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## Algebraic K-Theory and Determinants of Toeplitz Operators

### Efton Park

Texas Christian University

###
Thursday, May 18, 2006

L01 Carson Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: ** Let *H* be an infinite-dimensional Hilbert
space, and consider the set *B(H)* of operators on *H*; that is the
continuous linear maps from *H* to itself. For operators that are
``close'' to the identity map, one can define a determinant function
that behaves much like the usual determinant of a linear map on a
finite-dimensional vector space. This infinite dimensional
determinant is difficult to compute directly, so one might ask is
there is an easier way to proceed, at least for certain restricted
classes of operators. For a certain algebra of operators called the
Toeplitz operators, there is indeed a beautiful formula for computing
the determinant, and surprisingly, this formula is closely related to
the connecting map in algebraic K-theory.\par In my talk, I will
define the algebra of Toeplitz operators, state the determinant
formula for these operators, show how it relates to algebraic
K-theory, and discuss generalizations of these results. I will not
assume any prior knowledge of operator theory or algebraic K-theory,
and therefore the talk (at least the first two-thirds) should be
accessible to graduate students.

This talk will be accessible to graduate students.