Factorization problems in the invertible group of a homogeneousC^∗-algebra

Abstract

Let X be a compact metric space of dimension d. In previous work, we have shown that for all sufficiently large n, every element of the identity component Uo(C(X) 0 M n ) of the unitary group U(C(X)®M n ) is a product of at most 4 exponentials of skewadjoint elements. On the other hand, if X is a manifold then some elements of Uo(C(X) (8> M n ) require at least about d/n 2 exponentials. Similar qualitative behavior (with different bounds: 5 and d/(2n 2 )) holds for the problem of factoring elements of the identity component invo(C(X) 0M n ) of the invertible group as products of exponentials of arbitrary elements of the algebra. In this paper, we identify the sets of finite products of 10 other types of elements of invo(C(X)(g)M n ), and we show that the minimum lengths of factorizations have the same qualitative behavior as the two exponential factorization problems above (after a suitable minor modification in 3 of the 10 cases). We obtain upper bounds for large n that range from 5 to 22, and lower bounds approximately of the form rd/n 2 with r ranging from 1/16 to 2. The classes of elements we consider all make sense in general unital C*-algebras. They are: unipotents, positive invertibles, selfadjoint invertibles, symmetries, *-symmetries, commutators of elements of invo(^4) and Uo(A), accretive elements, accretive unitaries, and positive-stable elements (real part of spectrum positive). The last three classes are the ones requiring the slight modification; without it, lengths of factorization behave like exponential length rather than exponential rank.Introduction.

𝐶*-Algebras with the Approximate Positive Factorization Property

“…( If X is a compact Hausdorff space, then M n ⊗ C(X) has rank n. This follows from [21], and is also easy to check directly.…”

“…The study of factorization into positive and selfadjoint operators has been extended to several classes of C*-algebras: homogeneous C*-algebras (N.C. Phillips [21]), and unitized stable C*-algebras and purely infinite simple C*-algebras (M. Leen [15]). …”

Every Invertible Hilbert Space Operator is a Product of Seven Positive Operators

“…(*S 1 , M 6 )) such that g(x*ux)g~l = diag(af,..., a\) = d 2 . (The proof is the same as that of Lemma 2.5 of[5]. Also compare Lemma 4.2 of[2].)…”

“…For results on products of positive invertible elements in the C*-algebras C(X) (g) M", including a proof that in general there is no finite upper bound on the number of factors needed, see Section 2 of [5]. For recent results on limits of products of positive elements in AF algebras, algebras of measurable matrix valued functions, and other related algebras, see [6] and [7].…”

“…To prove the lemma, we have to show that u ~~ a for some a for which each a(9) is lower triangular with diagonal entries equal to 1. Taking c to be an appropriate (constant) permutation matrix obtained from a, we see that (5) u ~~ diag(exp(/a a( i)),..., exp(/a CT(6) )).…”