A generalization of Milnor’s formula

The Milnor number $$\mu _f$$ μ f of a holomorphic function $$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$ f : ( C n , 0 ) → ( C , 0 ) with an isolated singularity has several different characterizations as, for example: 1) the number of critical points in a morsification of f, 2) the middle Betti number of its Milnor fiber $$M_f$$ M f , 3) the degree of the differential $${\text {d}}f$$ d f at the origin, and 4) the length of an analytic algebra due to Milnor’s formula $$\mu _f = \dim _{\mathbb {C}}{\mathcal {O}}_n/{\text {Jac}}(f)$$ μ f = dim C O n / Jac ( f ) . Let $$(X,0) \subset ({\mathbb {C}}^n,0)$$ ( X , 0 ) ⊂ ( C n , 0 ) be an arbitrarily singular reduced analytic space, endowed with its canonical Whitney stratification and let $$f :({\mathbb {C}}^n,0) \rightarrow ({\mathbb {C}},0)$$ f : ( C n , 0 ) → ( C , 0 ) be a holomorphic function whose restriction f|(X, 0) has an isolated singularity in the stratified sense. For each stratum $${\mathscr {S}}_\alpha $$ S α let $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) be the number of critical points on $${\mathscr {S}}_\alpha $$ S α in a morsification of f|(X, 0). We show that the numbers $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) generalize the classical Milnor number in all of the four characterizations above. To this end, we describe a homology decomposition of the Milnor fiber $$M_{f|(X,0)}$$ M f | ( X , 0 ) in terms of the $$\mu _f(\alpha ;X,0)$$ μ f ( α ; X , 0 ) and introduce a new homological index which computes these numbers directly as a holomorphic Euler characteristic. We furthermore give an algorithm for this computation when the closure of the stratum is a hypersurface.

Applications of the asymptotic range to analytic subalgebras of groupoid C*-algebras

For a 1-cocycle c on a principal r-discrete groupoid G, that vanishes only on the unit space of G, we show that the asymptotic range of c, , is an invariant for the total order c−1([0, ∞]). It follows that is also an invariant (with respect to isometric isomorphisms) of the triangular analytic algebra supported on c−1([0, ∞]). We also prove that if and only if the analytic algebra has a certain maximality property.

Coordinates for analytic operator algebras

Let M be a σ-finite von Neumann algebra and α = {αt}t∈A be a representation of a compact abelian group A as *-automorphisms of M. Let Γ be the dual group of A and suppose that Γ is totally ordered with a positive semigroup Σ⊆Γ. The analytic algebra associated with α and Σ iswhere spα(a) is Arveson's spectrum. These algebras were studied (also for A not necessarily compact) by several authors starting with Loebl and Muhly [10].