1,095,936 research outputs found
Traces and Quasi-traces on the Boutet de Monvel Algebra
We construct an analogue of Kontsevich and Vishik's canonical trace for a
class of pseudodifferential boundary value problems in Boutet de Monvel's
calculus on compact manifolds with boundary.
For an operator A in the calculus (of class zero), and an auxiliary operator
B, formed of the Dirichlet realization of a strongly elliptic second-order
differential operator and an elliptic operator on the boundary, we consider the
coefficient C_0(A,B) of (-\lambda)^{-N} in the asymptotic expansion of the
resolvent trace Tr(A(B-\lambda)^{-N}) (with N large) in powers and log-powers
of \lambda as \lambda tends to infinity in a suitable sector of the complex
plane. C_0(A,B) identifies with the coefficient of s^0 in the Laurent series
for the meromorphic extension of the generalized zeta function \zeta(A,B,s)=
Tr(AB^{-s}) at s=0, when B is invertible.
We show that C_0(A,B) is in general a quasi-trace, in the sense that it
vanishes on commutators [A,A'] modulo local terms, and has a specific value
independent of B modulo local terms; and we single out particular cases where
the local ``errors'' vanish so that C_0(A,B) is a well-defined trace of A.
Our main tool is a precise analysis of the asymptotic expansion of the
resolvent trace, based on pseudodifferential calculations involving rational
functions (in particular Laguerre functions) of the normal variable.Comment: Final version to appear in Ann. Inst. Fourie
Traces in monoidal categories
The main result of this paper is the construction of a trace and a trace
pairing for endomorphisms satisfying suitable conditions in a monoidal
category. This construction is a common generalization of the trace for
endomorphisms of dualizable ob jects in a balanced monoidal category and the
trace of nuclear operators on a locally convex topological vector space with
the approximation property
Traces in braided categories
With any even Hecke symmetry R (that is a Hecke type solution of the
Yang-Baxter equation) we associate a quasitensor category. We formulate a
condition on R implying that the constructed category is rigid and its
commutativity isomorphisms R_{U,V} are natural. We show that this condition
leads to rescaling of the initial Hecke symmetry. We suggest a new way of
introducing traces as properly normalized categorical morphisms End(V) --> K
and deduce the corresponding normalization from categorical dimensions.Comment: Source: Revised version, a more attention is given to the problem of
trace definition and its proper normalization in braided categories with
Hecke type braidings. Minor corrections in Introduction. LaTex file, all
macros included, no figure
- …