) = a + b (conjugate linearity);
(ii). (ab) = b a ; and
(iii). (a ) = a
442C Banach algebras 200910
1
1.1
Introduction to Banach algebras
Denitions and examples
Let us adopt the convention that all vector spaces and Banach spaces are
over the eld of complex numbers.
1.1.1 Denition. A Banach algebra is a vector space A such th
2
2.1
A topological interlude
Topological spaces
Recall that a topological space is a set X with a topology: a collection T of
subsets of X, known as open sets, such that and X are open, and nite
intersections and arbitrary unions of open sets are open. W
3
Unital abelian Banach algebras
3.1
Characters and maximal ideals
Let A be a unital abelian Banach algebra.
3.1.1 Denition. A character on A is a non-zero homomorphism A C;
that is, a non-zero linear map : A C which satises (ab) = (a) (b) for
a, b A. We
UNIVERSITY OF DUBLIN
XMA442C
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
Trinity Term 2010
PL
MA442C Banach Algebras
E
SS Mathematics
SS Two Subject Moderatorship
Dr. R. Levene
Credit will be given for the best 3
Mathematics 442C
Suggested solutions to exercise sheet 5
1. Let A and B be Banach algebras, and let A0 be a dense subalgebra of A. If 0 : A0 B
is a continuous homomorphism, show that there is a unique continuous homomorphism
: A B such that |A0 = 0 .
Sol
Mathematics 442C
Suggested solutions to exercise sheet 4
1. (a) Since A is a nite dimensional vector subspace of the Banach space M2 (C) =
B(C2 ), it is closed. Moreover, the unit of M2 (C) is I, and I A. Since
T 2 = 0, for , , a, b C we have
(I + T )(aI
Mathematics 442C
Suggested solutions to exercise sheet 1
1. Suppose that an a and bn b. Let M = 1 + a + supn1 bn . Since (bn )
converges, we have 0 < M < , and a < M . Let > 0. There is n0 N
such that an a < 2M and bn b < 2M for all n n0 . Hence if n n0 ,
Dear Hiring Manager.
I'm applying for a position at your company. Based on
the posted description, I'm confident that I am fully
qualified for the position and will be a strong addition to
your team. I would appreciate a job interview at your
earliest con