These notes were prepared in conjunction with the N. S. F. regional
conference on measure algebras held at the University of Montana
during the week of June 19, 1972.
Our original objective in preparing these notes was to give a coherent
detailed and simplified presentation of a body of material on measure
algebras developed in a recent series of papers by the author (Taylor
[1] —[10]). This material has two main thrusts: the first concerns an
abstract characterization of Banach algebras which arise as algebras
of measures under convolution (convolution measure algebras) and a
semigroup representation of the spectrum (maximal ideal space) of such
an algebra; the second deals with a characterization of the cohomology
of the spectrum of a measure algebra and applications of this
characterization to the study of idempotents, logarithms, and
invertible elements.
As this project progressed the original concept broadened. The final
product is a more general treatment of measure algebras, although it
is still heavily slanted in the direction of our own work.
Chapter 1 contains a brief introductory discussion of convolution and
the structure of the algebras L1(G) and M(G), as well as an
introduction to several of the problems which will be solved or
partially solved in later chapters.
Chapters 2 and 3 are devoted to a development and discussion of
convolution measure algebras and to a representation theorem for the
spectrum of such an algebra. Several examples of convolution measure
algebras are discussed in Chapter 4. Much of the material of Chapters
2—4 is contained in Taylor [1] and can be skipped by readers familiar
with that paper. However, our discussion here is considerably more
detailed and does not assume familiarity with Kakutani’s L-space
theory or the theory of topological semigroups.
Chapters 5—9 are mainly concerned with a characterization of the
cohomology of the spectrum of a measure algebra and applications to
the study of idempotents, logarithms, and inverses in such an
algebra. This material originally appeared in Taylor [3] —[10]. The
development here has been considerably simplified.
Chapter 10 is largely independent of Chapters 4—9. In it we discuss
some results of Miller [1] on Gleason parts in a measure algebra, of
Taylor [2] and Johnson [3] on the Shilov boundary of M(G), and of
Brown and Moran [3] on infinite product measures.
Author(s): Taylor, Joseph L
Series: Regional conference series in mathematics, no. 16
Publisher: American Mathematical Society
Year: 1973
Language: English
Pages: 116
City: Providence, Rhode Island
Preface
Table of Contents
Chapter 1: Orientation
1.1. Measure algebras on semigroups
1.2. Convolution equations
1.3. The spectrum of M(G)
1.4. Idempotents
1.5. Wiener-Hopf equations
1.6. Logarithms
1.7. Other measure algebras
Chapter 2: L-spaces
2.1. L-subspaces of a measure space
2.2. L-homomorphisms
2.3. L-spaces and duals of L-spaces
2.4. The standard representation
2.5. Tensor products of L-spaces
Chapter 3: Convolution measure algebras
3.1. Definition and examples
3.2. The structure semigroup
3.3. Homomorphisms of measure algebras
3.4. Elementary properties of S and S
3.5. L-ideals of M
Chapter 4: Special examples
4.1. Compactifications and structure semigroups
4.2. Group algebras
4.3. The algebra M(G)
4.4. Arens-Singer algebras
4.5. An existence theorem for semicharacters
4.6. Idempotent semigroup algebras
Chapter 5: The structure of S
5.1. Topologies on S
5.2. Ideals, groups, and critical points
5.3. A covering lemma
Chapter 6: Cohomology of 8
6.1. Cohomology functors
6.2. Idempotents and logarithms
6.3. The cohomology of S
6.4. Cohomology and ciritcal points
Chapter 7: Critical points and group algebras
7.1. The main theorem
7.2. Linear equations and absolutely continuous measures in M(G0 x Rn)
7.3. Spectrum of a half-algebra
7.4. An alternate representation M
7.5. Proof of Theorem 7.1.4.
7.6. A characterization of group algebras
Chapter 8: Idempotents and logarithms
8.1. The groups H0(M) and H1(M)
8.2. The algebra M(G)
8.3. Logarithms and group algebras
8.4. Logarithms in M(R)
8.5. Weiner-Hopf equations
8.6. One-parameter groups in M(R)
Chapter 9: Invertible measures
9.1. The main theorem
9.2. The spectrum of a measure
9.3. The case M = M(G)
Chapter 10: Boundaries and Gleason parts
10.1. Analytic structure in S
10.2. Miller's theorem on parts
10.3. The strong boundary
10.4. Infinite convolution products
10.5. The Shilov boundary of M(G)
References:
