Projective Spectrum in Banach Algebras
نویسندگان
چکیده
For a tuple A = (A0, A1, ..., An) of elements in a unital Banach algebra B, its projective spectrum p(A) is defined to be the collection of z = [z0, z1, ..., zn] ∈ P such that A(z) = z0A0 + z1A1 + · · · + znAn is not invertible in B. The pre-image of p(A) in Cn+1 is denoted by P (A). When B is the k × k matrix algebra Mk(C), the projective spectrum is a projective hypersurface. In infinite dimensional cases, projective spectrums can be very complicated, but also have some properties similar to that of hypersurfaces. When A is commutative, P (A) is a union of hyperplanes. When B is reflexive or is a C∗-algebra, the projective resolvent set P (A) := Cn+1 \ P (A) is shown to be a disjoint union of domains of holomorphy. Later part of this paper studies Maurer-Cartan type B-valued 1-form A−1(z)dA(z) on P (A). As a consequence, we show that if B is a C∗-algebra with a trace φ, then φ(A−1(z)dA(z)) is a nontrivial element in the de Rham cohomology space H1 d (P (A), C). 0. Introduction The classical spectrum of an element A in a unital Banach algebra B is defined through the invertibility of A − λI. If A = (A0, A1, ..., An) is a commutative tuple of elements in B, then classical notions of joint spectrum are defined through the invertibility of (A0 − λ0I, A1 − λ1I, ..., An − λnI) in various senses (Hörmander [Hö] Ch3, and Taylor [Ta]). In all these cases, the identity I serves as a base against which the invertibilities of other elements are measured. The idea of projective spectrum, which we will define and study, is to set I free, and consider the invertibility of z0A0 + z1A1, or more generally, A(z) := z0A0+z1A1+ · · ·+znAn. This is a measurement of how the elements behave against 1991 Mathematics Subject Classification. Primary 47A13; Secondary 47L10.
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