A Fredholm module is a mathematical object used in the study of operator algebras. It consists of a C*-algebra, a Hilbert space, and a pair of operators that satisfy certain conditions. These modules are used to classify C*-algebras and understand their properties.
A Fredholm operator is a linear operator on a Hilbert space that has finite-dimensional kernel and cokernel. A Fredholm module, on the other hand, is a more general object that includes a C*-algebra and a Hilbert space, in addition to the pair of operators. While a Fredholm operator can be seen as a special case of a Fredholm module, the latter allows for a more comprehensive study of C*-algebras.
Fredholm modules have various applications in mathematics and physics. They are used in the study of index theory, which is a branch of mathematics that deals with the relationship between differential operators and topological invariants. They also have applications in quantum field theory and non-commutative geometry.
Fredholm modules are classified using K-theory, a mathematical tool that assigns invariants to C*-algebras. Specifically, they are classified by the K-theory group of the C*-algebra associated with the module. This classification allows for a better understanding of the properties of the C*-algebra and its associated Fredholm module.
One example of a Fredholm module is the Toeplitz algebra, which consists of all operators on the Hardy space that are multiplication by a bounded function. The Toeplitz module, which includes the Toeplitz algebra, the Hardy space, and the Toeplitz operator, is a Fredholm module. It has been extensively studied in the field of operator algebras and has applications in both mathematics and physics.