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 intersect
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
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 (conju
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
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
inte
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
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 Moderator
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
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
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
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